Solving mathematical programs with complementarity constraints as nonlinear programs

We investigate the possibility of solving mathematical programs with complemen tarity constraints MPCCs using algorithms and procedures of smooth nonlinear programming Although MPCCs do not satisfy a constraint quali cation we establish su cient conditions for their Lagrange multiplier set to be nonempty MPCCs that have nonempty Lagrange multiplier sets and that satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter In this context the elastic mode transforms MPCC into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem which in turn makes it approachable by sequential quadratic programming algorithms We also prove that a modi ed version of the elastic mode exhibits global convergence to C stationary points when applied to the optimization of parametric mixed P variational inequalities The robustness of the elastic mode when applied to MPCCs is demonstrated by several numerical examples Introduction Complementarity constraints can be used to model numerous economics or engineering applications Solving optimization problems with complementarity constraints may prove di cult for classical nonlinear optimization however given that at a solution x such problems cannot satisfy a constraint quali cation As a result algorithms based on the linearization of the feasible set such as sequential quadratic programming SQP algorithms may fail because feasibility of the linearization can no longer be guaranteed in a neighborhood of the solution Several methods have been recently proposed to accommodate such problems For example a nondi erentiable penalty term in the objective function can be used to replace the complementarity constraints while maintaining the same solution set Although the new problem may now satisfy the constraint quali cation the non di erentiability of the objective function is an obstacle to the e cient computation of an optimal point Another method is the disjunctive nonlinear programming dis junctive NLP approach though this may lead to a large number of subcases to account for the alternatives involving degenerate complementarity constraints If all constraint functions with the exception of the complementarity constraints are linear then e cient active set approaches can be de ned if the linear independence constraint quali cation holds Still other approaches have been de ned for prob lems whose complementarity constraints originate in equilibrium conditions A nonsmooth approach has been proposed in for MPCCs in which the un derlying complementarity constraints originate in a variational inequality with strong regularity properties A bundle trust region algorithm is de ned in which each ele ment of the bundle is generated from the generalized gradient of the reduced objective Thackeray Department of Mathematics University of Pittsburgh Pittsburgh PA anitescu math pitt edu This work was supported by the Mathematical Information and Com putational Sciences Division subprogram of the O ce of Advanced Scienti c Computing U S De partment of Energy under Contract W Eng This work was also supported by award DMS of the National Science Foundation function The key step is to produce an element of the generalized gradient Equa tions which may be quite costly for general cases at points where there are a substantial number of degenerate complementarity constraints In this work we investigate the possibility of solving MPCCs by applying cer tain SQP algorithms to their nonlinear programming formulation This endeavor is important because it allows one to extend the considerable body of analytical and computational expertise of smooth nonlinear programming to this new class of prob lems The advantage of such an approach over disjunctive programming for example is that it considers simultaneously all the alternatives involving degenerate comple mentarity constraints The disadvantage is that the description of the constraint set is considerably less well behaved Recognizing that the potential infeasibility of the subproblems with linearized constraints may prevent normal termination of SQP algorithms we discuss their use in conjunction with the elastic mode The elastic mode is a standard technique of approaching infeasible subproblems by relaxing the constraints and introducing a dif ferentiable penalty term in the objective function To show that such an approach can accommodate a large class of MPCCs we use the framework from to determine su cient conditions for MPCCs to have nonempty Lagrange multiplier sets As in the rst and second order optimality properties of an MPCC are compared with the similar properties of two nonlinear programs that involve no com plementarity constraints and may thus satisfy a constraint quali cation Here how ever we consider the optimality properties of an MPCC formulated as a nonlinear program with di erentiable data In MPCC is equivalently described with the complementarity constraints replaced by an equality involving the nondi erentiable function minfx x g The two formulations will ultimately have similar properties but the smooth description is important in anticipation of the use of a standard non linear programming algorithm to solve MPCCs The elastic mode approach we present here is di erent from other nonlinear pro gramming approaches for MPCC in the following important respect Virtually all smooth nonlinear programming approaches currently described in the literature for nding a solution x of MPCC consist of transforming it into another nonlinear pro gram depending on a parameter p MPCC p and then nding the solution x of the modi ed problem The problem MPCC p will have enough constraint regularity for x to be found reasonably e ciently The solution x is then obtained in the limit as p and x x for any p The program MPCC is unde ned or does not satisfy a constraint quali cation if the parameter is a penalty parameter c the same observation is valid by choosing p c For the elastic mode under conditions to be speci ed in the body of this work MPCC is transformed into a problem MPCC c that satis es a constraint quali ca tion and has x as a local solution for all c su ciently large but nite So MPCC is transformed by a nite procedure in a nonlinear program with the same solution that satis es a constraint quali cation which does not happen for the other approaches To our knowledge the developments presented here are the rst systematic approach of this type that is valid for a generic instance of mathematical programs with com plementarity constrains The paper is structured as follows In the remainder of Section we review the relevant nonlinear programming concepts In Section we discuss su cient condi tions for MPCC to have a nonempty Lagrange multiplier set in spite of not satisfying a constraint quali cation at any point This allows us to argue in Section that the elastic mode applied to an instance of the MPCC class will retrieve a local so lution of the problem for a nite value of the penalty parameter a point which is supported by several numerical examples In Section we show that an adaptive elastic mode approach can be guaranteed to retrieve a feasible C stationary point of an optimization problem whose complementarity constraints originate in a mixed P variational inequality To achieve this global convergence result we will allow the penalty parameter to grow to if necessary Optimality Conditions for General Nonlinear Programming We review the optimality conditions for a general nonlinear program min x f x subject to g x h x Here g Rn Rm h Rn Rr We assume that f g and h are twice continuously di erentiable In this work we will denote quantities connected to nonlinear programs such as by the superscript since f g and h will later denote the objective value and constraints of MPCC We call x a stationary point of if the Fritz John condition holds There exist multipliers m r Rm r such that rxL x h x i gi x for i m m X