Stephen M. Robinson

Strongly Regular Generalized Equations

This paper considers generalized equations, which are convenient tools for formulating problems in complementarity and in mathematical programming, as well as variational inequalities. We introduce a regularity condition for such problems and, with its help, prove existence, uniqueness and Lipschitz continuity of solutions to generalized equations with parametric data. Applications to nonlinear programming and to other areas are discussed, and for important classes of such applications the regularity condition given here is shown to be in a certain sense the weakest possible condition under which the stated properties will hold.

Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems

This paper develops a condition for stability of the solution set of a system of nonlinear inequalities over a closed convex set in a Banach space, when the functions defining the inequalities are ...

Regularity and Stability for Convex Multivalued Functions

Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. These properties are applied to develop a perturbation theory for convex inequalities and to extend results on the continuity of convex functions.

Normal maps inducted by linear transformations

We study a certain piecewise linear manifold, which we call the normal manifold, associated with a polyhedral convex set, and a family of continuous functions, called normal maps, that are induced on this manifold by continuous functions from Rn to Rn. These normal maps occur frequently in optimization and equilibrium problems, and the subclass of normal maps induced by linear transformations plays a key role. Our main result is that the normal map induced by a linear transformation is a Lipschitzian homeomorphism if and only if the determinant of the map in each n-cell of the normal manifold has the same nonzero sign.

Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms

This paper establishes quantitative bounds for the variation of an isolated local minimizer for a general nonlinear program under perturbations in the objective function and constraints. These bounds are then applied to establish rates of convergence for a class of recursive nonlinear-programming algorithms.

Analysis of Sample-Path Optimization

Sample-path optimization is a method for optimizing limit functions occurring in stochastic modeling problems, such as steady-state functions in discrete-event dynamic systems. It is closely related to retrospective optimization techniques and to M-estimation. The method has been computationally tested elsewhere on problems arising in production and in project planning, with apparent success. In this paper we provide a mathematical justification for sample-path optimization by showing that under certain assumptions---which hold for the problems just mentioned---the method will almost surely find a point that is, in a specified sense, sufficiently close to the set of optimizers of the limit function.

An implicit-function theorem for a class of nonsmooth functions

In this paper we introduce the concept of strong approximation of functions, and a related concept of strong Bouligand (B-) derivative, and we employ these ideas to prove an implicit-function theorem for nonsmooth functions. This theorem provides the same kinds of information as does the classical implicit-function theorem, but with the classical hypothesis of strong Frechet differentiability replaced by strong approximation, and with Lipschitz continuity replacing Frechet differentiability of the implicit function. Therefore it is applicable to a considerably wider class of functions than is the classical theorem. In the last part of the paper we apply this implicit function result to analyze local solvability and stability of perturbed generalized equations.

Sample-path solution of stochastic variational inequalities

Sample-path optimization is a simulation-based method for solving optimization problems that arise in the study of complex stochastic systems. In this paper we broaden its applicability to include the solution of stochastic variational inequalities. This formulation can model equilibrium phenomena in physics, economics, and operations research. We describe the method, provide general conditions for convergence, and present numerical results of an application of the method to a stochastic economic equilibrium model of the European natural gas market. We also point out some current limitations of the method and indicate areas in which research might help to remove those limitations.

Stability Theory for Systems of Inequalities. Part I: Linear Systems

This paper deals with the stability of systems of linear inequalities in partially ordered Banach spaces when the data are subjected to small perturbations. We show that a certain condition is necessary and sufficient for such stability. For some of the more important special cases, this condition is computationally verifiable; it reduces to the classical full-row-rank condition in the case of equations alone. In addition, we give quantitative estimates for the magnitudes of the changes in the solution sets in terms of the magnitudes of the perturbations.

A quadratically-convergent algorithm for general nonlinear programming problems

This paper presents an algorithm for solving nonlinearly constrained nonlinear programming problems. The algorithm reduces the original problem to a sequence of linearly-constrained minimization problems, for which efficient algorithms are available. A convergence theorem is given which states that if the process is started sufficiently close to a strict second-order Kuhn—Tucker point, then the sequence produced by the algorithm exists and convergesR-quadratically to that point.