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A prominent challenge to the safe and optimal operation of the modern power grid arises due to growing uncertainties in loads and renewables. Stochastic optimal power flow (SOPF) formulations provide a mechanism to handle these uncertainties by computing dispatch decisions and control policies that maintain feasibility under uncertainty. Most SOPF formulations consider simple control policies such as affine policies that are mathematically simple and resemble many policies used in current practice. Motivated by the efficacy of machine learning (ML) algorithms and the potential benefits of general control policies for cost and constraint enforcement, we put forth a deep neural network (DNN)-based policy that predicts the generator dispatch decisions in real time in response to uncertainty. The weights of the DNN are learnt using stochastic primal–dual updates that solve the SOPF without the need for prior generation of training labels and can explicitly account for the feasibility constraints in the SOPF. The advantages of the DNN policy over simpler policies and their efficacy in enforcing safety limits and producing near optimal solutions are demonstrated in the context of a chance constrained formulation on a number of test cases.

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... The maximum number of iterations was set to 1, 000, and we used the aer simulator statevector quantum simulation backend. For the dual update in (14), constraint violations were measured over the observables H m using the minimum eigenstate returned by VQE. The stopping criteria λ t − λ t−1 2 ≤ 1 · 10 −5 was utilized to ascertain the convergence of the dual updates (14). ...

... For the dual update in (14), constraint violations were measured over the observables H m using the minimum eigenstate returned by VQE. The stopping criteria λ t − λ t−1 2 ≤ 1 · 10 −5 was utilized to ascertain the convergence of the dual updates (14). ...

... This novel heuristic sets the foundation for further developments towards constrained discrete optimization. We are currently exploring several exciting directions: i) Coupling this approach with QAOA rather than VQE; ii) skipping the nested optimization in (15) through a primal-dual decomposition alternative as in [14,15]; and iii) dealing with mixed-binary setups. ...

Analytical and practical evidence indicates the advantage of quantum computing solutions over classical alternatives. Quantum-based heuristics relying on the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) have been shown numerically to generate high-quality solutions to hard combinatorial problems, yet incorporating constraints to such problems has been elusive. To this end, this work puts forth a quantum heuristic to cope with stochastic binary quadratically constrained quadratic programs (QCQP). Identifying the strength of quantum circuits to efficiently generate samples from probability distributions that are otherwise hard to sample from, the variational quantum circuit is trained to generate binary-valued vectors to approximately solve the aforesaid stochastic program. The method builds upon dual decomposition and entails solving a sequence of judiciously modified standard VQE tasks. Tests on several synthetic problem instances using a quantum simulator corroborate the near-optimality and feasibility of the method, and its potential to generate feasible solutions for the deterministic QCQP too.

... In [77], the relation between the LVDS feeder and the hosting capacity (HC) is mapped using a linear regression function based on the features of the feeder. Machine learning tools like neural networks are recently used for minimizing the cost of generation under uncertainties as in [78]. In [79], a deep neural network (DNN) is used to approximate the power flow calculation, making a surrogate model to reduce computational effort of MC based methods. ...

The increased penetration of renewable resources, such as photovoltaics (PV) and new loads, such as electric vehicles (EV) and heat pumps (HP), have increased uncertainty levels in the low voltage distribution system (LVDS). The traditional approach to planning such LVDS is the conservative ``fit and forget'' approach investing for the worst-case scenario, which means installing a lot of additional infrastructure. However, considering LVDS planning as a stochastic problem is deemed more efficient, reducing grid infrastructure investment cost. Determining the PV hosting capacity (HC) is one such planning problem, yielding the capacity of the system to incorporate new PV resources. Stochastic HC calculations are currently done using computationally expensive, iterative Monte Carlo (MC) based methods, which require solving the power flow equations thousands of times for each iteration of the PV installation scenario. To calculate the PV HC of a large service area consisting of thousands of LV feeders, a computationally tractable and accurate probabilistic power flow (PPF) tool is required. Various approaches exist to make the MC-based iterative HC calculation method computationally more efficient, e.g., linearizing the power flow equations, replacing some of the uncertain variables with deterministic values, etc. Alternatively, MC-based methods can be replaced by faster analytic methods, or the iterative HC calculation method can be replaced by stochastic optimization. This thesis aims to build analytical alternatives for MC-based methods and stochastic optimization tools to replace iterative LVDS HC calculation methods.
First, comparing the existing PV HC methods is challenging as they are demonstrated on different test feeders using different assumptions. Therefore, this thesis proposes a benchmark wherein the effects of assumptions made for calculating the HC are evaluated. The comparison of the PV HC obtained from existing methods shows a huge spread, mainly affected by the assumptions on the grid and stochastic limits and the size and number of the PV installations.
Standardizing these limits is a major step towards defining a realistic HC of LVDS feeders. As a benchmark, a stochastic limit of 5% and a grid voltage limit of 0.95-1.05 pu are recommended.
Second, MC simulations are the bottleneck in computing the stochastic PV HC of a large service area. This thesis proposes non-intrusive general Polynomial Chaos (gPC) expansion based PPF as an alternative. The gPC-based PPF outweighs MC and quasi-Monte Carlo (qMC) methods clearly in terms of computational effort with comparable accuracy, as the complex power flow equations are replaced by their polynomial surrogate. The proposed PPF, when used to calculate the congestion probability of a European LVDS, is ten times faster than the MC-based method with the same accuracy.
Third, stochastic HC calculations require an understanding of the uncertainties in LVDS. The operational uncertainties due to load and generation variations and PV scenario uncertainties due to size, type and phase of the PV installations are usually sampled together in MC-based probabilistic HC approaches. This dissertation presents a decoupled approach to calculate the stochastic PV HC, where the probability of violation of the operational limits is computed for possible PV planning scenarios. Then, the PV scenario that results in the highest total PV power installed without violating the stochastic operational limits of the LV feeder is proposed to be the stochastic HC of that feeder. Decoupling the PV (planning) scenarios from the operational uncertainties instead of sampling them together enables to study of the impact of planning policies and operational rules on the overall PV HC.
Fourth, the iterative approach of computing PV HC, where the PV size installed is increased in every iteration, is a computationally demanding process. The decoupled method is also a brute force approach where the probability of congestion is calculated for each possible scenario. In contrast, a chance-constrained stochastic optimal power flow (SOPF) based method reduces the decoupled HC calculation into a single-shot problem. This thesis proposes an intrusive gPC-based SOPF for calculating stochastic PV HC. This method reduces the computation time from days to seconds while giving the upper bound of the HC that can be identified using conventional methods.
Finally, when investigating the needs for a larger region, it is generally too hard to compute the stochastic HC for all individual LV feeders, as a small service area already can have hundreds of feeders. An approximation of the PV HC of the entire service area can be obtained by scaling up the HC of a small number of representative feeders. This thesis presents a clustering scheme that captures the most relevant features of LVDS feeders for PV HC to obtain such representative feeders. A case study shows that the PV HC of a small service area can be approximated by using only 3% of the carefully chosen representative feeders with an error of 20%.

Solving the optimal power flow (OPF) problem is a fundamental task to ensure the system efficiency and reliability in real-time electricity grid operations. We develop a new topology-informed graph neural network (GNN) approach for predicting the optimal solutions of real-time ac-OPF problem. To incorporate grid topology to the NN model, the proposed GNN-for-OPF framework innovatively exploits the locality property of locational marginal prices and voltage magnitude. Furthermore, we develop a physics-aware (ac-)flow feasibility regularization approach for general OPF learning. The advantages of our proposed designs include reduced model complexity, improved generalizability and feasibility guarantees. By providing the analytical understanding on the graph subspace stability under grid topology contingency, we show the proposed GNN can quickly adapt to varying grid topology by an efficient re-training strategy. Numerical tests on various test systems of different sizes have validated the prediction accuracy, improved flow feasibility, and topology adaptivity capability of our proposed GNN-based learning framework.

This paper considers the design of optimal resource allocation policies in wireless communication systems which are generically modeled as a functional optimization problem with stochastic constraints. These optimization problems have the structure of a learning problem in which the statistical loss appears as a constraint, motivating the development of learning methodologies to attempt their solution. To handle stochastic constraints, training is undertaken in the dual domain. It is shown that this can be done with small loss of optimality when using near-universal learning parameterizations. In particular, since deep neural networks (DNN) are near-universal their use is advocated and explored. DNNs are trained here with a model-free primal-dual method that simultaneously learns a DNN parametrization of the resource allocation policy and optimizes the primal and dual variables. Numerical simulations demonstrate the strong performance of the proposed approach on a number of common wireless resource allocation problems.

This paper jointly addresses two major challenges in power system operations: i) dealing with non-convexity in the power flow equations, and ii) systematically capturing uncertainty in renewable power availability and in active and reactive power consumption at load buses. To overcome these challenges, this paper proposes a two-stage adaptive robust optimization model for the multi-period AC optimal power flow problem (AC-OPF) with detailed modeling considerations such as reactive capability curves of conventional and renewable generators and transmission constraints. This paper then applies strong SOCP-based convex relaxations of AC-OPF combined with the use of an alternating direction method to identify worst-case uncertainty realizations, and also presents a speed-up technique based on screening transmission line constraints. Extensive computational experiments show that the solution method is efficient and that the robust AC OPF model has significant advantages both from the economic and reliability standpoints as compared to a deterministic AC-OPF model.

Smart distribution grids should efficiently integrate stochastic renewable resources while effecting voltage regulation. The design of energy management schemes is challenging, one of the reasons being that energy management is a multistage problem where decisions are not all made at the same timescale and must account for the variability during real-time operation. The joint dispatch of slow- and fast-timescale controls in a smart distribution grid is considered here. The substation voltage, the energy exchanged with a main grid, and the generation schedules for small diesel generators have to be decided on a slow timescale; whereas optimal photovoltaic inverter setpoints are found on a more frequent basis. While inverter and looser voltage regulation limits are imposed at all times, tighter bus voltage constraints are enforced on the average or in probability, thus enabling more efficient renewable integration. Upon reformulating the two-stage grid dispatch as a stochastic convex-concave problem, two distribution-free schemes are put forth. An average dispatch algorithm converges provably to the optimal two-stage decisions via a sequence of convex quadratic programs. Its non-convex probabilistic alternative entails solving two slightly different convex problems and is numerically shown to converge. Numerical tests on a real-world distribution feeder verify that both novel data-driven schemes yield lower costs over competing alternatives.

Due to the increasing amount of electricity generated from renewable sources,
uncertainty in power system operation will grow. This has implications for
tools such as Optimal Power Flow (OPF), an optimization problem widely used in
power system operations and planning, which should be adjusted to account for
this uncertainty. One way to handle the uncertainty is to formulate a Chance
Constrained OPF (CC-OPF) which limits the probability of constraint violation
to a predefined value. However, existing CC-OPF formulations and solutions are
not immune to drawbacks. On one hand, they only consider affine policies for
generation control, which are not always realistic and may be sub-optimal. On
the other hand, the standard CC-OPF formulations do not distinguish between
large and small violations, although those might carry significantly different
risk. In this paper, we introduce the Weighted CC-OPF (WCC-OPF) that can handle
general control policies while preserving convexity and allowing for efficient
computation. The weighted chance constraints account for the size of violations
through a weighting function, which assigns a higher risk to a higher
overloads. We prove that the problem remains convex for any convex weighting
function, and for very general generation control policies. In a case study, we
compare the performance of the new WCC-OPF and the standard CC-OPF and
demonstrate that WCC-OPF effectively reduces the number of severe overloads.
Furthermore, we compare an affine generation control policy with a more general
policy, and show that the additional flexibility allow for a lower cost while
maintaining the same level of risk.

When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely
used by the electric power industry to re-dispatch hourly controllable
generation (coal, gas and hydro plants) over control areas of transmission
networks, can result in grid instability, and, potentially, cascading outages.
This risk arises because OPF dispatch is computed without awareness of major
uncertainty, in particular fluctuations in renewable output. As a result, grid
operation under OPF with renewable variability can lead to frequent conditions
where power line flow ratings are significantly exceeded. Such a condition,
which is borne by simulations of real grids, would likely resulting in
automatic line tripping to protect lines from thermal stress, a risky and
undesirable outcome which compromises stability. Smart grid goals include a
commitment to large penetration of highly fluctuating renewables, thus calling
to reconsider current practices, in particular the use of standard OPF. Our
Chance Constrained (CC) OPF corrects the problem and mitigates dangerous
renewable fluctuations with minimal changes in the current operational
procedure. Assuming availability of a reliable wind forecast parameterizing the
distribution function of the uncertain generation, our CC-OPF satisfies all the
constraints with high probability while simultaneously minimizing the cost of
economic re-dispatch. CC-OPF allows efficient implementation, e.g. solving a
typical instance over the 2746-bus Polish network in 20 seconds on a standard
laptop.

MATPOWER is an open-source Matlab-based power system simulation package that provides a high-level set of power flow, optimal power flow (OPF), and other tools targeted toward researchers, educators, and students. The OPF architecture is designed to be extensible, making it easy to add user-defined variables, costs, and constraints to the standard OPF problem. This paper presents the details of the network modeling and problem formulations used by MATPOWER, including its extensible OPF architecture. This structure is used internally to implement several extensions to the standard OPF problem, including piece-wise linear cost functions, dispatchable loads, generator capability curves, and branch angle difference limits. Simulation results are presented for a number of test cases comparing the performance of several available OPF solvers and demonstrating MATPOWER's ability to solve large-scale AC and DC OPF problems.

We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given close to one probability, a system of randomly perturbed convex constraints. This problem may happen to be computationally intractable; our goal is to build its computationally tractable approximation, i.e., an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem. We construct a general class of such convex conservative approximations of the corresponding chance constrained problem. Moreover, under the assumptions that the constraints are a. ne in the perturbations and the entries in the perturbation vector are independent-of-each-other random variables, we build a large deviation-type approximation, referred to as "Bernstein approximation," of the chance constrained problem. This approximation is convex and efficiently solvable. We propose a simulation-based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the Bernstein and well-known scenario approximation approaches. Finally, we extend our construction to the case of ambiguous chance constrained problems, where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly, while the chance constraint should be satisfied for every distribution given by this set.

Coordinating inverters at scale under uncertainty is the desideratum for integrating renewables in distribution grids. Unless load demands and solar generation are telemetered frequently, controlling inverters given approximate grid conditions or proxies thereof becomes a key specification. Although deep neural networks (DNNs) can learn optimal inverter schedules, guaranteeing feasibility is largely elusive. Rather than training DNNs to imitate already computed optimal power flow (OPF) solutions, this work integrates DNN-based inverter policies into the OPF. The proposed DNNs are trained through two OPF alternatives that confine voltage deviations on the average and as a convex restriction of chance constraints. The trained DNNs can be driven by partial, noisy, or proxy descriptors of the current grid conditions. This is important when OPF has to be solved for an unobservable feeder. DNN weights are trained via back-propagation and upon differentiating the AC power flow equations. An alternative gradient-free variant is also put forth, which requires only a power flow solver and avoids computing gradients. Such variant is practically relevant when calculating gradients becomes cumbersome or prone to errors. Numerical tests compare the DNN-based inverter control schemes with the optimal inverter setpoints in terms of optimality and feasibility.

To shift the computational burden from real-time to offline in delay-critical power systems applications, recent works entertain the idea of using a deep neural network (DNN to predict the solutions of the AC optimal power flow (AC-OPF once presented load demands. As network topologies may change, training this DNN in a sample-efficient manner becomes a necessity. To improve data efficiency, this work utilizes the fact OPF data are not simple training labels, but constitute the solutions of a parametric optimization problem. We thus advocate training a sensitivity-informed DNN (SI-DNN to match not only the OPF optimizers, but also their partial derivatives with respect to the OPF parameters (loads . It is shown that the required Jacobian matrices do exist under mild conditions, and can be readily computed from the related primal/dual solutions. The proposed SI-DNN is compatible with a broad range of OPF solvers, including a non-convex quadratically constrained quadratic program (QCQP, its semidefinite program (SDP relaxation, and MATPOWER; while SI-DNN can be seamlessly integrated in other learning-to-OPF schemes. Numerical tests on three benchmark power systems corroborate the advanced generalization and constraint satisfaction capabilities for the OPF solutions predicted by an SI-DNN over a conventionally trained DNN, especially in low-data setups.

Electric power grids regularly experience uncertain fluctuations from load demands and renewables, which poses a risk of violating operational limits designed to safeguard the system. In this paper, we consider the robust AC OPF problem that minimizes the generation cost while requiring a certain level of system security in the presence of uncertainty. The robust AC OPF problem requires that the system satisfy operational limits for all uncertainty realizations within a specified uncertainty set. Guaranteeing robustness is particularly challenging due to the non-convex, nonlinear AC power flow equations, which may not always have a solution. In this work, we extend a previously developed convex restriction to a
robust convex restriction
, which is a convex inner approximation of the non-convex feasible region of the AC OPF problem that accounts for uncertainty in the power injections. We then use the robust convex restriction in an algorithm that obtains robust solutions to AC OPF problems by solving a sequence of convex optimization problems. We demonstrate our algorithm and its ability to control robustness versus operating cost trade-offs using PGLib test cases.

Growing uncertainty from renewable energy integration and distributed energy resources motivate the need for advanced tools to quantify the effect of uncertainty and assess the risks it poses to secure system operation. Polynomial chaos expansion (PCE) has been recently proposed as a tool for uncertainty quantification in power systems. The method produces highly accurate results, but has proved to be computationally challenging to scale to large systems. We propose a modified algorithm based on PCE with significantly improved computational efficiency that retains the desired high level of accuracy of the standard PCE. Our method uses computational enhancements by exploiting the sparsity structure and algebraic properties of the power flow equations. We show the scalability of the method on the 1354 pegase test system, assess the quality of the uncertainty quantification in terms of accuracy and robustness, and demonstrate an example application to solving the chance constrained optimal power flow problem.

There is an emerging need for efficient solutions to stochastic AC Optimal Power Flow (AC-OPF) to ensure optimal and reliable grid operations in the presence of increasing demand and generation uncertainty. This paper presents a highly scalable data-driven algorithm for stochastic AC-OPF that has extremely low sample requirement. The novelty behind the algorithm’s performance involves an iterative scenario design approach that merges information regarding constraint violations in the system with data-driven sparse regression. Compared to conventional methods with random scenario sampling, our approach is able to provide feasible operating points for realistic systems with much lower sample requirements. Furthermore, multiple sub-tasks in our approach can be easily paralleled and based on historical data to enhance its performance and application. We demonstrate the computational improvements of our approach through simulations on different test cases in the IEEE PES PGLib-OPF benchmark library.

Smart inverters have been advocated as a fast-responding mechanism for voltage regulation in distribution grids. Nevertheless, optimal inverter coordination can be computationally demanding, and preset local control rules are known to be subpar. Leveraging tools from machine learning, the design of customized inverter control rules is posed here as a multi-task learning problem. Each inverter control rule is modeled as a possibly nonlinear function of local and/or remote control inputs. Given the electric coupling, the function outputs interact to yield the feeder voltage profile. Using an approximate grid model, inverter rules are designed jointly to minimize a voltage deviation objective based on anticipated load and solar generation scenarios. Each control rule is described by a set of coefficients, one for each training scenario. To reduce the communication overhead between the grid operator and the inverters, we devise a voltage regulation objective that is shown to promote parsimonious descriptions for inverter control rules. Numerical tests using real-world data on a benchmark feeder demonstrate the advantages of the novel nonlinear rules and explore the trade-off between voltage regulation and sparsity in rule descriptions.

As the share of renewables in the grid increases, the operation of power systems becomes more challenging. The present paper proposes a method to formulate and solve chance-constrained optimal power flow while explicitly considering the full nonlinear AC power flow equations and stochastic uncertainties. We use polynomial chaos expansion to model the effects of arbitrary uncertainties of finite variance, which enables to predict and optimize the system state for a range of operating conditions. We apply chance constraints to limit the probability of violations of inequality constraints. Our method incorporates a more detailed and a more flexible description of both the controllable variables and the resulting system state than previous methods. Two case studies highlight the efficacy of the method, with a focus on satisfaction of the AC power flow equations and on the accurate computation of moments of all random variables.

This paper focuses on distribution systems featuring renewable energy sources (RESs) and energy storage systems, and presents an AC optimal power flow (OPF) approach to optimize system-level performance objectives while coping with uncertainty in both RES generation and loads. The proposed method hinges on a chance-constrained AC OPF formulation where probabilistic constraints are utilized to enforce voltage regulation with prescribed probability. A computationally more affordable convex reformulation is developed by resorting to suitable linear approximations of the AC power-flow equations as well as convex approximations of the chance constraints. The approximate chance constraints provide conservative bounds that hold for arbitrary distributions of the forecasting errors. An adaptive strategy is then obtained by embedding the proposed AC OPF task into a model predictive control framework. Finally, a distributed solver is developed to strategically distribute the solution of the optimization problems across utility and customers.

Distribution systems will be critically challenged by reverse power flows and voltage fluctuations due to the integration of distributed renewable generation, demand response, and electric vehicles. Yet the same transformative changes coupled with advances in microelectronics offer new opportunities for reactive power management in distribution grids. In this context and considering the increasing time-variability of distributed generation and demand, a scheme for stochastic loss minimization is developed here. Given uncertain active power injections, a stochastic reactive control algorithm is devised. Leveraging the recent convex relaxation of optimal power flow problems, it is shown that the subgradient of the power losses can be obtained as the Lagrange multiplier of the related second-order cone program (SOCP). Numerical tests on a 47-bus test feeder with high photovoltaic penetration corroborates the power efficiency and voltage profile advantage of the novel stochastic method over its deterministic alternative.

Distribution grids are critically challenged by the variability of renewable
energy sources. Slow response times and long energy management periods cannot
efficiently integrate intermittent renewable generation and demand. Yet
stochasticity can be judiciously coupled with system flexibilities to improve
efficiency of the grid operation. Voltage magnitudes for instance can
transiently exceed regulation limits, while smart inverters can be overloaded
over short time intervals. To implement such a mode of operation, an ergodic
energy management framework is developed here. Considering a distribution grid
with distributed energy sources and a feed-in tariff program, active power
curtailment and reactive power compensation are formulated as a stochastic
optimization problem. Tighter operational constraints are enforced in an
average sense, while looser margins are satisfied at all times. Stochastic dual
subgradient solvers are developed based on exact and approximate grid models of
varying complexity. Numerical tests on a real-world 56-bus distribution grid
relying on both grid models corroborate the advantages of the novel schemes
over its deterministic alternative.

We propose a probabilistic framework to design an secure day-ahead dispatch and determine the minimum cost reserves for power systems with wind power generation. We also identify a reserve strategy according to which we deploy the reserves in real-time operation, which serves as a corrective control action. To achieve this, we formulate a stochastic optimization program with chance constraints, which encode the probability of satisfying the transmission capacity constraints of the lines and the generation limits. To incorporate a reserve decision scheme, we take into account the steady-state behavior of the secondary frequency controller and, hence, consider the deployed reserves to be a linear function of the total generation-load mismatch. The overall problem results in a chance constrained bilinear program. To achieve tractability, we propose a convex reformulation and a heuristic algorithm, whereas to deal with the chance constraint we use a scenario-based-approach and an approach that considers only the quantiles of the stationary distribution of the wind power error. To quantify the effectiveness of the proposed methodologies and compare them in terms of cost and performance, we use the IEEE 30-bus network and carry out Monte Carlo simulations, corresponding to different wind power realizations generated by a Markov chain-based model.

In this paper, uncertainties from wind power in-feed are taken into account in a DC security-constrained optimal power flow (SCOPF) by formulating probabilistic constraints. The deviations from the wind power forecast are represented as Gaussian random variables and an analytical reformulation of the constraints is proposed, which is exact for the Gaussian distribution. The resulting formulation has the same computational complexity as the deterministic problem. Furthermore, a valuation framework to assess the cost of securing the system against fluctuations in wind power in-feed is proposed. The applicability of the method and the valuation framework is demonstrated on the IEEE RTS96 system. We show that the probabilistic formulation leads to lower probability of thermal overloads, and that it is less costly to secure the system against uncertain in-feed than to secure the system against failures in most cases.

This paper rigorously establishes that standard multilayer feedforward networks with as few as one hidden layer using arbitrary squashing functions are capable of approximating any Borel measurable function from one finite dimensional space to another to any desired degree of accuracy, provided sufficiently many hidden units are available. In this sense, multilayer feedforward networks are a class of universal approximators.

A smooth approximationp (x, α) to the plus function max{x, 0} is obtained by integrating the sigmoid function 1/(1 + e−αx
), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy forα sufficiently large. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finiteα. Speedup over MINOS 5.4 was as high as 1142 times for linear inequalities of size 2000 × 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems are converted into a system of smooth nonlinear equations and are solved by a quadratically convergent Newton method. For monotone LCPs with as many as 10 000 variables, the proposed approach was as much as 63 times faster than Lemke's method.

DC3: A learning method for optimization with hard constraints

- P L Donti
- D Rolnick
- J Z Kolter

DeepOPF-NGT: A fast unsupervised learning approach for solving AC-OPF problems without ground truth

- W Huang
- M Chen