Generalized Nash Equilibrium Problems, Bilevel Programming and MPEC (eBook)

The book discusses three classes of problems: the generalized Nash equilibrium problems, the bilevel problems and the mathematical programming with equilibrium constraints  (MPEC). These problems interact through their mathematical analysis as well as their applications. The primary aim of the book is to present the modern tool of variational analysis and optimization, which are used to analyze these three classes of problems. All contributing authors are respected academicians, scientists and researchers from around the globe. These contributions are based on the lectures delivered by experts at CIMPA School, held at the University of Delhi, India, from 25 November-6 December 2013, and peer-reviewed by international experts.

The book contains five chapters. Chapter 1 deals with nonsmooth, nonconvex bilevel optimization problems whose feasible set is described by using the graph of the solution set mapping of a parametric optimization problem. Chapter 2 describes a constraint qualification to MPECs considered as an application of calmness concept of multifunctions and is used to derive M-stationarity conditions for MPEC. Chapter 3 discusses the first- and second-order optimality conditions derived for a special case of a bilevel optimization problem in which the constraint set of the lower level problem is described as a general compact convex set. Chapter 4 concentrates the results of the modelization and analysis of deregulated electricity markets with a focus on auctions and mechanism design. Chapter 5 focuses on optimization approaches called reflection methods for protein conformation determination within the framework of matrix completion. The last chapter (Chap. 6) deals with the single-valuedness of quasimonotone maps by using the concept of single-directionality with a special focus on the case of the normal operator of lower semi-continuous quasiconvex functions.

D. AUSSEL is a Professor at the Department of Mathematics and Computer Science, University of Perpignan, France. He is an expert on the theoretical aspects of quasi-convex optimization and variational inequalities. In addition, his research also involves applications of optimization in engineering processes and mathematical economics, in particular for the modeling of electricity markets, a topic where Nash equilibrium and multi-leader-follower games play a central role. He has published over 50 research articles in several prominent mathematics journals such as Transactions of the American Mathematical Society, SIAM Journal on Control and Optimization, Journal of Optimization Theory and Applications, as well as physics journals including Energy Conversion and Management. He is an associate editor of the journal Optimization and has served nearly a decade as the Co-Director and subsequently as Director of the French CNRS Research Group on Mathematics of Optimization and Applications. He has supervised several Ph.D. students, mainly on topics concerning nonsmooth variational analysis and electricity markets. Deeply interested in conveying research knowledge to young generations, he has been actively involved in the organization of research schools and research courses all over the world, including countries such as Vietnam, India, Chile, Peru, Cuba, Taiwan and Saudi Arabia.

C.S. LALITHA is a Professor at the Department of Mathematics, University of Delhi, South Campus, New Delhi, India. Her areas of interest include optimization theory, nonsmooth analysis and variational inequalities. She has co-authored more than 50 research papers published in prominent journals such as Journal of Optimization Theory and Applications, Optimization, Optimization Letters, Journal of Global Optimization, and Journal of Mathematical Analysis and Applications. She has also co-authored a book entitled Generalized Convexity, Nonsmooth Inequalities and Nonsmooth Optimization and has co-edited a book Combinatorial Optimization: Some Aspects, published by Narosa. She is a recipient of the INSA Teacher Award 2016. She has supervised many MPhil and Ph.D. students at the University of Delhi and is a member of various learned scientific societies such as the American Mathematical Society, the Operational Research Society of India, the Indian Mathematical Society and the Ramanujan Mathematical Society. She has organized many training programs, seminars and conferences. In addition, she has presented papers and delivered talks at several national and international conferences and workshops.

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The book discusses three classes of problems: the generalized Nash equilibrium problems, the bilevel problems and the mathematical programming with equilibrium constraints  (MPEC). These problems interact through their mathematical analysis as well as their applications. The primary aim of the book is to present the modern tool of variational analysis and optimization, which are used to analyze these three classes of problems. All contributing authors are respected academicians, scientists and researchers from around the globe. These contributions are based on the lectures delivered by experts at CIMPA School, held at the University of Delhi, India, from 25 November-6 December 2013, and peer-reviewed by international experts. The book contains five chapters. Chapter 1 deals with nonsmooth, nonconvex bilevel optimization problems whose feasible set is described by using the graph of the solution set mapping of a parametric optimization problem. Chapter 2 describes a constraint qualification to MPECs considered as an application of calmness concept of multifunctions and is used to derive M-stationarity conditions for MPEC. Chapter 3 discusses the first- and second-order optimality conditions derived for a special case of a bilevel optimization problem in which the constraint set of the lower level problem is described as a general compact convex set. Chapter 4 concentrates the results of the modelization and analysis of deregulated electricity markets with a focus on auctions and mechanism design. Chapter 5 focuses on optimization approaches called reflection methods for protein conformation determination within the framework of matrix completion. The last chapter (Chap. 6) deals with the single-valuedness of quasimonotone maps by using the concept of single-directionality with a special focus on the case of the normal operator of lower semi-continuous quasiconvex functions.