The penalty interior-point method fails to converge

Equilibrium equations in the form of complementarity conditions often appear as constraints in optimization problems. Problems of this type are commonly referred to as mathematical programs with complementarity constraints (MPCCs). A popular method for solving MPCCs is the penalty interior-point algorithm (PIPA). This paper presents an example for which PIPA converges to a nonstationary point, providing a counterexample to the established theory. The reasons for this adverse behavior are discussed. †Preprint ANL/MCS-P1091-0903.

[1]  L. N. Vicente,et al.  Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems , 1998 .

[2]  Jonathan F. BARD,et al.  Convex two-level optimization , 1988, Math. Program..

[3]  Masao Fukushima,et al.  Some Feasibility Issues in Mathematical Programs with Equilibrium Constraints , 1998, SIAM J. Optim..

[4]  M. Anitescu.  On Solving Mathematical Programs With Complementarity Constraints As Nonlinear Programs , 2002 .

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

[6]  S. Dirkse,et al.  Frontiers in Applied General Equilibrium Modeling: Mathematical Programs with Equilibrium Constraints: Automatic Reformulation and Solution via Constrained Optimization , 2002 .

[7]  Hande Y. Benson,et al.  INTERIOR-POINT METHODS FOR NONCONVEX NONLINEAR PROGRAMMING: COMPLEMENTARITY CONSTRAINTS , 2002 .

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

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

[10]  Carey E. Priebe,et al.  A likelihood-MPEC approach to target classification , 2003, Math. Program..

[11]  Steven P. Dirkse,et al.  Mathematical Programs with Equilibrium Constraints : Automatic Reformulation and Solution via Constrained Optimization ∗ , 2002 .

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

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

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

[15]  R. Fletcher,et al.  Numerical experience with solving MPECs as NLPs , 2002 .

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

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

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

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

[20]  W. Murray,et al.  Algorithms for mathematical programs with equilibrium constraints with applications to deregulated electricity markets , 2001 .