Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications)
... In this paper, a bilevel programming formulation of the PWA regression problem is proposed, where the upper level fixes the polyhedral partition of the regressor domain and classifies the data points, and the lower level computes parameter estimates of the affine models in each region of the partition. The bilevel problem can then be recast as an equivalent mixed-integer program through the use of duality concepts [27]. Differently from [3], [10], the proposed framework allows one to formulate a general class of PWA regression problems, where parameter estimates in each region of the partition are based on a prediction error criterion, while the overall PWA model is selected according to a possibly different criterion. ...
... A bilevel problem is a mathematical program composed of two nested optimization problems, termed upper and lower level [27]. Formally, a bilevel problem looks like ...
... This results in a single level optimization problem that can be tackled in different ways, depending on its structure. The interested reader is referred to [27] for more details. ...
... A bilevel model is a mathematical program composed of two nested optimization problems, termed upper and lower level [27]. Formally, ...
... be the total profit of entity u within the considered community microgrid framework. In (26), the quantity J u energy takes into account the revenues and costs for entity u related to energy flows: (27) Notice that the energy exchanges with the community, e u t ...
... If the lower level solution is not unique, the bilevel problem can be tackled by recasting it as a single optimization program. The resulting model is nonlinear, due to the bilinear terms e u t u t , com , com and i u t u t , com , com appearing in (27). The recasting as a single optimization program relies on the fact that the lower level problem is a linear program, which can be replaced with its first-order necessary and sufficient Karush-Kuhn-Tucker conditions. ...
This work fits in the context of community microgrids, where members of a community can exchange energy and services among themselves, without going through the usual channels of the public electricity grid. We introduce and analyze a framework to operate a community microgrid, and to share the resulting revenues and costs among its members. A market-oriented pricing of energy exchanges within the community is obtained by implementing an internal local market based on the marginal pricing scheme. The market aims at maximizing the social welfare of the community, thanks to the more efficient allocation of resources, the reduction of the peak power to be paid, and the increased amount of reserve, achieved at an aggregate level. A community microgrid operator, acting as a benevolent planner, redistributes revenues and costs among the members, in such a way that the solution achieved by each member within the community is not worse than the solution it would achieve by acting individually. In this way, each member is incentivized to participate in the community on a voluntary basis. The overall framework is formulated in the form of a bilevel model, where the lower level problem clears the market, while the upper level problem plays the role of the community microgrid operator. Numerical results obtained on a real test case implemented in Belgium show around 54% cost savings on a yearly scale for the community, as compared to the case when its members act individually.
... Bilevel programs form a class of optimization problems that are suitable for modeling hierarchical settings with two independent decision-makers, namely, the leader and the follower, who are also often referred to as the upper-and lower-level decision-makers, respectively [5,15]. The involved decision-makers may be collaborative or conflicting. ...
... Note that if k = n 1 , then BMIP reduces to a bilevel linear program (BLP). Bilevel programming, in particular, BMIPs and BLPs, where for a given leader's decision the corresponding follower's problem reduces to a linear program (LP) as in (2), is a well-studied area of optimization with a host of algorithmic and theoretical developments; see, e.g., [2,4,5,13,15]. In particular, it is known that, in contrast to polynomially solvable single-level LPs, BLPs are NP-hard optimization problems [18]. ...
... The following statements hold for model (5): ...
We consider a class of bilevel linear mixed-integer programs (BMIPs), where the follower’s optimization problem is a linear program. A typical assumption in the literature for BMIPs is that the follower responds to the leader optimally, i.e., the lower-level problem is solved to optimality for a given leader’s decision. However, this assumption may be violated in adversarial settings, where the follower may be willing to give up a portion of his/her optimal objective function value, and thus select a suboptimal solution, in order to inflict more damage to the leader. To handle such adversarial settings we consider a modeling approach referred to as -pessimistic BMIPs. The proposed method naturally encompasses as its special classes pessimistic BMIPs and max–min (or min–max) problems. Furthermore, we extend this new modeling approach by considering strong-weak bilevel programs, where the leader is not certain if the follower is collaborative or adversarial, and thus attempts to make a decision by taking into account both cases via a convex combination of the corresponding objective function values. We study basic properties of the proposed models and provide numerical examples with a class of the defender–attacker problems to illustrate the derived results. We also consider some related computational complexity issues, in particular, with respect to optimistic and pessimistic bilevel linear programs.
... Bilevel linear programming (BLP) problem is an extension of the linear programming (LP) problem that consists of two levels of decision making stage (Bard 1998). After the decision maker at the upper level (leader) decides his choice, the decision maker at the lower level (follower) considers the leader's decision and makes her choice to optimize her objective function. ...
... Therefore, BLP problem formulation is very suitable for such problem involving a hierarchical relationship between two decision levels. In the conventional BLP problem formulation (Bard 1998; Dempe 2002; Colson et al. 2007; Labbé and Violin 2015) , each decision maker is assumed to have the complete information about the game. In fact, the precise information of counterpart are not always obtained, but the imprecise ones, to incorporate to BLP problems in realistic scenarios. ...
... Thus, the total summation of˜σof˜ of˜σ am y am represents the aggregate users' travel cost based on the imprecise data as in (4c) which is the objective function of lower-level problem. The constraints in the lower-level problem are the regular conditions in the minimal cost network flow problem (Bard 1998). The equation (4d) stands for the node-arc flow balancing equation. ...
A bilevel linear programming problem with ambiguous lower-level objective function is a sequential decision making under uncertainty of rational reaction. The ambiguous lower-level objective function is assumed that the coefficient vector of the follower lies in a convex polytope. We apply the maximin solution approach and formulate it as a special kind of three-level programming problem. Since an optimal solution exists at a vertex of feasible region, we adopt k-th best method to search an optimal solution. At each iteration of the k-th best method, we check rationality, local optimality and global optimality of the candidate solution. In this study, we propose a global optimality test based on an inner approximation method and compare its computational efficiency to other test methods based on vertex enumeration. We also extensively utilize the history of rationality tests to verify the rationality of the solution in the follower’s problem. Numerical experiments show the advantages of the proposed methods.
... The mathematical formulation of a simple bilevel optimization problem is given by: The learning problem that we will study in the next section is actually a simple bilevel optimization problem. Let us take a simple example extracted from [2] to better understand optimistic and pessimistic bilevel problems and how they prepare the leader to anticipate loss. ...
... In other words, there is no way for the leader to guarantee they achieve their minimum payoff. [2] Now let us consider how the leader would proceed in both instances of Optimistic and Pessimistic approach. ...
In this work, we developed a learning approach to selecting regularization parameter in Tikhonov regu-larization. It turns out that the learning problem is a bilevel optimization problem where the lower level problem is a Tikhonov regularized problem. The existence of possible solutions to the bilevel problem is discussed as well as the conditions that ensure the feasibility of such solutions. Furthermore, the optimality conditions is obtained based on the constraints of our chosen bilevel problem. Finally, a solution algorithm is investigated and tested on some real-world examples.
... Then during training, after each episode, the parameters of the NF are updated. We can frame this setup as a bi-level optimization problem [19] and derive a method for computing an approximate gradient through the latent MPC update. This involves treating MPC as a recurrent network, where the control distribution acts as a form of memory, and unrolling the computation to train with backpropagationthrough-time (BPTT). ...
... Learning the distribution π θ,λ amounts to solving a bi-level optimization problem [19], in which one optimization problem is nested in another. The lower-level optimization problem involves updating the latent distribution parameters at each time step, θ t , by minimizing the expected cost with DMD. ...
Sampling-based methods have become a cornerstone of contemporary approaches to Model Predictive Control (MPC), as they make no restrictions on the differentiability of the dynamics or cost function and are straightforward to parallelize. However, their efficacy is highly dependent on the quality of the sampling distribution itself, which is often assumed to be simple, like a Gaussian. This restriction can result in samples which are far from optimal, leading to poor performance. Recent work has explored improving the performance of MPC by sampling in a learned latent space of controls. However, these methods ultimately perform all MPC parameter updates and warm-starting between time steps in the control space. This requires us to rely on a number of heuristics for generating samples and updating the distribution and may lead to sub-optimal performance. Instead, we propose to carry out all operations in the latent space, allowing us to take full advantage of the learned distribution. Specifically, we frame the learning problem as bi-level optimization and show how to train the controller with backpropagation-through-time. By using a normalizing flow parameterization of the distribution, we can leverage its tractable density to avoid requiring differentiability of the dynamics and cost function. Finally, we evaluate the proposed approach on simulated robotics tasks and demonstrate its ability to surpass the performance of prior methods and scale better with a reduced number of samples.
... Therefore, how to establish a TLO model to solve the three-level quantitative optimization problem is a challenge. (3) Model solution: Bilevel programming is already NP-hard (Bard 1998), and the TLO model is more complex. For the solution of general TLO model, analytical methods or the methods that transforming the TLO model into bilevel programming can be used for simple TLO models when they are linear or meet some excellent conditions. ...
... Multi-level optimization model was first proposed by Bracken and McGill (Bracken and McGill 1974). Bilevel programming is a special branch of it that addresses the hierarchical optimization problems which involve two self-interested decision-makers who act and react in a non-cooperative and sequential manner (Bard 1998). The TLO model is an extension of the bilevel optimization model, which solves the hierarchical optimization problem involving three types of decision-makers ). ...
Product planning or product line design is critical for online service platform operations nowadays. There are limited studies on systematic optimization of service product planning (SPP) in light of emerging trend of online service platform that leverages upon multiple service agents and numerous service operations resource providers through outsourcing. This may be partially due to difficulties in quantitative optimization of service operations and complexity of service product fulfillment that are very different from those physical products in the manufacturing sector. The fulfillment of service products involves a complicated planning process that entails a dynamic interactive optimization problem involving multiple decision-makers at multiple levels of abstraction. This paper proposes a quantitative decision-making method with three-level leader-follower structure to deal with the dynamic SPP problem for online service platform to coordinate with service agents and their resource providers. The proposed solution framework includes designing a reasonable optimization structure, defining decision variables of three-level decision-makers, and establishing a 0–1 mixed three-level optimization (TLO) model for coordinated optimization of three decision-makers. A case study of applying the dynamic SPP method to the tourism industry is reported to illustrate the feasibility and potential.
... Indeed, game-theoretic scheduling softwares have been assisting the LAX police, the Federal Air Marshals service, and are under consideration by the TSA . They have been studied for patrolling (Agmon et al. 2008;Basilico, Gatti, and Amigoni 2009) and routing in networks (Kodialam and Lakshman 2003).At the backbone of these applications are attacker-defender Stackelberg games. The solution concept is to compute a strong Stackelberg equilibrium (SSE) (von Stengel and Zamir 2004;Conitzer and Sandholm 2006); specifically, the optimal mixed strategy for the defender. ...
... Maximum observation error for target t i Table 2: Notation expected utility of 4. Finding the optimal risk-averse strategy for large games remains difficult, as it is essentially a bi-level programming problem (Bard 2006). ...
Attacker-defender Stackelberg games have become a popular game-theoretic approach for security with deployments for LAX Police, the FAMS and the TSA. Unfortunately, most of the existing solution approaches do not model two key uncertainties of the real-world: there may be noise in the defender's execution of the suggested mixed strategy and/or the observations made by an attacker can be noisy. In this paper, we provide a framework to model these uncertainties, and demonstrate that previous strategies perform poorly in such uncertain settings. We also provide RECON, a novel algorithm that computes strategies for the defender that are robust to such uncertainties, and provide heuristics that further improve RECON's efficiency.
... We present the approach for Problem (25), since a very similar method follows for Problem (27). Considering Problem (25) with objective (28), the LPs formulating the support functions satisfy strong duality [34] since they are feasible and bounded for every bounded x ≥ 0 and w ≥ 0. This property is exploited in the penalty function algorithm to compute local optima. Introducing the optimal primal and dual variables ...
... We presented some numerical results to demonstrate the feasibility of the approach and two possible practical applications. Future research will further develop the solution algorithm by considering: (a) alternative solution methods such as, e.g., value function approaches [34]; (b) optimizing over matrices E and F . Extensions to feedback gain synthesis and system identification problems will be investigated. ...
Linear models with additive unknown-but-bounded input disturbances are extensively used to model uncertainty in robust control systems design. Typically, the disturbance set is either assumed to be known a priori or estimated from data through set-membership identification. However, the problem of computing a suitable input disturbance set in case the set of possible output values is assigned a priori has received relatively little attention. This problem arises in many contexts, such as in supervisory control, actuator design, decentralized control, and others. In this paper, we propose a method to compute input disturbance sets (and the corresponding set of states) such that the resulting set of outputs matches as closely as possible a given set of outputs, while additionally satisfying strict (inner or outer) inclusion constraints. We formulate the problem as an optimization problem by relying on the concept of robust invariance. The effectiveness of the approach is demonstrated in numerical examples that illustrate how to solve safe reference set and input-constraint set computation problems.
... Since a very similar method follows for the outer-approximation problems, we skip further details because of space constraints. Considering Problem (18) along with objective function (21), we note that all the LPs formulating the support functions are feasible and bounded for every bounded x ≥ 0 and w ≥ 0. Hence, they satisfy strong duality [26]. This property is exploited in the penalty function algorithm to compute local optima. ...
... Finally, we presented some numerical results to demonstrate the feasibility of the approach and two possible practical applications. Future research will further develop the solution algorithm by considering: (a) alternative solution methods such as, e.g., value function approaches [26]; (b) optimizing also over matrices E and F . Finally, the potential of this technique when applied to feedback controller synthesis and to system identification problems will be investigated. ...
Linear models with additive unknown-but-bounded input disturbances are extensively used to model uncertainty in robust control systems design. Typically, the disturbance set is either assumed to be known a priori or estimated from data through set-membership identification. However, the problem of computing a suitable input disturbance set in case the set of possible output values is assigned a priori has received relatively little attention. This problem arises in many contexts, such as in supervisory control, actuator design, decentralized control, and others. In this paper, we propose a method to compute input disturbance sets (and the corresponding set of states) such that the resulting set of outputs matches as closely as possible a given set of outputs, while additionally satisfying strict (inner or outer) inclusion constraints. We formulate the problem as an optimization problem by relying on the concept of robust invariance. The effectiveness of the approach is demonstrated in numerical examples that illustrate how to solve safe reference set and input-constraint set computation problems.
... Bilevel problems are nested optimization problems where an upper-level optimization problem is constrained by a lower-level optimization problem [2]. A common application of the bilevel problems is a static leader-follower game in economics [18], where the upper level decision maker (leader) has complete knowledge of the lower level problem (follower). ...
... Karush-Kuhn-Tucker (KKT) conditions [2]. The KKT conditions appear as dual and complementarity constraints, that is why KKT conditions require convexity, so this approach is limited to convex lower level problems. ...
In traditional machine learning techniques, the degree of closeness between true and predicted values generally measures the quality of predictions. However, these learning algorithms do not consider prescription problems where the predicted values will be used as input to decision problems. In this paper, we efficiently leverage feature variables, and we propose a new framework directly integrating predictive tasks under prescriptive tasks in order to prescribe consistent decisions. We train the parameters of predictive algorithm within a prescription problem via bilevel optimization techniques. We present the structure of our method and demonstrate its performance using synthetic data compared to classical methods like point-estimate-based, stochastic optimization and recently developed machine learning based optimization methods. In addition, we control generalization error using different penalty approaches and optimize the integration over validation data set.
... A non-linear auto regressive exogenous configuration of ANN is used to identify FDI attack on state estimation is formulated in [21]. Market decision making, non-linear optimization problems in electricity markets, along with Bi-level optimization problems in power system is presented in [22,23,24]. ...
... Moreover, the complementary slackness can be linearized using big M-method. This procedure is clearly explained in [23,24]. The KKT optimality conditions of the lower level market clearing problem is as follows (59) are injected into the problem as explained in [17]. ...
With the vast expansion of the grid network and enhancement of Communication system, Cybersecurity reinforcement is of paramount importance for reliable and secure power system operation. Cyber-attacks on electricity markets acquire financial profits to the adversary. In this paper, a modest attempt is made to model the cyber attacker’s objective of profit maximization by injecting false data into Day-ahead and Real Time electricity markets. It is considered that attacker runs a bi-level optimization problem which
includes the attacker’s profit maximization objective and market clearing problem for finding out the optimal attack measurements. While manipulating the measuring devices like RTUs, the attacker takes care to avoid being detected by the bad data detection (BDD) procedure run by the ISO. This paper focuses on financially motivated FDI attacks considering attacker as one of the virtual players in electricity market. A novel attacking model is designed using bi-level optimization problem where attacker aims to gain financial benefits by misleading market clearing problem. Potential impact of financially motivated False Data Injection (FDI) attacks on electricity markets is presented by considering PJM 5-bus system. The simulation results show the sharp impact on
Locational Marginal Prices (LMPs) in fulfilling the attacker’s objective, and the distinct relationship between LMPs and the market-clearing prices during the attack.
... The model is structured as a bilevel program (see e.g. [22]). The upper level is a long-term investment planning problem, which determines the lines to be expanded, while ensuring the recovery of both fixed and variable costs. ...
... Finally, Section V summarises the main conclusions. The proposed framework is structured as a bilevel model [22]. A bilevel model belongs to the class of hierarchical optimization problems, and it can be regarded as two nested optimization programs, termed upper and lower level problem. ...
The aim of the proposed framework is to show how flexible consumers and small generators paying nodal prices can coexist with traditional consumers paying fixed prices at the distribution grid level. The local grid is managed by a distribution network operator who also determines the lines to be built or expanded. A network tariff levied on grid users is optimally determined to ensure the recovery of both fixed and variable investment network costs. The model is structured as a non-linear integer bilevel program. The upper level represents a long-term network planning problem accounting for investment costs and network tariffs. The lower level is a market clearing problem, which considers the upper level investment decisions, and determines the cleared quantities and the distribution nodal prices. These values are used in turn by the upper level problem to determine the fixed price paid by traditional consumers and to ensure the recovery of the overall investment costs. The bilevel model is then recast as a mixed-integer quadratically constrained problem by using integer algebra and complementarity relations. The power flows at the distribution grid level are modelled by using a second-order cone relaxation. Numerical tests based on a 18-bus low-voltage distribution network are reported to demonstrate the effectiveness of the proposed approach. In particular, the results show that an increase of demand flexibility can be beneficial also for traditional consumers by triggering a reduction of fixed prices, and can mitigate the subsidising effect between them. Moreover, the optimal network planning shows that a significant welfare increase can be obtained, while ensuring the recovery of both fixed and variable costs through congestion rent and network tariffs levied on grid users.
... Similarly, Hearn and Yildirim (Hearn and Yildirim, 2002) proposed a toll piecing model, which later was generalized to solve elastic demand traffic assignment as well as combined distributionassignment problems (Yildirim and Hearn, 2005). Bard modeled the toll setting problem as a bilevel problem (Bard, 2006). After achieving promising result via toll pricing for congestion control, researchers tried to extend this method for controlling the risk of hazmat transportation. ...
... The DTP model (Equations (6) to (14)) defines different tolls for hazmat and regular vehicles for tollways to separate the hazmat and regular traffic flows (i.e., and , respectively). Equation (6), as the objective function, intends to maximize the difference between the hazmat toll price (i.e., ) and regular toll price (i.e., ) given the hazmat and regular traffic on each link. ...
Hazardous Materials (hazmat), although dangerous, are an irreplaceable aspect of everyday life. This paper is presenting an integrated traffic control policy for hazmat transportation to alleviate the risks associated with hazmat carriers. The proposed policy is devised based on dual toll pricing (DTP) and network design (ND) policies, where a two-stage simulation-based optimization framework is proposed to enhance public safety in highways. This integrated policy is devised to concurrently restrict hazmat carriers from freeways in densely populated areas via the ND policy, and control regular as well as hazmat traffic in tollways via the DTP policy. In the optimization module, mixed integer linear programming is employed to find the optimum integrated policy, where a linear-relaxation technique based on the Karush-Kuhn-Tucker (KKT) optimality conditions is applied to reduce the mathematical model. The simulation module of the proposed framework uses agent-based simulation (ABS) modeling to evaluate the suggested policies realistically. The proposed framework has been demonstrated with real traffic data of San Antonio, Texas under AnyLogic® ABS platform. The experimental results reveal that the proposed framework is able to efficiently find the optimum integrated policy which in return, effectively reduces the risk of hazmat transportation in highways.
... The solution methods proposed by Dutta et al. in [26] and by Dempe et al. in [27] assume a convex inner level optimization problem. Bard et al. [28], Dempe et al. [24] and Mitsos et al. [29] proposed solution methods considering a nonconvex inner level optimization problem. It is generally well known that there is a close connection between bilevel problem and semiinfinite programming (SIP) as discussed in [30]. ...
... The linearized ellipsoid clearly does not approximate the exact CRs well, where, as it can be expected, the approximation is looser w.r.t. p 2 that enters nonlinearly in output equation (28). It is an interesting observation that the presented linearized CRs are very similar to each other, despite the fact that they give significantly different exact CRs. ...
A model-based optimal experiment design (OED) of nonlinear systems is studied. OED represents a methodology for optimizing the geometry of the parametric joint-confidence regions (CRs), which are obtained in an a posteriori analysis of the least-squares parameter estimates. The optimal design is achieved by using the available (experimental) degrees of freedom such that more informative measurements are obtained. Unlike the commonly used approaches, which base the OED procedure upon the linearized CRs, we explore a path where we explicitly consider the exact CRs in the OED framework. We use a methodology for a finite parametrization of the exact CRs within the OED problem and we introduce a novel approximation technique of the exact CRs using inner-and outer-approximating ellipsoids as a computationally less demanding alternative. The employed techniques give the OED problem as a finite-dimensional mathematical program of bilevel nature. We use two small-scale illustrative case studies to study various OED criteria and compare the resulting optimal designs with the commonly used linearization-based approach. We also assess the performance of two simple heuristic numerical schemes for bilevel optimization within the studied problems.
... The solution methods proposed by Dutta et al. in [26] and by Dempe et al. in [27] assume a convex inner level optimization problem. Bard et al., Dempe et al. and Mitsos et al. in [28], [24], and [29] proposed solution methods considering a nonconvex inner level optimization problem. It is generally well known that there is a close connection between bilevel problem and semi-infinite programming (SIP) as discussed in [30]. ...
... The linearized ellipsoid clearly does not approximate the exact CRs well, where, as it can be expected, the approximation is looser w.r.t. p 2 that enters nonlinearly in output equation (28). It is an interesting observation that the presented linearized CRs are very similar to each other, despite the fact that they give significantly different exact CRs. Figure 3 shows the exact CRs for the classical ( ), ellipsoidal ( ) and the exact ( ) D designs using N = 4. ...
A model-based optimal experiment design (OED) of nonlinear systems is studied. OED represents a methodology for optimizing the geometry of the parametric joint-confidence regions (CRs), which are obtained in an a posteriori analysis of the least-squares parameter estimates. The optimal design is achieved by using the available (experimental) degrees of freedom such that more informative measurements are obtained. Unlike the commonly used approaches, which base the OED procedure upon the linearized CRs, we explore a path where we explicitly consider the exact CRs in the OED framework. We use a methodology for a finite parametrization of the exact CRs within the OED problem and we introduce a novel approximation technique of the exact CRs using inner- and outer-approximating ellipsoids as a computationally less demanding alternative. The employed techniques give the OED problem as a finite-dimensional mathematical program of bilevel nature. We use two small-scale illustrative case studies to study various OED criteria and compare the resulting optimal designs with the commonly used linearization-based approach. We also assess the performance of two simple heuristic numerical schemes for bilevel optimization within the studied problems.
... Bilevel problems are generally considered as nondeterministic polynomial-time (NP) hard problems (Hansen et al. 1992). Three workable approaches exist that can solve simple bilevel linear programming problems: vertex enumeration, replacing the follower problem with Karush-Kuhn-Tucker (KKT) conditions, and using branch-and-bound and penalty methods (Bard 1998;Sinha et al. 2018). These methods are usually able to analytically handle linear bilevel problems with a relatively small number of decision variables and constraints. ...
... However, when integer variables are included (e.g., as in MILP), the manageable problem size shrinks by nearly an order of magnitude (Amouzegar and Moshirvaziri 1999;Sinha et al. 2013). The main difficulty in solving such bilevel problems stems from their inherent nonconvexity and nondifferentiability (Bard 1998;Colson et al. 2007;Sinha et al. 2013). Therefore, in large-scale problems, applied studies usually develop hybrid methods combining heuristic search algorithms and analytical methods to solve complex bilevel programming problems. ...
Biofuel development to comply with the Renewable Fuel Standard (RFS) would alter conventional crop patterns in agricultural watersheds. As a result, the hydrologic response of the watersheds will exhibit different and often opposing effects on agrohydrological system variables such as riverine nitrate-N load and streamflow. Conventional modeling approaches treat those externalities as regulatory constraints, often fail to consider the hierarchical nature of the decision-making process, and end with unrealistic solutions. This study therefore proposes an alternative decision-modeling framework for biofuel development to optimize a water-quality objective under different levels of streamflow requirement in the watershed. A bilevel programming model is established to mimic the hierarchical decision-making process in environmental regulation. The model is applied to the Sangamon River basin, a typical agricultural watershed in central Illinois, to determine the optimal locations and type of ethanol biorefineries as policy instruments. The results show that the proposed instruments can effectively guide the decisions in biofuel development to meet the environmental objectives in the watershed, although adopting the proposed framework yields a lower profit than the conventional models, which is the price of a more realistic solution to the hierarchical decision problem. The results also highlight the importance of spatial heterogeneity and identifying an appropriate spatial scale to design effective environmental policies in biofuel development.
... Using this setup the bilevel problems (6)-(7) and (8)-(9) can easily be extended for aggregators with batteries. Only the upper level problems are affected: the costs for operating the batteries, which result from (13), are added to the objective function, the total demand Q(t) is defined using (14) and constraints (10)- (12) are added for describing the batteries. Because the aggregator plans over a time horizon much shorter than the whole life cycle of a battery, we use fixed upper and lower bounds for the battery level and do not model the decay of the capacity of batteries SoH as in [9]. ...
... Nevertheless, solution approaches for have been proposed in literature, see e.g. [12], [13] for an overview. The most common approach is the KKT transformation: If the lower level problem (given the decision of the upper level) is convex and Slater's condition holds ( [14, p. 190]), then q i (·) ∈ Φ i (p i (·)), if and only if the KKT conditions of the particular lower level problem are fulfilled. ...
We focus on economic aspects of grid management and model the interplay between a group of electricity consumers (respectively their smart equipment) and an aggregator. The aggregator utilizes gains from wholesale prices and sets internal prices for electricity in order to shape the consumption pattern of the whole group without compromising the comfort of participants. In addition, the aggregator may run batteries providing additional flexibility. We investigate a social-welfare-optimizing aggregator (acting in the interest of the prosumers) and a profit maximizing aggregator. We use bilevel optimization models to analyze the decisions and gains for consumers and the aggregator.
... During the past decades, some surveys and bibliographic reviews were given by several authors [11,12,15,41]. Reference books on bilevel programming and related issues have emerged [5,10,14,34,39]. ...
... (|x i | − y 2 i ) 2 without constrains at two levels 5,10], i = 1, 2, · · · , p; y i ∈ [−5, 10], i = 1, 2, · · · , q. ...
... Indeed the flow x represents the traces w such that (1) ∃w 1 , represented by y, with ww 1 ∈ L (A) ∩ W and (2) ∀w 2 , represented by u, with w aw 2 ∈ L (A) \ W. Note that if a flow x reveals some weakly liable principals, the minimisation carried on by γ t guarantees that the relevant transition t is found. Finding the weakly liable principals is a hard task, and belongs to the family of bilevel problems [4]. Basically, these problems contain two optimisation problems, one embedded in the other, and finding optimal solutions to them is still a hot research topic. ...
An approach to the formal description of service contracts is presented in terms of automata. We focus on the basic property of guaranteeing that in the multi-party composition of principals each of them gets his requests satisfied, so that the overall composition reaches its goal. Depending on whether requests are satisfied synchronously or asynchronously, we construct an orchestrator that at static time either yields composed services enjoying the required properties or detects the principals responsible for possible violations. To do that in the asynchronous case we resort to Linear Programming techniques. We also relate our automata with two logically based methods for specifying contracts.
... A detailed description of this classification with links to corresponding algorithms can be found in [70]. For solving bilevel linear problems, heuristic approaches such as evolution algorithms, tabu search, simulated annealing, grid search and other algorithms can be found in [71,72]. Computational results for the metaheuristics to the ( | )-centroid problem and other bilevel competitive facility location models can be found in [70,73]. ...
... There are different approaches to solving bi-level optimization problems, ranging from deterministic techniques (based on reformulation as equivalent single-level problems) to various heuristic procedures (Bard 1998). One class of techniques relies on the use of interactive, multistep heuristic algorithms to determine approximate Stackelberg solutions (Sinha et al. 2018). ...
Over 13% of the global population (most of which are rural communities) still lack access to electricity. A typical resolution to this would be to generate more electricity from existing power generation infrastructure. However, the urgency to meet net-zero global greenhouse gas emissions means that this resolution may not be the way forward. Instead, policymakers must consider decarbonization strategies such as renewable energy systems to generate more electricity in rural communities. As policymakers aim to encourage renewable energy generation, existing power plant operators may not share the same perspective. Operators typically wish to ensure profit margins in their operations as decarbonization efforts may be costly and reduce the profit. A balance must be struck between both parties so that the energy sector can continue to meet rising energy demands and decarbonization needs. This is a classic leader–follower situation where it involves the interplay between policymaker (as energy sector regulator) and industry (as energy sector investor). This work presents a bi-level optimization model to address the leader–follower interactions between policymakers and industry operators. The proposed model considers factors such as total investment, co-firing opportunities, incentives, disincentives, carbon emissions, scale, cost, and efficiency to meet electricity demands. To demonstrate the model, two Malaysian case studies were evaluated and presented. The first optimized networks is developed based on different energy demands. Results showed that when cost was minimized, the production capacity of the existing power plants was increased and renewable energy systems were not be selected. The second case study used bi-level optimization to determine an optimal trade-off $ 1.4 million in incentives per year, which serves as a monetary sum needed by policymakers to encourage industry operators to decarbonize their operations. Results from the second case were then compared to the ones in the first.
... Because the explicit expression of the objective in terms of processing times solely becomes far-fetched with gaps or idle times in the scheduling process on parallel machine as in Fig. 1. Against the backdrop of the bilevel programming [30]- [32], this leads to the suboptimality of the discrete and continuous subproblems from both levels after decomposition. The optimality structure of a partial solution to one subproblem may not be the part of the optimality structure of the complete solution to the primal problem [11]. ...
This paper studied a novel parallel machine rescheduling problem with controllable processing times under machine breakdown and precedence constraints. This problem is strongly
NP
-hard, and we modeled it as a mixed-integer problem (MIP). The primal problem is decomposed into the discrete subproblem and the continuous one. We treat the discrete with the dispatching rule and analyze the continuous in the terms of mathematical programming. Pros and cons from the analysis lead us to implement the commercial solver. The proposed method is capable of efficiency and nonzero initial state. We introduce transitive reduction to cull the redundant constraints out of its directed acyclic graph (DAG) representation. Transitive reduction extends the efficiency from the dispatching rule to the solver. The proposed method can do the predictive scheduling and pick up the partial solution left by the machine breakdown in the reactive session. As a result, the complete method solves the rescheduling problem with efficiency, and allows for large cases. At last, the computational results showed that this technique significantly brought down the time and RAM consumption in using the solver. This technique allows the scheduler to solve big instances with computational economy and efficiency.
... Typically, the optimum design of the lower level problem is decided independently from the upper level objective. 12 Therefore, when the regulation strategy is calculated through this type of nested optimization, there is the underlying assumption that the regulation strategy is designed for the only goal of power coefficient maximization. ...
In wind turbine optimization, the standard power regulation strategy follows a constrained trajectory based on the maximum power coefficient. It can be updated automatically during the optimization process by solving a nested maximization problem at each iteration. We argue that this model does not take advantage of the load alleviation potential of the regulation strategy and additionally requires significant computational effort. An alternative approach is proposed, where the rotational speed and pitch angle control points for the entire operation range are set as design variables, changing the problem formulation from nested to one‐level. The nested and one‐level formulations are theoretically and numerically compared on different aerodynamic blade design optimization problems for AEP maximization. The aerodynamics are calculated with a steady‐state blade element momentum method. The one‐level approach increases the design freedom of the problem and allows introducing a secondary objective in the design of the regulation strategy. Numerical results indicate that a standard regulation strategy can still emerge from a one‐level optimization. Second, we illustrate that novel optimal regulation strategies can emerge from the one‐level optimization approach. This is demonstrated by adding a thrust penalty term and a constraint on the maximum thrust. A region of minimal thrust tracking and a peak‐shaving strategy appear automatically in the optimal design.
... Bi-level Optimization: The concept of bi-level optimization has been discussed in (von Stackelberg et al., 1952;Bracken and McGill, 1973;Bard, 2006). Since then, the framework of bi-level optimization has been used in various machine learning applications like hyperparameter tuning (Mackay et al., 2019;Franceschi et al., 2018;Sinha et al., 2020), robust learning (Ren et al., 2018;Guo et al., 2020), meta-learning (Finn et al., 2017), efficient learning (Killamsetty et al., 2021) and continual learning (Borsos et al., 2020). ...
... Bilevel optimization was originally introduced in the 1930s by Stackelberg (von Stackelberg, 1934) in the context of two-players games with a leader and a follower, and later extensively studied in the field of optimization as a way to model optimization problems that contain different objectives (Bard, 1998). Closer to the learning formulation of interest to this article is bilevel optimization as introduced for the training of recurrent neural networks in the late 1980s. ...
This paper reviews gradient-based techniques to solve bilevel optimization problems. Bilevel optimization is a general way to frame the learning of systems that are implicitly defined through a quantity that they minimize. This characterization can be applied to neural networks, optimizers, algorithmic solvers and even physical systems, and allows for greater modeling flexibility compared to an explicit definition of such systems. Here we focus on gradient-based approaches that solve such problems. We distinguish them in two categories: those rooted in implicit differentiation, and those that leverage the equilibrium propagation theorem. We present the mathematical foundations that are behind such methods, introduce the gradient-estimation algorithms in detail and compare the competitive advantages of the different approaches.
... Bi-level Optimization: The concept of bi-level optimization was first discussed in (von Stackelberg et al., 1952;Bracken and McGill, 1973). Bard (2006) Since then, the framework of bi-level optimization has been used in various machine learning applications like hyperparameter tuning (Mackay et al., 2019;Franceschi et al., 2018;Sinha et al., 2020), robust learning (Ren et al., 2018;Guo et al., 2020), meta-learning (Finn et al., 2017), efficient learning and continual learning (Borsos et al., 2020). Previous applications of the bi-level optimization framework for robust learning have been limited to supervised and semi-supervised learning settings. ...
A critical bottleneck in supervised machine learning is the need for large amounts of labeled data which is expensive and time consuming to obtain. However, it has been shown that a small amount of labeled data, while insufficient to re-train a model, can be effectively used to generate human-interpretable labeling functions (LFs). These LFs, in turn, have been used to generate a large amount of additional noisy labeled data, in a paradigm that is now commonly referred to as data programming. However, previous approaches to automatically generate LFs make no attempt to further use the given labeled data for model training, thus giving up opportunities for improved performance. Moreover, since the LFs are generated from a relatively small labeled dataset, they are prone to being noisy, and naively aggregating these LFs can lead to very poor performance in practice. In this work, we propose an LF based reweighting framework \ouralgo{} to solve these two critical limitations. Our algorithm learns a joint model on the (same) labeled dataset used for LF induction along with any unlabeled data in a semi-supervised manner, and more critically, reweighs each LF according to its goodness, influencing its contribution to the semi-supervised loss using a robust bi-level optimization algorithm. We show that our algorithm significantly outperforms prior approaches on several text classification datasets.
... And it is straightforward to show that the pointwise maximum of a set of convex functions is a convex function. Since, VF(Pg i ,Pd i ) is a convex function in P g i and P d i , its maximum with respect to P g i and P d i occurs at the extreme points of P g i and P d i variables [58].■ By substituting (35) in our trilevel max-max-max problem, we have: ...
With the growing level of uncertainties in today's power systems, the vulnerability analysis of a power system with uncertain parameters becomes a must. This paper proposes a two-stage adaptive robust optimization (ARO) model for the vulnerability analysis of power systems. The main goal is to immunize the solutions against all possible realizations of the modeled uncertainty. In doing so, the uncertainties are defined by some predetermined intervals defined around the expected values of uncertain parameters. In our model, there are a set of first-stage decisions made before the uncertainty is revealed (attacker decision) and a set of second-stage decisions made after the realization of uncertainties (defender decision). This setup is formulated as a mixed-integer trilevel nonlinear program (MITNLP). Then, we recast the proposed trilevel program to a single-level mixed-integer linear program (MILP), applying the strong duality theorem (SDT) and appropriate linearization approaches. The efficient off-the-shelf solvers can guarantee the global optimum of our final MILP model. We also prove a lemma which makes our model much easier to solve. The results carried out on the IEEE RTS and modified Iran's power system show the performance of our model to assess the power system vulnerability under uncertainty.
... As shown by [10], atx = 1, bothȳ 1 ...
A new numerical method is presented for bilevel programs with a nonconvex follower’s problem. The basic idea is to piecewise construct convex relaxations of the follower’s problems, replace the relaxed follower’s problems equivalently by their Karush–Kuhn–Tucker conditions and solve the resulting mathematical programs with equilibrium constraints. The convex relaxations and needed parameters are constructed with ideas of the piecewise convexity method of global optimization. Under mild conditions, we show that every accumulation point of the optimal solutions of the sequence approximate problems is an optimal solution of the original problem. The convergence theorems of this method are presented and proved. Numerical experiments show that this method is capable of solving this class of bilevel programs.
... We can broadly categorize the solution algorithms for bilevel (3) single-level reformulation algorithms [45]. The third approach is widely used in the relevant literature to solve the bilevel optimization problems. ...
This paper examines the effects of reactive power dispatch, losses, and voltage profile on the results of the interdiction model to analyze the vulnerability of the power system. First, an attacker-defender Stackelberg game is introduced. The introduced game is modeled as a bilevel optimization problem where the attacker is modeled in the upper level and the defender is modeled in the lower level. The AC optimal power flow (ACOPF) is proposed as the defender’s tool in the lower-level problem to mitigate the attack consequences. Our proposed ACOPF-based mathematical framework is inherently a mixed-integer bilevel nonlinear program (MIBNLP) that is NP-hard and computationally challenging. This paper linearizes and then transforms it into a one-level mixed-integer linear program (MILP) using the duality theory and some proposed linearization techniques. The proposed MILP model can be solved to the global optimum using state-of-the-art solvers such as Cplex. Numerical results on two IEEE systems and Iran’s 400-kV transmission network demonstrate the performance of the proposed MILP for vulnerability assessment. We have also compared our MILP model with the DCOPF-based approach proposed in the relevant literature. The comparative results show that the reported damage measured in terms of load shedding for the DCOPF-based approach is always lower than or equal to that for the ACOPF-based approach and these models report a different set of critical lines, especially in more stressed and larger power systems. Also, the effectiveness and feasibility of the proposed MILP model for power-system vulnerability analysis are discussed and highlighted.
... A bilevel model is a mathematical program composed of two nested optimization problems, termed upper and lower level [14]. Formally, ...
... Problems under uncertainty often deal with making decisions under incomplete information. Many applications have a multilevel structure where phases in which information is revealed alternate with phases in which decisions are made; see, e.g., Ben-Tal et al. (2004) for a robust optimization perspective and Bard (1998) for a more general bilevel point of view. For example, betting on the outcome of a soccer match is typically a single-stage process. ...
Today, natural gas is one of the most important sources of energy and is regarded as a key instrument for achieving the politically set climate goals. Gas-fired power plants are valued as flexible buffers to compensate for fluctuations in electricity generation from renewable energy sources at short notice. Additionally, gas network operators face new challenges as a result of the liberalization of the European gas market. In the new entry-exit model, the gas network operators have to ensure that all possible market outcomes can be transported over the network. Hence, the operation of gas networks under uncertain conditions increasingly requires new aids for decision-making. To this end, this thesis investigates a class of general two-stage robust optimization problems whose second-stage variables are uniquely determined by non-convex constraints. This structure occurs, e.g., in gas network operations under uncertainty. Three general solution methods are developed for this problem class. The first two approaches use ideas from polynomial optimization to decide feasibility or infeasibility of a problem variant with an empty first stage. Both procedures use polynomial formulations that are approximated by semidefinite programs using the Lasserre relaxation hierarchy. The effectiveness of the methods is investigated on cyclic gas networks. It can be observed that often a low level of the Lasserre hierarchy is sufficient to decide robust feasibility or infeasibility. The third approach is based on a transformation of the two-stage problem into a normal, single-stage optimization problem. To this end, several subproblems have to be solved whose optimal values form the right-hand side of the transformed problem. An additional aggregation step can significantly reduce the number of subproblems that have to be considered. For a practical application to real-world gas network instances, mixed-integer linear relaxations of the subproblems are developed. Finally, the performance of the approach is demonstrated by benchmarks on several gas network instances, including a realistic model of the Greek natural gas network. Overall, robust feasible solutions for large networks under uncertainty can be found within a short time.
... Moreover, the contribution of this paper is threefold. First, we formulate the bilevel location-allocation problem for very general dimensional facilities and prove, under suitable conditions, the existence of optimal solutions (the reader is referred to [20,21] for further details on bilevel optimization and many of its applications). Secondly, we give an approximation scheme to solve the problem, discretizing some of its elements, providing convergence results to the optimal solution of the original problem [22][23][24]. ...
This paper deals with a bilevel approach of the location-allocation problem with dimensional facilities. We present a general model that allows us to consider very general shapes of domains for the dimensional facilities, and we prove the existence of optimal solutions under mild assumptions. To achieve these results, we borrow tools from optimal transport mass theory that allow us to give explicit solution structure of the considered lower level problem. We also provide a discretization approach that can approximate, up to any degree of accuracy, the optimal solution of the original problem. This discrete approximation can be optimally solved via a mixed-integer linear program. To address very large instance sizes, we also provide a GRASP heuristic that performs rather well according to our experimental results. The paper also reports some experiments run on test data.
... For the proposed method, as the travel time is discretised in the lower level optimisation and it is also the decision variable of the upper level optimisation, the number of decision variables and constraints in the lower level optimisation in the lower level optimisation are determined by the upper level optimisation. This also causes difficulties in applying KKT (Karush-Kuhn-Tucker) conditions on the lower level optimisation which is the most popular and efficient method for solving bilevel problems (Bard 1998;Lu et al. 2016). ...
Advances in autonomous and connected vehicles bring new opportunities for intelligent intersection control strategies. In this paper, we propose a centralised way to
jointly integrate an intersection control problem with vehicle trajectory planning. It
is formulated as a bilevel optimisation problem in which the upper level is designed
to minimise the total travel time by a mixed integer linear programming (MILP)
model. In contrast, the lower level is a linear programming (LP) model with an
objective function to maximise the total speed entering the intersection. The two
levels are coupled by the arrival time and terminal speed. By using the relationship
between the safe time headway and the process time, a novel platoon based method
is developed to reduce the computational burden. Finally, simulation tests are carried out to investigate the control performance under different demands, intersection
lengths, communication ranges and traffic compositions.
... In order to limit the complexity of this study, the airlines sectors are considered as a whole without taking into account competition among them. This leads to the formulation of two-levels nonlinear optimization problems [3][4][5] which can be treated using already well established bi-level programming techniques [6][7][8]. ...
This article addresses the problem of air traffic service (ATS) pricing over a domestic air transportation system with either private or public ATS providers. In both cases, to take into account feedback effects on the air transportation market, it is considered that the adopted pricing approaches can be formulated through optimization problems where an imbedded optimization problem is concerned with the supply of air transportation (offered seat capacity and tariffs for each connection). Under mild assumptions in both situations the whole problem can be reformulated as a mathematical program with linear objective function and quadratic constraints. A numerical application is performed to compare both pricing schemes when different levels of taxes are applied to air carriers and passengers.
... In this section, we model that by describing the internet congestion control as a nested optimization problem, that responds to the decided computing resource allocation (the decision variables p ij ). This leads to the following bilevel optimization problem [7], [8], a generalization of a Stackelberg game, which is in general non-convex. (5) is the upper level (leader) problem and represents the allocation layer. ...
... Indeed the flow x represents the traces w such that (1) ∃w 1 , represented by y, with ww 1 ∈ L (A) ∩ W and (2) ∀w 2 , represented by u, with w aw 2 ∈ L (A) \ W. Note that if a flow x reveals some weakly liable principals, the minimisation carried on by γ t guarantees that the relevant transition t is found. Finding the weakly liable principals is a hard task, and belongs to the family of bilevel problems [4]. Basically, these problems contain two optimisation problems, one embedded in the other, and finding optimal solutions to them is still a hot research topic. ...
An approach to the formal description of service contracts is presented in terms of automata. We focus on the basic property of guaranteeing that in the multi-party composition of principals each of them gets his requests satisfied, so that the overall composition reaches its goal. Depending on whether requests are satisfied synchronously or asynchronously, we construct an orchestrator that at static time either yields composed services enjoying the required properties or detects the principals responsible for possible violations. To do that in the asynchronous case we resort to Linear Programming techniques. We also relate our automata with two logically based methods for specifying contracts.
... DR1 is a bi-level model. Readers could consult [5] for a thorough treatment of bi-level optimization models, whereas [10] provides a useful survey on this subject. The lower level, i.e., (14) to (18), is an economic dispatch model (ED1) that takes the demand response variable r as given. ...
We propose a bi-level optimization model for demand response in organized wholesale energy markets. In this model, the lower level performs the economic dispatch of energy and generates the price and the upper level minimizes the total amount of demand response subject to a net benefit requirement. In an economic sense, demand response is a trade of ‘consuming rights’ instead of a sale of energy. Therefore it must be traded separately from the energy market. Although a bi-level optimization model is very hard to solve in general, we demonstrate that realistic power networks have characteristics that can be exploited to reduce the effective size of the problem instance. In particular, we transform the nonconvex net benefit test constraint to an equivalent linear form, and reformulate the nonconvex complementarity conditions of doubly bounded variables using SOS2 constraints. For realistic instances of the MPEC, we employ a three-phase approach that exploits the fast local solution from a nonlinear programming solver as well as LP-based bound strengthening within a mixed integer/SOS2 formulation. The model is tested against various data cases and settings, and generates useful insight for demand response dispatch operations in practice.
In this paper, we mainly focus on the solving approach for the linear trilevel programming (LTP) problem. Firstly, based on the lower-level problem’s Karush–Kuhn–Tucker (K-K-T) optimality conditions, we transform the LTP problem into a bilevel programming (BP) problem with complementary constraints. Secondly, taking the complementary constraints as penalties and appending them to the upper-level objective, a penalized BP problem is obtained. Thirdly, for the penalized BP problem, we use K-K-T optimality conditions again and append the corresponding complementary conditions to the upper level as penalties. Then, an overall penalized problem for the LTP problem is formed; we analyze the characteristics of the optimal solutions of the overall penalized problem and propose a penalty function algorithm. The numerical results show that the penalty function approach is feasible and effective.
Facing a significant increase in connected distributed energy systems, the optimal coordination strategy among multiple distributed energy systems is vitally important to explore. In this paper, an energy management system is introduced and operated by an energy service company, which is responsible for managing the interaction of multiple distributed energy systems. To optimize the day-ahead scheduling of the distributed energy systems, a coordination scheme with a bilevel framework is proposed. The energy interaction between the energy management system and distributed energy systems contains electricity and heat, which is a Stackelberg problem. Two types of internal price schemes, namely, real-time pricing and time-of-use pricing, are discussed. Moreover, the uncertainties of renewable energy resources, energy demand, and energy prices are considered within both upper- and lower-level problems. The problem is formulated as a nonlinear bilevel robust optimization model and transformed into a single-level mixed-integer linear problem. Numerical cases illustrate how the energy management system coordinates with distributed energy systems and show the effectiveness of the coordination strategy such that all participators benefit from the proposed strategy and create a win-win situation. The model and results can serve as references for the business managers of companies that provide energy services for building clusters.
The problem of determining the effective (optimal) heat supply radius is considered. Heat supply radius is transportation distance of heat energy in the district heating systems (DHS), under which the highest indices of economic efficiency of district heating to consumers are respected. To solve this difficult and multifactorial problem, a bi-level approach has been proposed. This approach allows finding the optimal frameworks of territorial areas of district heating while fulfilling the necessary requirements for thermal-hydraulic modes in heat networks and for reliability of heating to consumers. Methodology for solving the formulated problem is based on bi-level programming methods, models of Theory of hydraulic circuits, nonlinear optimization methods, nodal reliability indices (availability factor, failure-free operation probability), Markov random processes models and other methods and models. A case study has been conducted using the developed methodological apparatus for the actual DHS scheme of the Irkutsk city (Russia, Siberia).
Bilevel optimization refers to a challenging optimization problem which contains two levels of optimization problems. The task of bilevel optimization is to find the optimum of the upper-level problem, subject to the optimality of the corresponding lower-level problem. This nested nature introduces many difficulties such as non-convexity and disconnectedness, and poses great challenges to traditional optimization methods. Using evolutionary algorithms in bilevel optimization has been demonstrated to be very promising in recent years. However, these algorithms suffer from low efficiency since they usually require a huge number of function evaluations. This paper proposes a bilevel covariance matrix adaptation evolution strategy (BL-CMA-ES) to handle bilevel optimization problems. A search distribution sharing mechanism is designed so that we can extract a priori knowledge of the lower-level problem from the upper-level optimizer, which significantly reduces the number of function evaluations. We also propose a refinement based elite preservation mechanism to trace the elite and avoid inaccurate solutions. Comparisons with five state-of-the-art algorithms on twenty-two benchmark problems and two real-world applications are carried out to test the performance of the proposed approach. The experimental results have shown the effectiveness of the proposed approach in keeping a good trade-off between solution quality and computational efficiency. IEEE
As an effective way to solicit useful information from the crowd, crowdsourcing has emerged as a popular paradigm to solve challenging tasks. However, the data provided by the participating workers are not always trustworthy. In real world, there may exist malicious workers in crowdsourcing systems who conduct the data poisoning attacks for the purpose of sabotage or financial rewards. Although data aggregation methods such as majority voting are conducted on workers» labels in order to improve data quality, they are vulnerable to such attacks as they treat all the workers equally. In order to capture the variety in the reliability of workers, the Dawid-Skene model, a sophisticated data aggregation method, has been widely adopted in practice. By conducting maximum likelihood estimation (MLE) using the expectation maximization (EM) algorithm, the Dawid-Skene model can jointly estimate each worker»s reliability and conduct weighted aggregation, and thus can tolerate the data poisoning attacks to some degree. However, the Dawid-Skene model still has weakness. In this paper, we study the data poisoning attacks against such crowdsourcing systems with the Dawid-Skene model empowered. We design an intelligent attack mechanism, based on which the attacker can not only achieve maximum attack utility but also disguise the attacking behaviors. Extensive experiments based on real-world crowdsourcing datasets are conducted to verify the desirable properties of the proposed mechanism.
A direct-search derivative-free Matlab optimizer for bound-constrained problems is described, whose remarkable features are its ability to handle a mix of continuous and discrete variables, a versatile interface as well as a novel self-training option. Its performance compares favorably with that of NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search), a well-known derivative-free optimization package. It is also applicable to multilevel equilibrium- or constrained-type problems. Its easy-to-use interface provides a number of user-oriented features, such as checkpointing and restart, variable scaling, and early termination tools.
In this paper, we formulate a dynamic transportation network design model in which traffic dynamics are modeled by the cell transmission model. In the formulation, transportation planners decide on the optimal capacity expansion policies of existing transportation network infrastructure with limited resources, while road users react to the capacity changes by selfishly choosing routes to maximize their own profit. Owing to the problem complexity, a majority of the research efforts have focused on tackling this problem using meta-heuristics. In this study, we incorporate a series of dual-variable approximation techniques into the paradigm of a quantum-inspired genetic algorithm (QIGA) and devise an efficient evaluation function based on these techniques. The proposed QIGA contains a series of enhancements compared to conventional genetic algorithms (GAs) and can be considered as a better alternative when solving problems with a complex solution space. The QIGA is applied to a synthetic network, a subnetwork of a real-world road network, and a realistic network to justify its theoretical and practical value. From the numerical results, it is found that in the same computational time, the QIGA outperforms the conventional GA by 3.86–5.63% in terms of the objective value, which can be significant, especially when network expansion of a large urban area is considered. Technical, computational, and practical issues involved in developing the QIGA are investigated and discussed.
Bilevel optimization problems are referred to as having a nested inner optimization problem as a constraint to a outer optimization problem in the domain of mathematical programming. It is also known as Stackelberg problems in game theory. In the recent past, bilevel optimization problems have received a growing attention because of its relevance in practice applications. However, the hierarchical structure makes these problems difficult to handle and they are commonly optimized with a deterministic setup. With presence of constrains, bilevel optimization problems are considered for finding reliable solutions which are subjected to a possess a minimum reliability requirement under decision variable uncertainties. Definition of reliable bilevel solution, the effect of lower and upper level uncertainties on reliable bilevel solution, development of efficient reliable bilevel evolutionary algorithm, and supporting simulation results on test and engineering design problems amply demonstrate their further use in other practical bilevel problems.
In this paper a biobjective Swap Body Inventory Routing Problem (SB-IRP) is considered: A combination of the Swap Body Vehicle Routing Problem (SB-VRP), which minimizes fixed and variable routing costs, and the Inventory Routing Problem (IRP). The problem is based on the context of the VeRoLog Solver Challenge 2014, where our proposed VeRoLog Solver was ranked third, and our previous work on the IRP. Since we are investigating a multi-period problem, an additional objective function is formulated which includes inventory levels at the customers. Dealing with the allocation of scarce optimization time to the VeRoLog Solver is an essential topic, since an alternating approach of the determination of a replenishment strategy and the routing is considered. We propose an Iterative Variable Neighborhood Search and analyze the allocation of the computational time by extended VeRoLog test instances.
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