A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be solved in order to check feasibility of the upper level minimization problem. The current paper discusses several strategies to reformulate this class of problems into equivalent finite minimization problems by exploiting the concept of Wolfe duality for convex lower level problems. Here, the main contribution is the discussion of the non-degeneracy of the corresponding formulations under various assumptions. Finally, these non-degenerate reformulations of the original GSIP allow us to apply standard nonlinear optimization algorithms.

[1]  Hubertus Th. Jongen,et al.  General Semi-Infinite Programming: Symmetric Mangasarian-Fromovitz Constraint Qualification and the Closure of the Feasible Set , 2010, SIAM J. Optim..

[2]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[3]  Georg Still,et al.  Discretization in semi-infinite programming: the rate of convergence , 2001, Math. Program..

[4]  Arkadi Nemirovski,et al.  Robust Truss Topology Design via Semidefinite Programming , 1997, SIAM J. Optim..

[5]  E. Polak.  On the mathematical foundations of nondifferentiable optimization in engineering design , 1987 .

[6]  Alexander Shapiro,et al.  First-Order Optimality Conditions in Generalized Semi-Infinite Programming , 1999 .

[7]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[8]  Paul I. Barton,et al.  Interval Methods for Semi-Infinite Programs , 2005, Comput. Optim. Appl..

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

[10]  Oliver Stein.  First-Order Optimality Conditions for Degenerate Index Sets in Generalized Semi-Infinite Optimization , 2001, Math. Oper. Res..

[11]  Arkadi Nemirovski,et al.  Robust solutions of uncertain linear programs , 1999, Oper. Res. Lett..

[12]  Rainer Tichatschke,et al.  A Branch-and-Bound Approach for Solving a Class of Generalized Semi-infinite Programming Problems , 1998, J. Glob. Optim..

[13]  Oliver Stein,et al.  Bi-Level Strategies in Semi-Infinite Programming , 2003 .

[14]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[15]  Paul I. Barton,et al.  Relaxation-Based Bounds for Semi-Infinite Programs , 2008, SIAM J. Optim..

[16]  Hubertus Th. Jongen,et al.  On the closure of the feasible set in generalized semi-infinite programming , 2007, Central Eur. J. Oper. Res..

[17]  Paul I. Barton,et al.  Global solution of bilevel programs with a nonconvex inner program , 2008, J. Glob. Optim..

[18]  Oliver Stein,et al.  On Linear and Linearized Generalized Semi-Infinite Optimization Problems , 2001, Ann. Oper. Res..

[19]  Kenneth O. Kortanek,et al.  Semi-Infinite Programming: Theory, Methods, and Applications , 1993, SIAM Rev..

[20]  Gerhard-Wilhelm Weber,et al.  On generalized semi-infinite optimization of genetic networks , 2007 .

[21]  Yuval Rabani,et al.  Linear Programming , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[22]  Oliver Stein,et al.  Feasible Method for Generalized Semi-Infinite Programming , 2010 .

[23]  R. P. Hettich,et al.  Semi-infinite programming: Conditions of optimality and applications , 1978 .

[24]  G. Still,et al.  Second order optimality conditions for generalized semi-infinite programming problems , 1995 .

[25]  Rembert Reemtsen,et al.  Numerical Methods for Semi-Infinite Programming: A Survey , 1998 .

[26]  Oliver Stein,et al.  Solving Semi-Infinite Optimization Problems with Interior Point Techniques , 2003, SIAM J. Control. Optim..

[27]  Paul I. Barton,et al.  Global solution of semi-infinite programs , 2004 .

[28]  Oliver Stein,et al.  How to solve a semi-infinite optimization problem , 2012, Eur. J. Oper. Res..

[29]  P.A. Parrilo,et al.  Polynomial games and sum of squares optimization , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[30]  Marco A. López,et al.  Semi-infinite programming , 2007, Eur. J. Oper. Res..

[31]  Alexander Shapiro,et al.  Second-Order Optimality Conditions in Generalized Semi-Infinite Programming , 2001 .

[32]  Oliver Stein,et al.  Generalized semi-infinite programming: A tutorial , 2008 .

[33]  Hubertus Th. Jongen,et al.  Generalized semi-infinite optimization: A first order optimality condition and examples , 1998, Math. Program..

[34]  Georg J. Still,et al.  Generalized semi-infinite programming: numerical aspects , 2001 .

[35]  Johannes O. Royset,et al.  On the Use of Augmented Lagrangians in the Solution of Generalized Semi-Infinite Min-Max Problems , 2005, Comput. Optim. Appl..

[36]  C. Kanzow,et al.  A Fritz John Approach to First Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints , 2003 .

[37]  Laurent El Ghaoui,et al.  Robust Optimization , 2021, ICORES.

[38]  Oliver Stein,et al.  The Adaptive Convexification Algorithm: A Feasible Point Method for Semi-Infinite Programming , 2007, SIAM J. Optim..

[39]  L El Ghaoui,et al.  ROBUST SOLUTIONS TO LEAST-SQUARE PROBLEMS TO UNCERTAIN DATA MATRICES , 1997 .

[40]  Guerra VázquezF.,et al.  Generalized semi-infinite programming , 2008 .

[41]  Rainer Hettich,et al.  Numerische Methoden der Approximation und semi-infiniten Optimierung , 1982 .

[42]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[43]  Gerhard-Wilhelm Weber,et al.  Generalized semi-infinite optimization and related topics , 1999 .

[44]  Oliver Stein,et al.  The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets , 2012, Math. Program..