Global convergence of a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programs without constraint qualifications

In this paper, we propose a new sequential quadratic semidefinite programming (SQSDP) method for solving nonlinear semidefinite programs (NSDPs), in which we produce iteration points by solving a sequence of stabilized quadratic semidefinite programming (QSDP) subproblems, which we derive from the minimax problem associated with the NSDP. Differently from the existing SQSDP methods, the proposed one allows us to solve those QSDP subproblems just approximately so as to ensure global convergence. One more remarkable point of the proposed method is that any constraint qualifications (CQs) are not required in the global convergence analysis. Specifically, under some assumptions without CQs, we prove the global convergence to a point satisfying any of the following: the stationary conditions for the feasibility problem; the approximate-Karush-Kuhn-Tucker (AKKT) conditions; the trace-AKKT conditions. The latter two conditions are the new optimality conditions for the NSDP presented by Andreani et al. (2018) in place of the Karush-Kuhn-Tucker conditions. Finally, we conduct some numerical experiments to examine the efficiency of the proposed method

A Primal-Dual Augmented Lagrangian

Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both the primal and the dual variables simultaneously. A benefit of this approach is that the quality of the dual variables is monitored explicitly during the solution of the subproblem. Moreover, each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of conventional primal methods are proposed: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual $\ell$1 linearly constrained Lagrangian (pd$\ell$1-LCL) method

A Preconditioned Inexact Active-Set Method for Large-Scale Nonlinear Optimal Control Problems

We provide a global convergence proof of the recently proposed sequential homotopy method with an inexact Krylov--semismooth-Newton method employed as a local solver. The resulting method constitutes an active-set method in function space. After discretization, it allows for efficient application of Krylov-subspace methods. For a certain class of optimal control problems with PDE constraints, in which the control enters the Lagrangian only linearly, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems.Comment: 26 page