Tullio Zolezzi

Extended well-posedness of optimization problems

The well-posedness concept introduced in Ref. 1 for global optimization problems with a unique solution is generalized here to problems with many minimizers, under the name of extended well-posedness. It is shown that this new property can be characterized by metric criteria, which parallel to some extent those known about generalized Tikhonov well-posedness.

Well-Posedness and Stability Analysis in Optimization

Publisher Summary This chapter synthesizes some results about well-posedness and stability analysis in abstract minimum problems and optimal control of ordinary differential inclusions. The chapter describes well-posedness of convex minimum problems in Banach spaces by using their optimal value functions. The chapter also deals with optimal control problems for differential inclusions. Neither convexity nor existence of optimal solutions is assumed. Further relates the continuous behavior of the optimal value function with respect to perturbations acting on the data, with its stable behavior when passing to the (unperturbated) relaxed problem. The approach behind these theorems is based on the variational convergence. The relations between variational convergence (including epi-convergence) and some results reported in the chapter are considered in with an eye on mathematical programming problems.

Condition Number Theorems in Linear-Quadratic Optimization

The condition number of a given mathematical problem is often related to the reciprocal of its distance from ill-conditioning. Such a property is proved here in the infinite-dimensional setting for linear-quadratic convex optimization of two types: linearly constrained convex quadratic problems, and minimum norm least squares solutions. A uniform version of such theorem is obtained in both cases for suitably equi-bounded classes of optimization problems. An application to the conditioning of a Ritz method is presented. For least squares problems it is shown that the semi-Fredholm property of the operators involved determines the validity of a condition number theorem.

Wellposed convex integral functionals

Sufficient conditions are obtained for wellposedness of convex minimum problems of the calculus of variations for multiple integrals under strong or weak perturbations of the boundary data. Problems with a unique minimizer as well as problems with several solutions are treated. Wellposedness under weak convergence of the boundary data in W1 p ω is proved if p >2 and a counterexample is exhibited if p =2.

On Condition Number Theorems in Mathematical Programming

A condition number of nonconvex mathematical programming problems is defined as a measure of the sensitivity of their global optimal solutions under canonical perturbations. A (pseudo-)distance among problems is defined via the corresponding augmented Kojima functions. A characterization of well-conditioning is obtained. In the nonconvex case, we prove that the distance from ill-conditioning is bounded from above by a multiple of the reciprocal of the condition number. Moreover, a lower bound of the distance from a special class of ill-conditioned problems is obtained in terms of the condition number. The proof is based on a new theorem about the permanence of the Lipschitz character of set-valued inverse mappings. A uniform version of the condition number theorem is proved for classes of convex problems defined through bounds of some constants available from problem’s data.