This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and approximate solutions in situations where the set of Lagrange multipliers may be unbounded, or even empty. We give rather complete results for nonlinear programming problems, and describe some partial extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.

. Recently, Wright proposed a stabilized sequential quadratic programming algorithm for inequality constrained optimization. Assuming the MangasarianFromovitz constraint qualification and the existence of a strictly positive multiplier (but possibly dependent constraint gradients), he proved a local quadratic convergence result. In this paper, we establish quadratic convergence in cases where both strict complementarity and the Mangasarian-Fromovitz constraint qualification do not hold. The constraints on the stabilization parameter are relaxed, and linear convergence is demonstrated when the parameter is kept fixed. We show that the analysis of this method can be carried out using recent results for the stability of variational problems. Key words. Sequential quadratic programming, quadratic convergence, superlinear convergence, degenerate optimization, stabilized SQP, error estimation To appear in Computational Optimization and Applications This paper is dedicated to Olvi L. Manga...

Based on an inverse function theorem for a system of semismooth equations, this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular/stable. We show further that for a parametric complementarity problem of this kind, if a solution corresponding to a base parameter is strongly stable, then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution. Similar results are also derived for a complementarity problem defined on the Lorentz cone. The analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the Lorentz cone. 1

For a general category of variational inclusions in finite dimensions, a class of parameterizations, called “ample” parameterizations, is identified that is rich enough to provide a full theory of Lipschitz-type properties of solution mappings without the need for resorting to the auxiliary introduction of canonical parameters. Ample parameterizations also support a detailed description of the graphical geometry that underlies generalized differentiation of solutions mappings. A theorem on proto-derivatives is thereby obtained. The case of a variational inequality over a polyhedral convex set is given special treatment along with an application to minimizing a parameterized function over such a set.

Metric regularity is a central concept in variational analysis for the study of solution mappings associated with “generalized equations,” including variational inequalities and parameterized constraint systems. Here it is employed to characterize the distance to irregularity or infeasibility with respect to perturbations of the system structure. Generalizations of the Eckart-Young theorem in numerical analysis are obtained in particular.

this paper, we prove that the Graves theorem is a consequence of the following general result: the openness with linear rate of a locally closed set-valued map F around a point (x 0 ; y 0 ) of its graph is invariant with respect to a perturbation of the form f +F provided that the strict derivative of f at x 0 is zero. In 1934 L. A. Lyusternik [19] published the following fundamental geometric result: if a function f from Banach space X into a Banach space Y is Fr'echet differentiable near x 0 , its derivative rf is continuous at x 0 , and rf(x 0 ) is onto, then the tangent manifold to

The behavior of a minimizing point when an objective function is tilted by adding a small linear term is studied from the perspective of second-order conditions for local optimality. The classical condition of a positive-definite Hessian in smooth problems without constraints is found to have an exact counterpart much more broadly in the positivity of a certain generalized Hessian mapping. This fully characterizes the case where tilt perturbations cause the minimizing point to shift in a lipschitzian manner.

. A very general optimization problem with a variational inequality constraint, inequality constraints and an abstract constraint is studied. Fritz John type and Kuhn-Tucker type necessary optimality conditions involving Mordukhovich coderivatives are derived. Several constraint qualifications for the Kuhn-Tucker type necessary optimality conditions involving Mordukhovich coderivatives are introduced and their relationships are studied. Applications to bilevel programming problems are also given. Key words. optimization problems, variational inequality constraints, necessary optimality conditions, constraint qualifications, coderivatives of set-valued maps, nonsmooth analysis. AMS subject classifications. 49K99, 90C, 90D65 1. Introduction. An optimization problem with variational inequality constraints (OPVIC) is a special class of an optimization problem over variables x and y in which some or all of its constraints are defined by a parametric variational inequality with y as its...

We discuss first and second order optimality conditions for nonlinear second-order cone programming problems, and their relation with semidefinite programming problems. For doing this we extend in an abstract setting the notion of optimal partition. Then we state a characterization of strong regularity in terms of second order optimality conditions.

informs ® doi 10.1287/moor.1040.0115 © 2005 INFORMS In this paper we discuss local uniqueness, continuity, and differentiability properties of solutions of parameterized variational inequalities (generalized equations). To this end we use two types of techniques. One approach consists in formulating variational inequalities in a form of optimization problem based on regularized gap functions, and applying a general theory of perturbation analysis of parameterized optimization problems. Another approach is based on a theory of contingent (outer graphical) derivatives and some results about differentiability properties of metric projections. Key words: variational inequalities; gap functions; sensitivity analysis; second order regularity; quadratic growth condition; locally upper Lipschitz and Hölder continuity; directional differentiability; prox-regularity; graphical derivatives