An SOS1-Based Approach for Solving MPECs with a Natural Gas Market Application

This paper presents a new method for solving mathematical programs with equilibrium constraints. The approach uses a transformation of the original problem via Schur’s decomposition coupled with two separate formulations for modeling related absolute value functions. The first formulation, based on SOS1 variables, when solved to optimality will provide a global solution to the MPEC. The second, penalty-based formulation is used to heuristically obtain local solutions to large-scale MPECs. The advantage of these methods over disjunctive constraints for solving MPECs is that computational time is much lower, which is corroborated by numerical examples. Finally, an application of the method to an MPEC representing the United States natural gas market is given.

[1]  E. M. L. Beale.  Some Uses of Mathematical Programming Systems to Solve Problems that are not Linear , 1975 .

[2]  José Fortuny-Amat,et al.  A Representation and Economic Interpretation of a Two-Level Programming Problem , 1981 .

[3]  R. Gibbons.  Game theory for applied economists , 1992 .

[4]  J. Renegar.  Some perturbation theory for linear programming , 1994, Math. Program..

[5]  Knut Einar Rosendahl,et al.  Effects of Liberalizing the Natural Gas Markets in Western Europe* , 1995 .

[6]  Jane J. Ye,et al.  Optimality conditions for bilevel programming problems , 1995 .

[7]  James Renegar,et al.  Incorporating Condition Measures into the Complexity Theory of Linear Programming , 1995, SIAM J. Optim..

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

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

[10]  Christodoulos A. Floudas,et al.  Global Optimization of Nonlinear Bilevel Programming Problems , 2001, J. Glob. Optim..

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

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

[13]  Benjamin F. Hobbs,et al.  Leader-Follower Equilibria for Electric Power and NOx Allowances Markets , 2006, Comput. Manag. Sci..

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

[15]  Antonio J. Conejo,et al.  A practical approach to approximate bilinear functions in mathematical programming problems by using Schur's decomposition and SOS type 2 variables , 2006, J. Oper. Res. Soc..

[16]  Katta G. Murty,et al.  Nonlinear Programming Theory and Algorithms , 2007, Technometrics.

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

[18]  Jing Hu,et al.  On the Global Solution of Linear Programs with Linear Complementarity Constraints , 2008, SIAM J. Optim..

[19]  Richard W. Cottle,et al.  Linear Complementarity Problem , 2009, Encyclopedia of Optimization.

[20]  A. Uderzo,et al.  Exact penalty functions and calmness for mathematical programming under nonlinear perturbations , 2010 .

[21]  M. Ulbrich,et al.  A New Relaxation Scheme for Mathematical Programs with Equilibrium Constraints , 2010, SIAM J. Optim..

[22]  Alexander Mitsos,et al.  Global solution of nonlinear mixed-integer bilevel programs , 2010, J. Glob. Optim..

[23]  Steven A. Gabriel,et al.  Solving discretely-constrained MPEC problems with applications in electric power markets , 2010 .

[24]  Steven A. Gabriel,et al.  Cartelization in gas markets: Studying the potential for a “Gas OPEC” , 2012 .