Sensitivity analysis of the optimal solutions to Huff-type competitive location and design problems

A chain wants to set up a single new facility in a planar market where similar facilities of competitors, and possibly of its own chain, are already present. Fixed demand points split their demand probabilistically over all facilities in the market proportionally with their attraction to each facility, determined by the different perceived qualities of … Read more

Two theoretical results for sequential semidefinite programming

We examine the local convergence of a sequential semidefinite programming approach for solving nonlinear programs with nonlinear semidefiniteness constraints. Known convergence results are extended to slightly weaker second order sufficient conditions and the resulting subproblems are shown to have local convexity properties that imply a weak form of self-concordance of the barrier subproblems. Citation Preprint, … Read more

Sensitivity analysis in linear semi-infinite programming via partitions

This paper provides sufficient conditions for the optimal value function of a given linear semi-infinite programming problem to depend linearly on the size of the perturbations, when these perturbations are directional, involve either the cost coefficients or the right-hand-side function or both, and they are sufficiently small. Two kinds of partitions are considered. The first … Read more

Stability and Sensitivity Analysis for Optimal Control Problems with a First-order State Constraint having (nonessential) Touch Points

The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control. In presence of (nonessential) touch points, the arc structure of the trajectory is not stable. We show how to perform a sensitivity analysis that predicts which touch points will, under a small perturbation, become inactive, remain touch … Read more

Prox-Regularity and Stability of the Proximal Mapping

Fundamental insights into the properties of a function come from the study of its Moreau envelopes and Proximal point mappings. In this paper we examine the stability of these two objects under several types of perturbations. In the simplest case, we consider tilt-perturbations, i.e. perturbations which correspond to adding a linear term to the objective … Read more

Generalized Support Set Invariancy Sensitivity Analysis

Support set invariancy sensitivity analysis deals with finding the range of the parameter variation where there are optimal solutions with the same positive variables for all parameter values throughout this range. This approach to sensitivity analysis has been studied for Linear Optimization (LO) and Convex Quadratic Optimization (CQO) problems, when they are in standard form. … Read more

The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications

For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson’s constraint qualification, the following conditions are proved to be equivalent: the strong second order sufficient condition and constraint nondegeneracy; the nonsingularity of Clarke’s Jacobian of the Karush-Kuhn-Tucker system; the strong regularity of the Karush-Kuhn-Tucker point; and others. Citation Technical Report, Department of … Read more

Sensitivity analysis in convex quadratic optimization: simultaneous perturbation of the objective and right-hand-side vectors

In this paper we study the behavior of Convex Quadratic Optimization problems when variation occurs simultaneously in the right-hand side vector of the constraints and in the coefficient vector of the linear term in the objective function. It is proven that the optimal value function is piecewise-quadratic. The concepts of transition point and invariancy interval … Read more

Rigorous Error Bounds for the Optimal Value in Semidefinite Programming

A wide variety of problems in global optimization, combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how … Read more

Computational Experience with Rigorous Error Bounds for the Netlib Linear Programming Library

The Netlib library of linear programming problems is a well known suite containing many real world applications. Recently it was shown by Ordonez and Freund that 71% of these problems are ill-conditioned. Hence, numerical difficulties may occur. Here, we present rigorous results for this library that are computed by a verification method using interval arithmetic. … Read more