Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints

Abstract

Recently, nonlinear programming solvers have been used to solve a range of mathematical programs with equilibrium constraints (MPECs). In particular, sequential quadratic programming (SQP) methods have been very successful. This paper examines the local convergence properties of SQP methods applied to MPECs. SQP is shown to converge superlinearly under reasonable assumptions near a strongly stationary point. A number of examples are presented that show that some of the assumptions are difficult to relax.

Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints

“…The case of violation of classical constraint qualifications has been a subject of considerable interest in the past decade, both in the general case (e.g., [2,6,11,13,14,20,21,24,25,39]) and in the special case of equilibrium or complementarity constraints (e.g., [3,4,12,22,29,[33][34][35]). …”

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

“…It is known that introducing slacks can be advantageous for numerical solution by SQP [7,8] (see [15] for details of our implementation of this algorithm). The SQP methods were implemented without any tools for tackling possible infeasibility of subproblems, and without any tools for avoiding the Maratos effect.…”

“…It is therefore meaningful to ask "how much" the inclusion (8) can be violated without destroying the primal superlinear rate (assuming convergence, i.e., in a posteriori analysis). Another motivation for considering possible violation of the inclusion (8) is related to globalization of SQP via linesearch for a penalty function.…”

“…Indeed, as is well-known, the matrices in ∂ x ∂L ∂x (x k , λ k ) need not be positive definite under any natural assumptions, while to obtain a descent direction for the penalty function H k has to be positive definite. This again motivates considering H k that possibly does not satisfy (8).…”

“…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:…”