Semidefinite relaxation for linear programs with equilibrium constraints

Fampa, Marcia; Melo, Wendel; Maculan, Nelson

doi:10.1111/j.1475-3995.2012.00869.x

Cited by 6 publications

(3 citation statements)

References 17 publications

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“…As an option, we can smoothen the valuation function by using a smooth approximation function. The model ( 28) is an MPEC, and many efficient numerical algorithms have been proposed for solving MPEC (see, e.g., Facchinei et al, 1999;Fampa et al 2013;Fletcher et al, 2006;Gabriel et al, 2010;Hoheisel et al, 2012Hoheisel et al, , 2013Kim et al, 2020;Luo et al, 1996;Scholtes, 2001). Dempe and Dutta (2012) study the equivalence relationship between the bi-level program and its MPEC reformulation at global/local optimal solutions.…”

Section: Solutions To Focus Programming Models Under the Negative Eva...mentioning

confidence: 99%

Focus programming: a bi‐level programming approach to static stochastic optimization problems

Guo

Zhu

2023

Int Trans Operational Res

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Static stochastic optimization problems are formulated with the focus theory of choice where the optimal solution is determined as per which solution's focus (the most salient realization of a random vector) is the most preferred. The new formulation that we call the focus programming is a bi‐level programming approach in which the lower‐level program is used to choose the focus of each feasible solution and the upper‐level program is to determine the optimal solution. Since in focus programming models upper‐level and lower‐level programs are maximin or minimax problems, they are nonsmooth and sometimes even nonconvex so that the existing optimization methods cannot solve such bi‐level programming problems. We propose several single‐level reformulation methods for such problems.

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Section: Solutions To Focus Programming Models Under the Negative Eva...mentioning

confidence: 99%

Focus programming: a bi‐level programming approach to static stochastic optimization problems

Guo

Zhu

2023

Int Trans Operational Res

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“…The research in this field is still active as shown, for example, in the recent works of Anstreicher (), Burer and Vandenbussche (), Fampa et al. (), Rendl et al. (), Saxena et al.…”

Section: Introductionmentioning

confidence: 99%

Linear programing relaxations for a strategic pricing problem in electricity markets

Fampa

Pimentel

2016

Int Trans Operational Res

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Strategic bidding problems in energy markets have been vastly studied in the literature due to their economic importance and mathematical difficulty. In this paper, we consider this problem under the natural uncertainty associated to the competitors' bids and the system demand, and formulate it as a nonconvex quadratically constrained quadratic program. In order to solve this nonconvex maximization problem or to evaluate the quality of heuristic solutions, it is important to know good upper bounds to its optimal value. With this goal, we apply a cutting plane algorithm (CPA) to solve an extended linear relaxation of the problem, which is an outer approximation of a semidefinite relaxation. Valid inequalities are added to an initial linear relaxation at each iteration of the CPA improving the bound computed, including inequalities derived from reformulation linearization techniques (RLTs) and semidefinite programing (SDP) cuts that enforce the positive semidefiniteness of the matrix variable. SDP cuts are well known to considerably delay the solution of the relaxation because of their density, and therefore, we apply a sparsification procedure for the cuts, introduced by Margot et al. in 2012. We propose a modification to the algorithm in Margot et al. to better control the density of the SDP cuts; and after tuning the parameters of our algorithm, we show the trade-off between the bounds and the density of the cuts for test instances derived from the Brazilian electric system.

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“…Other strategies to solve bilevel problems include: bundle type algorithms [89], semi-definite relaxations [90], penalty function based methods [91,92,93], Benders decomposition [94]. Cutting-plane approaches [95,96,97] have received attention recently because they can handle lower-level problems with integer variables -solvers were developed by [98,99].…”

Section: Solving Bilevel Optimizationmentioning

confidence: 99%

The Effectiveness of Bilevel Optimization in Large-Scale Power Systems Problems: A Bilevel Optimization Toolbox, a Framework for Application-Driven Learning, and a Market Simulator

GARCIA¹

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First and foremost, I would like to thank Beatriz for all the companionship and support of my friend, partner, girlfriend, and wife, Beatriz, who helped me with absolutely everything during my doctoral years. Your support, affection, attention, help, and lots of love made this journey much more peaceful and pleasant.I would like to thank my family for their constant support, motivation, excitement, interest, and much love. My parents, Patricia and Joaquim, gave me every opportunity and always encouraged me. My grandparents are all examples that contributed a lot to my path. Helena always motivates my curiosity and willingness to explore everything. Maria Olivia always brings the whole family together to give more strength to everyone. Carlos José, the most creative person I know always excited about my academic developments. Luiz Antônio, is an incredible example of kindness and goodness, unfortunately, he could not see the end of this journey here on earth for a few months, but he has certainly seen it from wherever he is. My brother João Pedro who was always by my side at all times, always ready to do anything for me. Luiz Gabriel and Rafael who are my hope for the future of this world. Maria Teresa and Steve who have been so supportive over the years. My cousins Maria, Luiza and Carol for their company and support.Thanks to my dear friends who have always been by my side on this journey. Especially when I needed to work on this thesis.I would like to thank my advisor and great friend Alexandre Street. Thank you for your company and guidance on this long journey that began as a scientific initiation project in the electrical engineering graduation. Since then, we've worked hard, and I've learned how to write, present, research, converse, and much more. He was a sensational guide on this trip, and I could not have chosen someone better to develop this work with.I thank Mario Veiga, my advisor, supervisor, friend, and one of my heroes. Mario shows every day how to do top-quality science. Since my first day of internship at PSR, Mario has taught me to be a better researcher, developer, professional, and person. Our daily discussions motivated the works of this thesis, among many others. Thank you for introducing me to JuMP, which was essential throughout the thesis.Thanks to my incredible colleagues at PUC and LAMPS. Special thanks to

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Semidefinite relaxation for linear programs with equilibrium constraints

Cited by 6 publications

References 17 publications

Focus programming: a bi‐level programming approach to static stochastic optimization problems

Focus programming: a bi‐level programming approach to static stochastic optimization problems

Linear programing relaxations for a strategic pricing problem in electricity markets

The Effectiveness of Bilevel Optimization in Large-Scale Power Systems Problems: A Bilevel Optimization Toolbox, a Framework for Application-Driven Learning, and a Market Simulator

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