A Truncated SQP Method Based on Inexact Interior-Point Solutions of Subproblems

Abstract

Abstract. We consider sequential quadratic programming (SQP) methods applied to optimization problems with nonlinear equality constraints and simple bounds. In particular, we propose and analyze a truncated SQP algorithm in which subproblems are solved approximately by an infeasible predictor-corrector interior-point method, followed by setting to zero some variables and some multipliers so that complementarity conditions for approximate solutions are enforced. Verifiable truncation conditions based on the residual of optimality conditions of subproblems are developed to ensure both global and fast local convergence. Global convergence is established under assumptions that are standard for linesearch SQP with exact solution of subproblems. The local superlinear convergence rate is shown under the weakest assumptions that guarantee this property for pure SQP with exact solution of subproblems, namely, the strict Mangasarian-Fromovitz constraint qualification and second-order sufficiency. Local convergence results for our truncated method are presented as a special case of the local convergence for a more general perturbed SQP framework, which is of independent interest and is applicable even to some algorithms whose subproblems are not quadratic programs. For example, the framework can also be used to derive sharp local convergence results for linearly constrained Lagrangian methods. Preliminary numerical results confirm that it can be indeed beneficial to solve subproblems approximately, especially on early iterations. 1. Introduction. In this paper, we are concerned with truncated and perturbed versions of sequential quadratic programming (SQP) methods [5,27] for constrained optimization. We refer to an algorithm as a truncated SQP method if the solver for the SQP subproblem may be terminated early, producing an inexact solution. This is, in fact, a special case of the more general class of perturbed SQP methods, which we define as any approach where the iterates can be viewed, perhaps a posteriori, as approximate solutions to relevant SQP subproblems. We thus regard truncation as a special way of inducing perturbations. Our local convergence analysis allows other forms of perturbations as well and is also applicable to some methods that are not modifications of SQP as such. One example is the linearly constrained (augmented) Lagrangian method [42,39,18,33].We shall consider problems with equality constraints and simple bounds:

Global Convergence of Augmented Lagrangian Methods Applied to Optimization Problems with Degenerate Constraints, Including Problems with Complementarity Constraints

“…We invoke SQP methods because it is known [23] that they are quite robust and effective when applied to MPCC. The use of the linearly constrained (augmented) Lagrangian algorithm implemented in MINOS is motivated by the fact that (in a certain sense) it is related to both augmented Lagrangian methods and SQP; see [35,36].…”

Some composite-step constrained optimization methods interpreted via the perturbed sequential quadratic programming framework

“…Then the iteration subproblem of the pSQP framework [16], when specialized for the problem setting of (1.1), has the form of perturbed KKT conditions of the problem (2.1). Specifically,…”

“…Otherwise, it is the functions ω 1 and ω 2 that define each specific algorithm within the pSQP framework (they represent "the difference" between the pure SQP iteration and that of the given algorithm). We note that in general, in the last line of (2.2) the inequality constraints in primal variables can be perturbed too (in (2.2) they are not), see [16,17]; but complementarity relations have to be maintained exactly. We do not need the extra generality of perturbing inequality constraints for the applications in this paper.…”

“…This framework has been developed in [17,16,7] and it proved useful for analyzing, in a unified manner, a number of different Newtonian and Newton-related algorithms for constrained optimization (truncated and augmented Lagrangian modifications of SQP itself, sequential quadratically constrained quadratic programming, and linearly constrained Lagrangian methods, to mention some of the applications); see [20,Chapter 4]. In this paper we continue this line of reasoning and show that in addition to the above, local convergence properties of the inexact restoration methods [22,21,23,3,10,8,4] and of composite-step SQP methods [26,24], [6,Section 15.4], can also be derived from the pSQP theory.…”

Sharp Primal Superlinear Convergence Results for Some Newtonian Methods for Constrained Optimization

“…We shall consider the class of algorithms for solving (1.1) (or (1.2)) described by a perturbed sequential quadratic programming (pSQP) framework [24], to some extent related to inexact sequential quadratic programming in [37]; see also [21]. To this end, recall that given a primal-dual iterate (x k , λ k , μ k ) ∈ R n × R l × R m , the SQP subproblem [4] has the form…”