Results 1  10
of
13
Robust Solutions To Uncertain Semidefinite Programs
 SIAM J. OPTIMIZATION
, 1998
"... In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible value of paramet ..."
Abstract

Cited by 82 (8 self)
 Add to MetaCart
In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hölderstable) with respect to the unperturbed problem's data. The approach can thus be used to regularize illconditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.
Robust Solutions To Uncertain Semidefinite Programs
, 1998
"... In this paper we consider semidenite programs (SDPs) whose data depends on some unknownbutbounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible values of paramet ..."
Abstract

Cited by 57 (2 self)
 Add to MetaCart
In this paper we consider semidenite programs (SDPs) whose data depends on some unknownbutbounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible values of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist, as SDPs. When the perturbation is "full", our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique, and continuous (Hölderstable) with respect to the unperturbed problems' data. The approach can thus be used to regularize illconditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation and integer programming.
First and Second Order Analysis of Nonlinear Semidefinite Programs
 Mathematical Programming
, 1997
"... In this paper we study nonlinear semidefinite programming problems. Convexity, duality and firstorder optimality conditions for such problems are presented. A secondorder analysis is also given. Secondorder necessary and sufficient optimality conditions are derived. Finally, sensitivity analysi ..."
Abstract

Cited by 47 (11 self)
 Add to MetaCart
In this paper we study nonlinear semidefinite programming problems. Convexity, duality and firstorder optimality conditions for such problems are presented. A secondorder analysis is also given. Secondorder necessary and sufficient optimality conditions are derived. Finally, sensitivity analysis of such programs is discussed. Key words: Semidefinite programming, cone constraints, convex programming, duality, secondorder optimality conditions, tangent cones, optimal value function, sensitivity analysis. AMS subject classification: 90C25, 90C30, 90C31 1 Introduction In this paper we consider the following optimization problem (P ) min x2IR m f(x) subject to G(x) 0: Here G : IR m ! S n is a mapping from IR m into the space S n of n \Theta n symmetric matrices and, for A; B 2 S n , the notation A B (the notation A B) means that the matrix A \Gamma B is positive semidefinite (negative semidefinite). Consider the cone K ae S n of positive semidefinite matrices. Then the co...
Optimization Problems with perturbations, A guided tour
 SIAM REVIEW
, 1996
"... This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and app ..."
Abstract

Cited by 46 (10 self)
 Add to MetaCart
This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and approximate solutions in situations where the set of Lagrange multipliers may be unbounded, or even empty. We give rather complete results for nonlinear programming problems, and describe some partial extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
Sensitivity Analysis of Optimization Problems Under Second Order Regular Constraints
, 1996
"... We present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition. We derive Lipschitz and Holder expansions of approximate optimal solutions, under a directional constraint qualification hypothesis and vari ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
We present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition. We derive Lipschitz and Holder expansions of approximate optimal solutions, under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem. We also show how the theory applies to semidefinite optimization and, more generally, to semiinfinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds. As a final application we provide a result on differentiability of metric projections in finite dimensional spaces.
Quadratic Growth and Stability in Convex Programming Problems With Multiple Solutions
, 1995
"... Given a convex program with C² functions and a convex set S of solutions to the problem, we give a second order condition which guarantees that the problem does not have solutions outside of S. This condition is interpreted as a characterization for the quadratic growth of the cost function. The cr ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
Given a convex program with C² functions and a convex set S of solutions to the problem, we give a second order condition which guarantees that the problem does not have solutions outside of S. This condition is interpreted as a characterization for the quadratic growth of the cost function. The crucial role in the proofs is played by a theorem describing a certain uniform regularity property of critical cones in smooth convex programs. We apply these results to the discussion of stability of solutions of a convex program under possibly nonconvex perturbations.
On differentiability of symmetric matrix valued functions
"... With every real valued function, of a real argument, can be associated a matrix function mapping a linear space of symmetric matrices into itself. In this paper we study directional differentiability properties of such matrix functions associated with directionally differentiable real valued functio ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
With every real valued function, of a real argument, can be associated a matrix function mapping a linear space of symmetric matrices into itself. In this paper we study directional differentiability properties of such matrix functions associated with directionally differentiable real valued functions. In particular, we show that matrix valued functions inherit semismooth properties of the corresponding real valued functions. Key words: matrix function, eigenvalues and eigenvectors, directional derivatives, semismooth mappings
Amenable functions in optimization
 IN NONSMOOTH OPTIMIZATION METHODS AND APPLICATIONS
, 1992
"... ..."
Asymptotic Analysis of Congested Communication Networks
, 1997
"... : This paper is devoted to the study of a routing problem in telecommunication networks, when the cost function is the average delay. We establish asymptotic expansions for the value function and solutions in the vicinity of a congested nominal problem. The study is strongly related to the one of a ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
: This paper is devoted to the study of a routing problem in telecommunication networks, when the cost function is the average delay. We establish asymptotic expansions for the value function and solutions in the vicinity of a congested nominal problem. The study is strongly related to the one of a partial inverse barrier method for linear programming. Keywords: Telecommunication networks, multicommodity flows, asymptotic expansions, linear programming, perturbation analysis, barrier functions, penalty methods. (R'esum'e : tsvp) INRIA, B.P. 105, 78153 Rocquencourt, France. Email: Frederic.Bonnans@inria.fr. y INRIA, B.P. 105, 78153 Rocquencourt, France. Email: Mounir.Haddou@inria.fr. Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Téléphone : (33 1) 39 63 55 11  Télécopie : (33 1) 39 63 53 Analyse asymptotique des r'eseaux de communications congestionn'es R'esum'e : Nous 'etudions le probl`eme de minimisation d...
Second Order Necessary and Sufficient Optimality Conditions Under Constraints
, 1996
"... In this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints. Our analysis is based on various concepts of second order tangent sets and parametric duality. We introduce a condition, called second order regularity, under which there is no gap b ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
In this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints. Our analysis is based on various concepts of second order tangent sets and parametric duality. We introduce a condition, called second order regularity, under which there is no gap between the corresponding second order necessary and second order sufficient conditions. We show that the second order regularity always holds in the case of semidefinite programming.