Some notes on applying computational divided differencing in optimization

We consider the problem of accurate computation of the finite difference $f(\x+\s)-f(\x)$ when $\Vert\s\Vert$ is very small. Direct evaluation of this difference in floating point arithmetic succumbs to cancellation error and yields 0 when $\s$ is sufficiently small. Nonetheless, accurate computation of this finite difference is required by many optimization algorithms for a “sufficient decrease” … Read more

Full Stability in Finite-Dimensional Optimization

The paper is devoted to full stability of optimal solutions in general settings of finite-dimensional optimization with applications to particular models of constrained optimization problems including those of conic and specifically semidefinite programming. Developing a new technique of variational analysis and generalized differentiation, we derive second-order characterizations of full stability, in both Lipschitzian and H\”olderian … Read more

Full stability of locally optimal solutions in second-order cone programming

The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to problems of second-order cone programming (SOCP) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sucient conditions under the corresponding constraint quali cations. We also establish close relationships between … Read more

A practicable framework for distributionally robust linear optimization

We developed a modular framework to obtain exact and approximate solutions to a class of linear optimization problems with recourse with the goal to minimize the worst-case expected objective over an ambiguity set of distributions. The ambiguity set is specified by linear and conic quadratic representable expectation constraints and the support set is also linear … Read more

On the Separation of Split Inequalities for Non-Convex Quadratic Integer Programming

We investigate the computational potential of split inequalities for non-convex quadratic integer programming, first introduced by Letchford and further examined by Burer and Letchford. These inequalities can be separated by solving convex quadratic integer minimization problems. For small instances with box-constraints, we show that the resulting dual bounds are very tight; they can close a … Read more

Polynomial solvability of variants of the trust-region subproblem

The trust region subproblem concerns the minimization of a general quadratic over the unit ball in R^n. Extensions to this problem are of interest because of applications to, for example, combinatorial optimization. However the extension obtained by adding an arbitrary family of linear side constraints is NP-hard. In this paper we consider variants of the … Read more

An Inexact Proximal Method for Quasiconvex Minimization

In this paper we propose an inexact proximal point method to solve constrained minimization problems with locally Lipschitz quasiconvex objective functions. Assuming that the function is also bounded from below, lower semicontinuous and using proximal distances, we show that the sequence generated for the method converges to a stationary point of the problem. Citation July … Read more

A Unified View on Relaxations for a Nonlinear Network Flow Problem

We consider a nonlinear nonconvex network flow problem that arises, for example, in natural gas or water transmission networks. Given is such network with active and passive components, that is, valves, compressors, pressure regulators (active) and pipelines (passive), and a desired amount of flow at certain specified entry and exit nodes of the network. Besides … Read more

A polynomial projection algorithm for linear programming

We propose a polynomial algorithm for linear programming. The algorithm represents a linear optimization or decision problem in the form of a system of linear equations and non-negativity constraints. Then it uses a procedure that either fi nds a solution for the respective homogeneous system or provides the information based on which the algorithm rescales the … Read more

On Equilibrium Problems Involving Strongly Pseudomonotone Bifunctions

We study equilibrium problems with strongly pseudomonotone bifunctions in real Hilbert spaces. We show the existence of a unique solution. We then propose a generalized strongly convergent projection method for equilibrium problems with strongly pseudomonotone bifunctions. The proposed method uses only one projection without requiring Lipschitz continuity. Application to variational inequalities is discussed. Citation Institute … Read more