The Fermat Rule for Set Optimization Problems with Lipschitzian Set-Valued Mappings

n this paper, we consider set optimization problems with respect to the set approach. Specifically, we deal with the lower less and the upper less set relations. First, we derive properties of convexity and Lipschitzianity of suitable scalarizing functionals, under the same assumption on the set-valued objective mapping. We then obtain upper estimates of the … Read more

Optimality conditions for nonlinear second-order cone programming and symmetric cone programming

Nonlinear symmetric cone programming (NSCP) generalizes important optimization problems such as nonlinear programming, nonlinear semidefinite programming and nonlinear second-order cone programming (NSOCP). In this work, we present two new optimality conditions for NSCP without constraint qualifications, which implies the Karush-Kuhn-Tucker conditions under a condition weaker than Robinson’s constraint qualification. In addition, we show the relationship … Read more

Relations Between Abs-Normal NLPs and MPCCs Part 2: Weak Constraint Qualifications

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie’s and Guignard’s kink qualification and … Read more

Relations Between Abs-Normal NLPs and MPCCs Part 1: Strong Constraint Qualifications

This work is part of an ongoing effort of comparing non-smooth optimization problems in abs-normal form to MPCCs. We study the general abs-normal NLP with equality and inequality constraints in relation to an equivalent MPCC reformulation. We show that kink qualifications and MPCC constraint qualifications of linear independence type and Mangasarian-Fromovitz type are equivalent. Then … Read more

Local minimizers of semi-algebraic functions

Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so–called {\em tangency variety} of $f$ at $\bar{x},$ we first provide necessary and sufficient conditions for $\bar{x}$ to be a local minimizer of $f,$ and then in the case where $\bar{x}$ is an isolated local minimizer of … Read more

Optimality conditions and global convergence for nonlinear semidefinite programming

Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush-Kuhn-Tucker … Read more

A Newton-CG Algorithm with Complexity Guarantees for Smooth Unconstrained Optimization

We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton’s method and linear conjugate gradient, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity … Read more

Optimality Conditions and Constraint Qualifications for Generalized Nash Equilibrium Problems and their Practical Implications

Generalized Nash Equilibrium Problems (GNEPs) are a generalization of the classic Nash Equilibrium Problems (NEPs), where each player’s strategy set depends on the choices of the other players. In this work we study constraint qualifications and optimality conditions tailored for GNEPs and we discuss their relations and implications for global convergence of algorithms. Surprisingly, differently … Read more

Two-level value function approach to nonsmooth optimistic and pessimistic bilevel programs

The authors’ paper in Ref. [5], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analyzed in Ref. [4], for the case of optimistic bilevel programs. One of the basic assumptions in both of these … Read more

Inner Conditions for Error Bounds and Metric Subregulerity of Multifunctions

We introduce a new class of sets, functions and multifunctions which is shown to be large and to enjoy some nice common properties with the convex setting. Error bounds for objects attached to this class are characterized in terms of inner conditions of Abadie’s type, that is conditions bearing on normal cones and coderivatives at … Read more