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

Abstract

We consider the sequential quadratic programming algorithm (SQP) applied to equalityconstrained optimization problems, where the problem data is differentiable with Lipschitzcontinuous 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 DennisMoré 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.

Frequency-sparse optimal quantum control

“…This would be required for local superlinear convergence, see e.g. [18]. Since the dimension of the control space can become quite large with our approach we chose the memory efficient L-BFGS method, see [25].…”

An algorithm based on continuation techniques for minimization problems with highly non-linear equality constraints

“…One of the ways consists in using a sequence of quadratic programming subproblems (SQP) which have to be solved successively. Since SQP methods were developed [1][2][3], a great variety of research has been carried out focusing on this technique [4][5][6][7][8]. Many SQP algorithms are combined with trust region methods.…”

“…where x is the Euclidean norm of x ∈ R n and each p j (x) is a polynomial of degree d j . More precisely, we are interested in computing points that locally minimize the Euclidean norm x under the polynomial constraints (4). We note that the problems that we have in mind are too complex to apply algorithms that guarantee the computation of all local minimizers of the Euclidean norm.…”

A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems

“…This problem setting with restricted smoothness requirements has multiple applications: e.g., in stochastic programming and optimal control (the so-called extended linear-quadratic problems [26,28,29]), and in semi-infinite programming and in primal decomposition procedures (see [21,25] and references therein). Once but not twice differentiable functions arise also when reformulating complementarity constraints as in [12] or in the lifting approach [10,11,30]. Other possible sources are subproblems in penalty or augmented Lagrangian methods with lower-level constraints treated directly and upper-level inequality constraints treated via quadratic penalization or via augmented Lagrangian, which gives rise to certain terms that are not twice differentiable in general; see, e.g., [1].…”

“…Using the terminology of [4], the multiplier (λ,μ) being noncritical also means that the cone-matrix pair (C(x), ∂Ψ ∂ x (x,λ,μ)) has the so-called R 0 property (see the discussion following (3.3.18) in [4]). Finally, note that multiplying the first equality in (9) by ξ and using the other two equalities in (9) and the relations in (10), it can be seen that a sufficient condition for (λ,μ) to be noncritical is the following second-order sufficiency condition (SOSC):…”