We discuss in this paper continuity and differentiability properties of solutions of parameterized variational inequalities (generalized equations). To this end we use an approach of formulating variational inequalities in a form of optimization problems and applying a general theory of perturbation analysis of parameterized optimization problems. CitationSchool of Industrial and Systems Engineering, Georgia Institute of … Read more
Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for … Read more
The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z_* := min_x {c’x | Ax-b \in C_Y, x \in C_X }, to the more general non-conic format: (GP_d): z_* := min_x {c’x | Ax-b \in C_Y, x \in P}, where P is … Read more
We discuss in this paper a class of nonsmooth functions which can be represented, in a neighborhood of a considered point, as a composition of a positively homogeneous convex function and a smooth mapping which maps the considered point into the null vector. We argue that this is a sufficiently rich class of functions and … Read more
We study convex conic optimization problems in which the right-hand side and the cost vectors vary linearly as a function of a scalar parameter. We present a unifying geometric framework that subsumes the concept of the optimal partition in linear programming (LP) and semidefinite programming (SDP) and extends it to conic optimization. Similar to the … Read more
We study the asymptotic behavior of the interior-point bounds arising from the work of Yildirim and Todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds. For perturbations of the right-hand side vector and the cost matrix, we show that the interior-point bounds evaluated on the central path using the Monteiro-Zhang … Read more
We consider the interior-point approach to sensitivity analysis in linear programming (LP) developed by the authors. We investigate the quality of the interior-point bounds under degeneracy. In the case of a special degeneracy, we show that these bounds have the same nice relationship with the optimal partition bounds as in the nondegenerate case. We prove … Read more