Assessing the reliability of general-purpose Inexact Restoration methods

Abstract

Inexact Restoration methods have been proved to be effective to solve constrained optimization problems in which some structure of the feasible set induces a natural way of recovering feasibility from arbitrary infeasible points. Sometimes natural ways of dealing with minimization over tangent approximations of the feasible set are also employed. A recent paper [N. Banihashemi and C. Y. Kaya, Inexact Restoration for Euler discretization of box-constrained optimal control problems, Journal of Optimization Theory and Applications 156, pp. 726-760, 2013] suggests that the Inexact Restoration approach can be competitive with well-established nonlinear programming solvers when applied to certain control problems without any problem-oriented procedure for restoring feasibility. This result motivated us to revisit the idea of designing general-purpose Inexact Restoration methods, especially for large-scale problems. In this paper we introduce an affordable algorithm of Inexact Restoration type for solving arbitrary nonlinear programming problems and we perform the first experiments that aim to assess its reliability.

Inexact Restoration for Minimization with Inexact Evaluation both of the Objective Function and the Constraints

“…In the case of success (so that (10) and ( 11) hold) update k ← k + 1 and go to Step 1. Else, continue at Step 4.…”

“…The so far generated trial point is accepted or not as new iterate according to the decrease of a merit function or using filter criteria [32,36,25,26,46]. Theoretical papers concerning Inexact Restoration methods for constrained optimization include [41,10,24]. Algorithmic variations are discused in [41,27,4,22,25,26,28], and applications may be found in [43,1,23,38,15,37,5,31,30,6,29,14,47].…”

Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization

“…One of the attractiveness of AGP is that it does not involve Lagrange multipliers estimates. AGP is the natural optimality condition that fits stopping criteria for algorithms based on inexact restoration [27,29,19,16,13], and is strictly stronger than the usual AKKT condition. Consequently, the stopping criteria based on AGP are more reliable that those based on AKKT.…”

A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles

“…While performance profiles of Dolan and Moré [DM02] are popular benchmarking tools, they require that a binary success/failure criterion be applied to each problem in a test set. However, even in convex settings, performance profiles (and thus any conclusions made from them) can be sensitive to how exactly they are created, which is of particular concern since the success/failure criteria employed must typically be chosen arbitrarily and is often sensitive to the choice of each method's stopping parameters, difficulties which are discussed in detail in [BM14,BG12,BBM15]. The additional considerations of nonconvex, nonsmooth constrained problems only intensify the difficulties of implementing a fair and informative benchmark.…”