On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions

Abstract

Assuming that the primal part of the sequence generated by a Newtontype (e.g., SQP) method applied to an equality-constrained problem converges to a solution where the constraints are degenerate, we investigate whether the dual part of the sequence is attracted by those Lagrange multipliers which satisfy second-order sufficient condition (SOSC) for optimality, or by those multipliers which violate it. This question is relevant at least for two reasons: one is speed of convergence of standard methods; the other is applicability of some recently proposed approaches for handling degenerate constraints. We show that for the class of damped Newton methods, convergence of the dual sequence to multipliers satisfying SOSC is unlikely to occur. We support our findings by numerical experiments. We also suggest a simple auxiliary procedure for computing multiplier estimates, which does not have this

Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints

“…The purpose of this paper, therefore, is to discuss possible scenarios of dual behaviour of Newton methods applied to degenerate optimization problems, and to put together a representative library of small examples that illustrate the possibilities and that can be used for future development of algorithms for degenerate problems. For the case of equality constraints only (i.e., when there are no inequality constraints in (1.1)), we believe that the overall picture is quite clear [25]. The case of mixed constraints, including complementarity constraints, is more complex and requires further study.…”

“…The case of violation of classical constraint qualifications has been a subject of considerable interest in the past decade, both in the general case (e.g., [2,6,11,13,14,20,21,24,25,39]) and in the special case of equilibrium or complementarity constraints (e.g., [3,4,12,22,29,[33][34][35]). …”

“…Among various scenarios of this behaviour, one of the prominent ones is dual convergence to multipliers violating SOSC (see, e.g., [38,Sect. 6], [25]). Understanding this phenomenon better is one of the purposes for putting together a collection of examples presented in this paper.…”

“…We next recall the notion of critical multipliers, which plays a central role in understanding dual behaviour of Newton-type schemes on degenerate problems, see [25]. It is convenient to do this for the case of equality-constrained problems first.…”

Combining stabilized SQP with the augmented Lagrangian algorithm

“…In the case of degenerate problems it is known [24,29,35] that, for both sSQP and for Aug-L methods, often there still exist rather large areas of attraction to critical multipliers (thus violating SOSC). Granted, the tendency of attraction to such multipliers is much weaker for sSQP than for the usual SQP and SQPrelated methods [28,29]; see also [31,Chapter 7]. Nevertheless, this attraction can still be observed with certain frequency, and in such cases the convergence rate of sSQP is also usually only linear.…”