An Active-Set Newton Method for Mathematical Programs with Complementarity Constraints

Abstract

Abstract. For a mathematical program with complementarity constraints (MPCC), we propose an active-set Newton method, which has the property of local quadratic convergence under the MPCC linear independence constraint qualification (MPCC-LICQ) and the standard second-order sufficient condition (SOSC) for optimality. Under MPCC-LICQ, this SOSC is equivalent to the piecewise SOSC on branches of MPCC, which is weaker than the special MPCC-SOSC often employed in the literature. The piecewise SOSC is also more natural than MPCC-SOSC because, unlike the latter, it has an appropriate second-order necessary condition as its counterpart. In particular, our assumptions for local quadratic convergence are weaker than those required by standard SQP when applied to MPCC and are equivalent to assumptions required by piecewise SQP for MPCC. Moreover, each iteration of our method consists of solving a linear system of equations instead of a quadratic program. Some globalization issues of the local scheme are also discussed, and illustrative examples and numerical experiments are presented.

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

“…From (15), by the upper semicontinuity of the generalized Jacobian and the results in [11], it follows that∆…”

“…where ∆ : R n × R l → 2 R n×n is a given multifunction satisfying (15), and that (21) holds with∆ defined according to (17). Then for each k there exists…”

“…On the other hand, taking into account (15), local uniform boundedness of generalized Jacobians of Lipschitzian mappings, (17), and the inclusion W k ∈ ∆(x k , λ k ), condition (21) implies (31). Therefore, superlinear rate of convergence of {x k } follows from Remark 2.…”

“…Under ULSCC, various relevant second-order sufficient optimality conditions (see [15]) at a strongly stationary pointx of problem (2) for an associated MPCCmultiplierμ reduce to the following:…”

“…We refer the reader to [7,8,14,15,20,23,28,31] for motivations and theoretical and computational developments for this problem class. As suggested in [32], the constraints of (2) can be written as smooth equalities by introducing an auxiliary variable y ∈ R m and using the lifted reformulation:…”