A global optimization algorithm for generalized semi-infinite, continuous minimax with coupled constraints and bi-level problems

We propose an algorithm for the global optimization of three problem classes: generalized semi-infinite, continuous coupled minimax and bi-level problems. We make no convexity assumptions. For each problem class, we construct an oracle that decides whether a given objective value is achievable or not. If a given value is achievable, the oracle returns a point with a value better than or equal to the target. A binary search is then performed until the global optimum is obtained with the desired accuracy. This is achieved by solving a series of appropriate finite minimax and min-max-min problems to global optimality. We use Laplace’s smoothing technique and a simulated annealing approach for the solution of these problems. We present computational examples for all three problem classes.

[1]  George E. Monahan Finding saddle points on polyhedra : Solving certain continuous minimax problems , 1993 .

[2]  Berç Rustem,et al.  Linearly Constrained Global Optimization and Stochastic Differential Equations , 2006, J. Glob. Optim..

[3]  Efstratios N. Pistikopoulos,et al.  Flexibility analysis and design of linear systems by parametric programming , 2000 .

[4]  Armen Der Kiureghian,et al.  On an Approach to Optimization Problems with a Probabilistic Cost and or Constraints , 2000 .

[5]  Berç Rustem,et al.  An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems , 2009 .

[6]  K. C. Kiwiel,et al.  A direct method of linearization for continuous minimax problems , 1987 .

[7]  Efstratios N. Pistikopoulos,et al.  A Decomposition-Based Global Optimization Approach for Solving Bilevel Linear and Quadratic Programs , 1996 .

[8]  Anton Winterfeld,et al.  Application of general semi-infinite programming to lapidary cutting problems , 2008, Eur. J. Oper. Res..

[9]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[10]  Yoshihiro Maruyama,et al.  Generalized Constrained Games in Farm Planning , 1972 .

[11]  Armen Der Kiureghian,et al.  Adaptive Approximations and Exact Penalization for the Solution of Generalized Semi-infinite Min-Max Problems , 2003, SIAM J. Optim..

[12]  Hitoshi Sasai,et al.  An Interior Penalty Method for Minimax Problems with Constraints , 1974 .

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

[14]  Georg Still,et al.  Generalized semi-infinite programming: Theory and methods , 1999, Eur. J. Oper. Res..

[15]  Michael I. Jordan,et al.  A Robust Minimax Approach to Classification , 2003, J. Mach. Learn. Res..

[16]  H. Luthi Algorithms for Worst-Case Design and Applications to Risk Management:Algorithms for Worst-Case Design and Applications to Risk Management , 2002 .

[17]  J. E. Falk,et al.  Infinitely constrained optimization problems , 1976 .

[18]  B. Rustem,et al.  Convergence of an Interior Point Algorithm for Continuous Minimax , 2008 .

[19]  Johannes O. Royset,et al.  Algorithms with Adaptive Smoothing for Finite Minimax Problems , 2003 .

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

[21]  E. Aiyoshi,et al.  Necessary conditions for min-max problems and algorithms by a relaxation procedure , 1980 .

[22]  Berç Rustem,et al.  Semi-Infinite Programming and Applications to Minimax Problems , 2003, Ann. Oper. Res..

[23]  Efstratios N. Pistikopoulos,et al.  Global Optimization Issues in Multiparametric Continuous and Mixed-Integer Optimization Problems , 2004, J. Glob. Optim..

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

[25]  Berç Rustem,et al.  An interior point algorithm for continuous minimax: implementation and computation , 2008, Optim. Methods Softw..

[26]  Berç Rustem,et al.  Parametric global optimisation for bilevel programming , 2007, J. Glob. Optim..

[27]  Oliver Stein,et al.  On generalized semi-infinite optimization and bilevel optimization , 2002, Eur. J. Oper. Res..

[28]  Johannes O. Royset,et al.  Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques , 2003 .

[29]  Alexander Kaplan,et al.  On a class of terminal variational problems , 1995 .