Semismooth SQP method for equality-constrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints

We consider the sequential quadratic programming (SQP) algorithm applied to equality-constrained optimization problems, where the problem data is differentiable with Lipschitz-continuous first derivatives. For this setting, Dennis–Moré-type analysis of primal superlinear convergence is presented. Our main motivation is a special modification of SQP tailored to the structure of the lifted reformulation of mathematical programs with complementarity constraints (MPCC). For this problem, we propose a special positive definite modification of the matrices in the generalized Hessian, which is suitable for globalization of SQP based on the penalty function, and at the same time can be expected to satisfy our general Dennis–Moré-type conditions, thus preserving local superlinear convergence. (Standard quasi-Newton updates in the SQP framework require twice differentiability of the problem data at the solution for superlinear convergence.) Preliminary numerical results comparing a number of quasi-Newton versions of semismooth SQP applied to MPCC are also reported.