A Primal-Dual Interior-Point Method for Nonlinear Programming with Strong Global and Local Convergence Properties

Abstract

A scheme|inspired from an old idea due to Mayne and Polak Math. Prog., vol. 11, 1976, pp. 67 80|is proposed for extending to general smooth constrained optimization problems a previously proposedfeasible interior-point method for inequality constrained problems. It is shown that the primal-dual interior point framework allows for a signi cantly more e ective implementation of the Mayne-Polak idea than that discussed an analyzed by the originators in the context of rst order methods of feasible direction. Strong global and local convergence results are proved under mild assumptions. In particular, the proposed algorithm does not su er the W achter-Biegler e ect.

An interior algorithm for nonlinear optimization that combines line search and trust region steps

“…The difficulties caused by rank deficient constraint Jacobians are sometimes addressed at the linear algebra level by introducing perturbations during the factorization of the KKT matrix [1]. Other approaches include the use of 1 or 2 penalizations of the constraints, which provide regularization [16,24], and the use of a feasibility restoration phase [15,27]. In this paper we describe a mechanism for stabilizing the line search iteration that is different from those proposed in the literature.…”

Local analysis of the feasible primal-dual interior-point method

“…Subsection 5.1 completes the algorithm description, namely, we explain there how to compute, without too much effort, a stepsize that satisfies condition (11). As a baseline for comparison, we have also tested the infeasible primal-dual interior point method (details are explained in Subsect.…”

“…Consider a sequence {w k = (x k , y k , z k )} generated by the feasible primal-dual interior-point method described in Algorithm 2.1. If α k satisfies (8)(9) and (11) and τ k ∈ (0, 1) and μ k > 0 satisfy (12) and (13), then there exist positive constants ε and κ independent of k such that, when…”

“…The feasible primal-dual interior-point method of Tits et al [11] achieves a quadratic rate of local convergence. In this algorithm, the multipliers corresponding to the inequality constraints are updated according to an appropriate formula and do not result directly from the Newton related primal-dual step on the perturbed KKT system of first-order optimality conditions.…”

A note on the use of vector barrier parameters for interior-point methods

“…These methods have proved to be very successful for the solution of linear and general convex problems. Recently, a significant amount of effort has been devoted to extending these procedures to non-convex problems, see for example El-Bakry et al [2], Gajulapalli [4], Gay et al [5], Vanderbei and Shanno [15], Yamashita [17], Tits et al [14], Moguerza and Prieto [10], among others. These methods proceed by (approximately) solving a sequence of equality-constrained problems of the form An appropriate choice of values for the parameter l may have a significant impact on the practical performance of the algorithm, both on its convergence (a sequence that converges to zero at an excessively fast rate may imply numerical difficulties and lack of convergence), and its rate of convergence (a slowly convergent sequence will imply a slow algorithm).…”