Mathieu Van Vyve

Algorithms, Applied Mathematics, Energy Economics

PhD
16.78

Publications

  • Mehdi Madani · Mathieu Van Vyve
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    ABSTRACT: We examine the specific problem of clearing day-ahead electricity market auctions where each bidder, whether a producer or consumer, can specify a minimum profit or maximum payment condition constraining the acceptance of a set of bid curves spanning multiple time periods in locations connected through a transmission network with linear constraints. This helps describing e.g. the recovery of start-up costs of a power plant, or analogously for a large consumer, utility reduced by a constant term. We propose here a new market model with a corresponding MILP formulation for uniform locational price day-ahead auctions, handling bids with a minimum profit or maximum payment condition, which we call MP bids, in a uniform and computationally-efficient way. We also propose an exact decomposition procedure with sparse strengthened Benders cuts derived from the MILP formulation. Both the MILP formulation and the exact decomposition procedure are similar to computationally-efficient approaches previously proposed to handle indivisible bids (so-called block bids) according to European market rules, though the clearing conditions could appear different at first sight. Indeed, the approach and the cuts proposed are also valid to deal with both kinds of bids simultaneously, as block bids with a minimum acceptance ratio, which generalize regular (fully indivisible) block bids, turn out to be a special case of the MP bids introduced here which include minimum power output level information. We compare these MP bids to previous models for minimum profit conditions proposed in the academic literature, and to the model for minimum income conditions used by the Spanish power exchange OMIE, now integrated to the pan-European PCR market.
    No preview · Article · Aug 2015
  • Mehdi Madani · Mathieu Van Vyve
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    ABSTRACT: It is well known that a market equilibrium with uniform prices often does not exist in non-convex day-ahead electricity auctions. We consider the case of the non-convex, uniform-price Pan-European day-ahead electricity market “PCR” (Price Coupling of Regions), with non-convexities arising from so-called complex and block orders. Extending previous results, we propose a new primal-dual framework for these auctions, which has applications in both economic analysis and algorithm design. The contribution here is threefold. First, from the algorithmic point of view, we give a non-trivial exact (i.e., not approximate) linearization of a non-convex ‘minimum income condition’ that must hold for complex orders arising from the Spanish market, avoiding the introduction of any auxiliary variables, and allowing us to solve market clearing instances involving most of the bidding products proposed in PCR using off-the-shelf MIP solvers. Second, from the economic analysis point of view, we give the first MILP formulations of optimization problems such as the maximization of the traded volume, or the minimization of opportunity costs of paradoxically rejected block bids. We first show on a toy example that these two objectives are distinct from maximizing welfare. Third, we provide numerical experiments on realistic large-scale instances. They illustrate the efficiency of the approach, as well as the economics trade-offs that may occur in practice.
    No preview · Article · Aug 2015 · EURO Journal on Computational Optimization
  • Mehdi Madani · Mathieu Van Vyve
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    ABSTRACT: We consider the optimization problem implementing current market rules for European day-ahead electricity markets. We propose improved algorithmic approaches for that problem. First, a new MIP formulation is presented which avoids the use of complementarity constraints to express market equilibrium conditions, and also avoids the introduction of auxiliary continuous or binary variables. Instead, we rely on strong duality theory for linear or convex quadratic optimization problems to recover equilibrium constraints. When so-called stepwise bid curves are considered to describe continuous bids, the new formulation allows to take full advantage of state-of-the-art MILP solvers, and in most cases, an optimal solution including market prices can be computed for large-scale instances without any further algorithmic work. Second, the new formulation suggests a Benders-like decomposition procedure. This helps in the case of piecewise linear bid curves that yield quadratic primal and dual objective functions leading to a dense quadratic constraint in the formulation. This procedure essentially strengthens classical Benders cuts locally. Computational experiments using 2011 historical instances for the Central Western Europe region show excellent results. In the linear case, both approaches are very efficient, while for quadratic instances, only the decomposition procedure is appropriate. Finally, when most orders are block orders, and instances are combinatorially very hard, the direct MILP approach is substantially more efficient.
    No preview · Article · Apr 2015 · European Journal of Operational Research
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    Mehdi Madani · Mathieu Van Vyve
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    ABSTRACT: It is well-known that a market equilibrium with uniform prices often does not exist in non-convex day-ahead electricity auctions. We consider the case of the new Pan-European day-ahead electricity market PCR (Price Coupling of Regions), with non-convexities arising from so-called complex and block orders. We propose a new primal-dual framework for these auctions, which has applications for both economic analysis and algorithm design. The contribution is threefold. First, extending previous results, we give a non-trivial exact (i.e. not approximate) linearisation of a non-convex minimum income condition that must hold for complex orders arising from the Spanish market, avoiding the introduction of any auxiliary variables. Second, we give the first and tractable MILP formulations of optimization problems such as the maximization of the traded volume, or the minimization of opportunity costs of paradoxically rejected orders. Third, numerical experiments are presented which show the tradeoffs that may occur in practice, as well as the efficiency of the approach.
    Full-text · Article · Oct 2014
  • Mehdi Madani · Mathieu Van Vyve
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    ABSTRACT: We provide with the first formulation of an optimization problem that aims at minimizing total opportunity costs of so-called paradoxically rejected block orders (PRB) in non-convex European day-ahead electricity markets with uniform prices such as the CWE market. This is done in a way avoiding complementarity constraints to model market rules, and without binary variables to linearize them, instead relying on strong duality results. This key property is of much importance from an algorithmic viewpoint. A toy example shows that a welfare maximizing solution (WMS) may not be a solution minimizing opportunity costs (OCMS). However, computations on real data kindly provided by main European power exchanges show that most of the time, the WMS is at least very close to the OCMS. The present work is the first providing with mathematical programming tools to assess the real tradeoff between welfare optimality and opportunity costs of PRB in such non-convex markets.
    No preview · Conference Paper · May 2014
  • Mathieu Van Vyve · Laurence A. Wolsey · Hande Yaman
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    ABSTRACT: We consider several variants of the two-level lot-sizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stock-dominant relaxation. We further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables. We consider several variants: uncapacitated at both levels with or without start-up costs, uncapacitated at the upper level and constant capacity at the lower level, constant capacity at both levels. We finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products.
    No preview · Article · Aug 2013 · Mathematical Programming
  • Sebastian Pokutta · Mathieu Van Vyve
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    ABSTRACT: We show that there are 0–1 and unbounded knapsack polytopes with super-polynomial extension complexity. More specifically, for each n∈Nn∈N we exhibit 0–1 and unbounded knapsack polytopes in dimension nn with extension complexity Ω(2n).
    No preview · Article · Jul 2013 · Operations Research Letters
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    Mathieu Van Vyve · Laurence A. Wolsey
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    ABSTRACT: For certain integer programs, one way to obtain a strong dual bound is to use an extended formulation and then solve the associated linear programming relaxation. The classical way to obtain a bound of the same value in the original variable space is through the use of Benders’ algorithm. Here, we propose an alternative approach based on a decomposition of the dual optimal solution of the extended formulation linear program. An example of the approach using the multi-commodity formulation of a two-level production/transportation problem is presented.
    Full-text · Article · Jan 2013 · EURO Journal on Computational Optimization
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    ABSTRACT: During the life period of Small and Medium Enterprises (SMEs) in incubators they need some training programs to acquire the required knowledge in order to survive and succeed in the business environment. This paper presents a heuristic method based on an optimization model to schedule these programs at the most suitable times. Based on the proposed heuristic, each training program is implemented in a suitable time by considering the SMEs’ requirements and some other logical constraints. The proposed heuristic is described in detail, and its implementation is demonstrated via a real-life numerical example. The numerical results of the heuristic are compared with other methods.
    Full-text · Article · Mar 2012 · European Journal of Operational Research
  • Mathieu Van Vyve
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    ABSTRACT: The fixed-charge transportation problem is a fixed-charge network flow problem on a bipartite graph. This problem appears as a subproblem in many hard transportation problems, and has also strong links with the challenging big-bucket multi-item lot-sizing problem. We provide a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path. We describe a new class of inequalities that we call “path-modular” inequalities. We give two distinct proofs of their validity. The first one is direct and crucially relies on sub- and super-modularity of an associated set function, thereby providing an interesting link with flow-cover type inequalities. The second proof is by projecting a tight extended formulation, therefore also showing that these inequalities suffice to describe the convex hull of the feasible solutions to this problem. We finally show how to solve the separation problem associated to the path-modular inequalities in O(n3)\mathcal{O}(n^3) time.
    No preview · Conference Paper · Jan 2011
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    Miguel Constantino · Andrew J. Miller · Mathieu Van Vyve
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    ABSTRACT: This paper is a polyhedral study of a generalization of the mixing set where two different, divisible coefficients are allowed for the integral variables. Our results generalize earlier work on mixed integer rounding, mixing, and extensions. They also directly apply to applications such as production planning problems involving lower bounds or start-ups on production, when these are modeled as mixed-integer linear programs. We define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed-integer set. We give a characterization of each of the maximal faces of the convex hull, as well as a closed form description of its extreme points and rays, and show how to separate over this set in O(n log n). Finally, we give several extended formulations of polynomial size, and study conditions under which adding certain simple constraints on the integer variables preserves our main result.
    Full-text · Article · Jun 2010 · Mathematical Programming
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    Mathieu VAN VYVE
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    ABSTRACT: The fixed-charge transportation problem is an interesting problem in its own right. This paper further motivates its study by showing that it is both a special case and a strong relaxation of the big-bucket multi-item lot-sizing problem. We then provide a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path. We give a O(n^3)-time optimization algorithm and two O(n^2)-size linear programming extended formulation. We describe a new class of inequalities that we call "path-modular" inequalities. We give two distinct proofs of their validity. The first one is direct and crucially relies on sub- and super- modularity of an associated set function. The second proof is by showing that the projection of one of the extended linear programming formulations onto the original variable space yields exactly the polyhedron described by the path- modular inequalities. Thus we also show that these inequalities suffice to describe the convex hull of the set of feasible solutions. We finally report on computational experiments comparing extended LP formulation, valid inequalities separation and a standard MIP solver.
    Full-text · Article · Jan 2010 · Mathematical Programming
  • Yves Pochet · Mathieu Van Vyve · Laurence A. Wolsey
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    ABSTRACT: Much progress has been made in recent years in solving certain classes of production planning problems using mixed integer programming. One of the major challenges is how to make this expertise available and relatively easy to use for the non-specialist and the practitioner. Here we describe a modeling approach and tool LS-LIB. LS-LIB is a library of primitives to declare procedures/subroutines/global constraints in a high-level modeling language that we believe offers an interesting partial answer to this challenge. LS-LIB provides routines for problem reformulation, cut generation, and heuristic solution of instances. The user must provide an initial formulation of his problem in the chosen modeling language MOSEL. Then using knowledge of the problem the user must first classify each product or sku according to a simple three field scheme: [production type, capacity type, variant]. Then it is a simple matter to use the global constraints of LS-LIB by adding a few lines to the initial modeling language formulation to get a tightened formulation and/or call the appropriate cut generation routines. The heuristic procedures are called in a similar fashion. The result is a tool that allows researchers and end-users to improve the solution time and quality of a variety of production planning problems within minutes. The library incorporates much of the modeling knowledge concerning lot-sizing problems derived over the last twenty years, and is also easy to maintain and extend. We illustrate the use of LS-LIB on an intractable two-level problem, and a difficult multi-level problem.
    No preview · Chapter · Nov 2008
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    Mathieu Van Vyve
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    ABSTRACT: The main result of this paper is an O(n3) algorithm for the single-item lot-sizing problem with constant batch size and backlogging. We consider a general number of installable batches, i.e., in each time period t we may produce up to mt batches, where the mt are given and time-dependent. This generalizes earlier results as we consider backlogging and a general number of maximum batches. We also give faster algorithms for three special cases of this general problem. When backlogging is not allowed and the costs satisfy the Wagner-Whitin property, the problem is solvable in O(n2 log n) time. When the production in each period is required to be either zero or equal to the installed capacity, it is possible to solve the problem with and without backlogging in O(n2) and O(n log n) time, respectively.
    Full-text · Article · Aug 2007 · Mathematics of Operations Research
  • Mathieu Van Vyve
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    ABSTRACT: Recently, several authors [8, 10] have argued for the use of extended formulations to tighten production planning models. In this work we present two linear-programming extended formulations of the constant-capacity lot-sizing problem with backlogging. The first one applies to the problem with a general cost function and has O(n3) variables and constraints. This improves on the more straightforward O(n4) Florian and Klein [2] type formulation. The second one applies when the costs satisfy the Wagner-Whitin property but it has O(n2) variables and O(n3) constraints. As a by-product, we positively answer an open question of Miller and Wolsey [4] about the tightness of an extended formulation for the continuous mixing set.
    No preview · Article · Aug 2006 · Mathematical Programming
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    Mathieu Van Vyve · Laurence A. Wolsey
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    ABSTRACT: Mixed integer programming (MIP) formulations are typically tightened through the use of a separation algorithm and the addition of violated cuts. Using extended formulations involving new variables is a possible alternative, but this often results in prohibitively large MIPs where even the linear programming relaxations are hard or impossible to solve. In this paper, we demonstrate how, in certain cases, it is possible and interesting to define ``approximate'' extended formulations. In all the examples considered, our description involves a single control parameter K. Large values of K result in strong but large formulations. In particular, when K takes its maximum value, the approximate formulation is identical to the complete extended formulation. Through this approximation parameter, the user has control over the tradeoff between the strength and the size of the formulation. Approximate extended formulations are proposed for a variety of lot-sizing problems and to the travelling salesman problem. We report computational results for several problems, including an industrial application and several small instances from the TSPLIB. The significant conclusion is that small values of the approximation parameter K are often sufficient to obtain excellent bounds.
    Full-text · Article · Feb 2006 · Mathematical Programming
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    Yves POCHET · Mathieu VAN VYVE · Laurence A. Wolsey
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    ABSTRACT: Much progress has been made in recent years in solving certain classes of production planning problems using mixed integer programming. One of the major challenges is how to make this expertise available and easy to use to the non-specialist and to the practitioners. Here we describe a modeling approach and tool LS-LIB, and report on computational results. LS-LIB is a library of primitives to declare procedures/subroutines/global constraints in a high-level modeling language that we believe offers an interesting partial answer to this challenge. LS-LIB provides routines for problem reformulation, cut generation, and heuristics to find good feasible solutions quickly. The user must provide an initial formulation of his problem in the modeling language MOSEL. Then using his knowledge of the problem he must first classify each product or sku according to a simple three field scheme: [production type, capacity type, variant] proposed recently. Then it is a simple matter to use the global constraints of LS-LIB by adding a few lines to his initial MOSEL formulation to get a tightened formulation and/or call the appropriate cut separation routines. The heuristic procedures are called in a similar fashion. We illustrate the use of LS-LIB on an intractable two-level problem, and a hard multi-level problem.
    Full-text · Article · Jul 2005 · SSRN Electronic Journal
  • Mathieu Van Vyve
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    ABSTRACT: We analyze the polyhedral structure of the sets P CMIX ={(s,r,z)∈ℝ×ℝ + n ×Z n ∣s+r j +z j ≥f j ,j=1,⋯,n} and P + CMIX =P CMIX ∩{s≥0}. The set P + CMIX is a natural generalization of the mixing set studied by Y. Pochet and L. A. Wolsey [Math. Oper. Res. 18, No. 4, 767–785 (1993; Zbl 0808.90058), Math. Program. 67, No. 3 (A), 297–323 (1994; Zbl 0822.90049)] and O. Günlük and Pochet [Math. Program. 90, No. 3 (A), 429–457 (2001; Zbl 1041.90033)] and recently has been introduced by A. J. Miller and L. A. Wolsey [Math. Program. 98, No. 1-3 (B), 73–88 (2003; Zbl 1047.90035)]. We introduce a new class of valid inequalities that has proven to be sufficient for describing conv(P CMIX ). We give an extended formulation of size O(n)×O(n 2 ) variables and constraints and indicate how to separate over conv(P CMIX ) in O(n 3 ) time. Finally, we show how the mixed integer rounding (MIR) inequalities of G. L. Nemhauser and L. A. Wolsey [Integer and combinatorial optimization. New York etc.: Wiley (1988; Zbl 0652.90067)] and the mixing inequalities of Günlük and Pochet (loc. cit.) constitute special cases of the cycle inequalities.
    No preview · Article · May 2005 · Mathematics of Operations Research
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    Mathieu Van Vyve
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    ABSTRACT: We survey the main results presented in the authors PhD Thesis presented in June 2003 at the Universit catholique de Louvain and supervised by Y. Pochet and L. A. Wolsey. The dissertation is written in English and is available from the author. In the first part of the thesis, we investigate the complexity and the polyhedral structure of various extensions of the uncapacitated single-item lot-sizing problem (Barany etal. 1984). In particular, we study models involving fixed charges on stocks, constant capacity and backlogging, and lower bounds on production. We describe algorithms, extended formulations, (facet-defining) valid inequalities and separation algorithms. Emphasis is placed on compact (i.e. of polynomial size) exact extended formulations. In a second part, we show how such extended reformulations for single-item problems can help to improve the solution of much more general production planning problems.
    Full-text · Article · Jan 2004 · 4OR quarterly journal of the Belgian, French and Italian Operations Research Societies
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    Mathieu VAN VYVE · Francisco ORTEGA
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    ABSTRACT: In this paper we examine a variant of the uncapacitated lot-sizing model of Wagner-Whitin that includes fixed charges on the stocks. Such a model is natural in a production environment where stocking is a complex operation. The problem can also be seen as a single source uncapacitated fixed charge network flow problem on a simple graph. Extended formulations, a dynamic program, the convex hull of integer solutions and a separation algorithm are presented. All these turn out to be very natural extensions of the corresponding results of Barany, Van Roy and Wolsey [?] for the uncapacitated lot-sizing problem. The convex hull proof is based on showing that an extended facility location formulation is tight and by pro jecting it onto the original space of variables.
    Full-text · Article · Jan 2003 · Discrete Optimization

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