A New Relaxation Scheme for Mathematical Programs with Equilibrium Constraints: Theory and Numerical Experience

In this thesis we are concerned with a specific solution approach for Mathematical Programs with Complementarity Constraints. The approach we suggest is based on the successive solution of a sequence of newly relaxed programs. The new relaxation scheme we propose differs from existing ones in the fact that it combines an exact and a relaxed reformulation of the complementarity conditions. A positive parameter determines to what extend the complementarity conditions are relaxed. In this thesis we not only study the various properties of the programs that are relaxed in this way but we also develop convergence results concerning a corresponding sequence of solutions. Moreover, we consider the new relaxation scheme in connection with Sequential Quadratic Programming and Interior Point methods. Finally we give a detailed report and an associated analysis of the numerical results that we obtained for the new solution approach.

[1]  Xing-De Mo,et al.  A Strongly and Superlinearly Convergent SQP Algorithm for Optimization Problems with Linear Complementarity Constraints , 2006 .

[2]  Georg Still,et al.  Mathematical Programs with Complementarity Constraints: Convergence Properties of a Smoothing Method , 2007, Math. Oper. Res..

[3]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[4]  Michael P. Friedlander,et al.  A two-sided relaxation scheme for Mathematical Programs with Equilibrium Constraints , 2005, SIAM J. Optim..

[5]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[6]  Walter Schlee.  Einführung in die Spieltheorie , 2004 .

[7]  R. Janin.  Directional derivative of the marginal function in nonlinear programming , 1984 .

[8]  Stephan Dempe,et al.  Optimization with Multivalued Mappings , 2006 .

[9]  Stephan Dempe,et al.  Foundations of Bilevel Programming , 2002 .

[10]  M. Fukushima,et al.  Smoothing methods for mathematical programs with equilibrium constraints , 2004, International Conference on Informatics Research for Development of Knowledge Society Infrastructure, 2004. ICKS 2004..

[11]  Lorenz T. Biegler,et al.  An Interior Point Method for Mathematical Programs with Complementarity Constraints (MPCCs) , 2005, SIAM J. Optim..

[12]  D. Ralph,et al.  Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints , 2004 .

[13]  Mihai Anitescu,et al.  On Using the Elastic Mode in Nonlinear Programming Approaches to Mathematical Programs with Complementarity Constraints , 2005, SIAM J. Optim..

[14]  Stefan Scholtes,et al.  Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity , 2000, Math. Oper. Res..

[15]  Manfred J. Holler,et al.  Einführung in die Spieltheorie , 1993 .

[16]  S. Scholtes,et al.  Exact Penalization of Mathematical Programs with Equilibrium Constraints , 1999 .

[17]  S. Leyffer.  Complementarity constraints as nonlinear equations: Theory and numerical experience , 2006 .

[18]  Christian Kanzow,et al.  Theorie und Numerik restringierter Optimierungsaufgaben , 2002 .

[19]  Stefan Scholtes,et al.  Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints , 2000, SIAM J. Optim..

[20]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[21]  Peter Spellucci,et al.  An SQP method for general nonlinear programs using only equality constrained subproblems , 1998, Math. Program..

[22]  Yoshihiro Kanno,et al.  Arc-Length Method for Frictional Contact Problems Using Mathematical Programming with Complementarity Constraints , 2006 .

[23]  Zhi-Quan Luo,et al.  Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints , 1996, Math. Program..

[24]  Mihai Anitescu,et al.  Global Convergence of an Elastic Mode Approach for a Class of Mathematical Programs with Complementarity Constraints , 2005, SIAM J. Optim..

[25]  Jane J. Ye,et al.  Optimality Conditions for Optimization Problems with Complementarity Constraints , 1999, SIAM J. Optim..

[26]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[27]  Francisco Facchinei,et al.  A smoothing method for mathematical programs with equilibrium constraints , 1999, Math. Program..

[28]  Jane J. Ye,et al.  Merit-Function Piecewise SQP Algorithm for Mathematical Programs with Equilibrium Constraints , 2007 .

[29]  Christian Kanzow,et al.  On M-stationary points for mathematical programs with equilibrium constraints , 2005 .

[30]  Daniel Ralph,et al.  QPECgen, a MATLAB Generator for Mathematical Programs with Quadratic Objectives and Affine Variational Inequality Constraints , 1999, Comput. Optim. Appl..

[31]  Patrice Marcotte,et al.  An overview of bilevel optimization , 2007, Ann. Oper. Res..

[32]  Michael C. Ferris,et al.  Expressing Complementarity Problems in an Algebraic Modeling Language and Communicating Them to Solvers , 1999, SIAM J. Optim..

[33]  S. K. Mishra,et al.  Nonconvex Optimization and Its Applications , 2008 .

[34]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[35]  Gui-Hua Lin,et al.  A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints , 2002, Ann. Oper. Res..

[36]  Jie Sun,et al.  Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints , 2004, Math. Program..

[37]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[38]  Sven Leyffer,et al.  Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints , 2006, SIAM J. Optim..

[39]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[40]  Jiří V. Outrata,et al.  Optimality Conditions for Disjunctive Programs with Application to Mathematical Programs with Equilibrium Constraints , 2007 .

[41]  Hiroshi Yamashita,et al.  Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization , 1996, Math. Program..

[42]  Christian Kanzow,et al.  Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints , 2005 .

[43]  S. Y. Wang,et al.  A globally convergent approximately active search algorithm for solving mathematical programs with linear complementarity constraints , 2004, Numerische Mathematik.

[44]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[45]  P. Toint,et al.  A brief history of filter methods. , 2007 .

[46]  Stephen J. Wright,et al.  Some properties of regularization and penalization schemes for MPECs , 2004, Optim. Methods Softw..

[47]  Joseph N. Prashker,et al.  The applicability of non-cooperative game theory in transport analysis , 2006 .

[48]  M. Padberg.  Linear Optimization and Extensions , 1995 .

[49]  Michael C. Ferris,et al.  Engineering and Economic Applications of Complementarity Problems , 1997, SIAM Rev..

[50]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[51]  Sven Leyffer,et al.  Solving mathematical programs with complementarity constraints as nonlinear programs , 2004, Optim. Methods Softw..

[52]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[53]  Jane J. Ye,et al.  Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints , 2005 .

[54]  Stephen J. Wright,et al.  Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties , 2007, Math. Program..

[55]  Florian Jarre,et al.  Optimal Truss Design by Interior-Point Methods , 1998, SIAM J. Optim..

[56]  J. Pang,et al.  Strategic gaming analysis for electric power systems: an MPEC approach , 2000 .

[57]  Gui-Hua Lin,et al.  Some Exact Penalty Results for Nonlinear Programs and Mathematical Programs with Equilibrium Constraints , 2003 .

[58]  P. Venkataraman,et al.  Applied Optimization with MATLAB Programming , 2001 .

[59]  M. Ferris,et al.  On the solution of a minimum weight elastoplastic problem involving displacement and complementarity constraints , 1999 .

[60]  Masao Fukushima,et al.  A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints , 1998, Comput. Optim. Appl..

[61]  Donald W. Hearn,et al.  An MPEC approach to second-best toll pricing , 2004, Math. Program..

[63]  Daniel Ralph,et al.  Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints , 1999, SIAM J. Optim..

[64]  罗志军,et al.  A new smoothing technique for mathematical programs with equilibrium constraints , 2007 .

[65]  Masao Fukushima,et al.  Complementarity Constraint Qualifications and Simplified B-Stationarity Conditions for Mathematical Programs with Equilibrium Constraints , 1999, Comput. Optim. Appl..

[66]  Jorge Nocedal,et al.  Interior Methods for Mathematical Programs with Complementarity Constraints , 2006, SIAM J. Optim..