ref_equilibrium.bib

@book{luoMathematicalProgramsEquilibrium1996,
  title = {Mathematical {{Programs}} with {{Equilibrium Constraints}}},
  author = {Luo, Zhi-Quan and Pang, Jong-Shi and Ralph, Daniel},
  year = {1996},
  publisher = {Cambridge University Press},
  address = {Cambridge},
  doi = {10.1017/CBO9780511983658},
  url = {https://www.cambridge.org/core/books/mathematical-programs-with-equilibrium-constraints/03981C32ABDD55A4001BF58BA0C57444},
  urldate = {2022-03-22},
  abstract = {This book provides a solid foundation and an extensive study for an important class of constrained optimization problems known as Mathematical Programs with Equilibrium Constraints (MPEC), which are extensions of bilevel optimization problems. The book begins with the description of many source problems arising from engineering and economics that are amenable to treatment by the MPEC methodology. Error bounds and parametric analysis are the main tools to establish a theory of exact penalisation, a set of MPEC constraint qualifications and the first-order and second-order optimality conditions. The book also describes several iterative algorithms such as a penalty-based interior point algorithm, an implicit programming algorithm and a piecewise sequential quadratic programming algorithm for MPECs. Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.},
  isbn = {978-0-521-57290-3}
}

@article{raghunathanMathematicalProgramsEquilibrium2003,
  title = {Mathematical Programs with Equilibrium Constraints ({{MPECs}}) in Process Engineering},
  author = {Raghunathan, Arvind U and Biegler, Lorenz T},
  year = {2003},
  month = oct,
  journal = {Computers \& Chemical Engineering},
  volume = {27},
  number = {10},
  pages = {1381--1392},
  issn = {0098-1354},
  doi = {10.1016/S0098-1354(03)00092-9},
  url = {https://www.sciencedirect.com/science/article/pii/S0098135403000929},
  urldate = {2022-03-22},
  abstract = {Mathematical programs with equilibrium constraints (MPECs) form a relatively new and interesting subclass of nonlinear programming problems. In this paper we propose a novel method of solving MPECs by appropriate reformulation of the equilibrium conditions. The reformulation can be easily incorporated in a certain class of interior point algorithms for nonlinear optimization. The algorithm used in the study follows a primal-dual interior point approach and shows encouraging results on a test suite of MPECs. The algorithm is also able to perform optimization of distillation columns with phase changes and tray optimization using only continuous variables. We also consider a number of topics to improve performance of the algorithm and to identify classes of process engineering problems that can be handled as MPECs.},
  langid = {english}
}

@phdthesis{allendeMathematicalProgramsEquilibrium2006,
  title = {Mathematical Programs with Equilibrium Constraints: Solution Techniques from Parametric Optimization},
  author = {Allende, Gamayqzel Bouza},
  year = {2006},
  address = {Enschede, NL},
  url = {https://ris.utwente.nl/ws/files/6042785/thesis_Allende.pdf},
  abstract = {Equilibrium constrained problems form a special class of mathematical programs where the decision variables satisfy a finite number of constraints together with an equilibrium condition. Optimization problems with a variational inequality constraint, bilevel problems and semi-infinite programs can be seen as particular cases of equilibrium constrained problems. Such models appear in many practical applications. Equilibrium constraint problems can be written in bilevel form with possi- bly a finite number of extra inequality constraints. This opens the way to solve these programs by applying the so-called Karush-Kuhn-Tucker approach. Here the lower level problem of the bilevel program is replaced by the Karush-Kuhn- Tucker condition, leading to a mathematical program with complementarity con- straints (MPCC). Unfortunately, MPCC problems cannot be solved by classical algorithms since they do not satisfy the standard regularity conditions. To solve MPCCs one has tried to conceive appropriate modifications of standard methods. For example sequential quadratic programming, penalty algorithms, regulariza- tion and smoothing approaches. The aim of this thesis is twofold. First, as a basis, MPCC problems will be investigated from a structural and generical viewpoint. We concentrate on a special parametric smoothing approach to solve these programs. The convergence behavior of this method is studied in detail. Although the smoothing approach is widely used, our results on existence of solutions and on the rate of convergence are new. We also derive (for the first time) genericity results for the set of minimizers (generalized critical points) for one-parametric MPCC. In a second part we will consider the MPCC problem obtained by applying the KKT-approach to equilibrium constrained programs and bilevel problems. We will analyze the generic structure of the resulting MPCC programs and adapt the related smoothing method to these particular cases. All corresponding results are new.},
  school = {Universiteit Twente}
}

@article{andreaniSolutionMathematicalProgramming2001,
  title = {On the Solution of Mathematical Programming Problems with Equilibrium Constraints},
  author = {Andreani, Roberto and Mart{\i}{\textasciiacute}nez, Jos{\'e} Mario},
  year = {2001},
  month = dec,
  journal = {Mathematical Methods of Operations Research},
  volume = {54},
  number = {3},
  pages = {345--358},
  issn = {1432-5217},
  doi = {10.1007/s001860100158},
  url = {https://doi.org/10.1007/s001860100158},
  urldate = {2022-03-22},
  abstract = {Mathematical programming problems with equilibrium constraints (MPEC) are nonlinear programming problems where the constraints have a form that is analogous to first-order optimality conditions of constrained optimization. We prove that, under reasonable sufficient conditions, stationary points of the sum of squares of the constraints are feasible points of the MPEC. In usual formulations of MPEC all the feasible points are nonregular in the sense that they do not satisfy the Mangasarian-Fromovitz constraint qualification of nonlinear programming. Therefore, all the feasible points satisfy the classical Fritz-John necessary optimality conditions. In principle, this can cause serious difficulties for nonlinear programming algorithms applied to MPEC. However, we show that most feasible points do not satisfy a recently introduced stronger optimality condition for nonlinear programming. This is the reason why, in general, nonlinear programming algorithms are successful when applied to MPEC.},
  langid = {english}
}

@unpublished{ramosMathematicalProgramsEquilibrium2017,
  title = {Mathematical {{Programs}} with {{Equilibrium Constraints}}: {{A}} Sequential Optimality Condition, New Constraint Qualifications and Algorithmic Consequences},
  author = {Ramos, A.},
  year = {2017},
  month = may,
  address = {Department of Mathematics, Federal University of Paran{\' }a, Curitiba, PR, Brazil.},
  url = {http://www.optimization-online.org/DB_FILE/2016/04/5423.pdf},
  abstract = {Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, are a dif- ficult class of constrained optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications (CQs). Thus, the Karush-Kuhn-Tucker (KKT) conditions are not necessarily satisfied by minimizers and the convergence assumptions of many methods for solving constrained optimization problems are not fulfilled. Therefore it is necessary, both from a theoretical and numerical point of view, to consider suitable optimality conditions, tailored CQs and specially designed al- gorithms for solving MPECs. In this paper, we present a new sequential optimality condition useful for the convergence analysis for several methods of solving MPECs, such as relaxations schemes, complementarity- penalty methods and interior-relaxation methods. We also introduce a variant of the augmented Lagrangian method for solving MPEC whose stopping criterion is based on this sequential condition and it has strong convergence properties. Furthermore, a new CQ for M-stationary which is weaker than the recently intro- duced MPEC relaxed constant positive linear dependence (MPEC-RCPLD) associated to such sequential condition is presented. Relations between the old and new CQs as well as the algorithmic consequences will be discussed.}
}