On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

Journal of Optimization Theory and Applications - We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values...     

Vector Variational Inequalities for Nondifferentiable Convex Vector Optimization Problems

In this paper, we consider a nondifferentiable convex vector optimization problem (VP), and formulate several kinds of vector variational inequalities with subdifferentials. Here we examine relations among solution sets of such vector variational inequalities and (VP). Mathematics Subject classification (2000). 90C25, 90C29, 65K10 This work was supported by the Brain Korea 21Project in 2003. The authors wish to express their appreciation to the anonymous referee for giving valuable comments.     

Well-Posedness and Optimization under Perturbations

Estimates of the size of sets of approximate solutions are obtained for well-posed optimization problems in a Banach space, and extended to problems subject to perturbations of a general form. An estimate of the perturbations guaranteeing a prescribed level of suboptimality is presented.     

Applications of optimization methods to robust stability of linear systems

We study the robust stability problem for a family of polynomials. We allow for all the coefficients of the polynomials to be affinely perturbed, where the size of the perturbation is measured by an arbitrary convex function. We apply optimization techniques, and in particular convex duality methods, to derive simple formulas for the stability radius, to find a minimal perturbation which destroys stability, and to obtain necessary and sufficient conditions for robust stability. Our framework is general enough to cover many applications. As special cases, we obtain many results recently reported in the literature.The work of the first author was partially supported by AFOSR Grant 91-008 and NSF Grant DMS-92-01297.     

Continuity of the Solution Map in Quadratic Programs under Linear Perturbations

It is well known that the solution map of a quadratic program where only the linear part of the data is subject to perturbation is an upper Lipschitz multifunction. This paper characterizes the continuity and lower semicontinuity of that solution map.This work was supported by the Brain Korea 21 Project in 2003, the APEC Postdoctoral Fellowships Program, and the KOSEF Brain Pool Program. The authors thank Professor F. Giannessi and two anonymous referees for helpful comments.Communicated by F. Giannessi     

First and Second-Order Optimality Conditions for Convex Composite Multiobjective Optimization

Multiobjective optimization is a useful mathematical model in order to investigate real-world problems with conflicting objectives, arising from economics, engineering, and human decision making. In this paper, a convex composite multiobjective optimization problem, subject to a closed convex constraint set, is studied. New first-order optimality conditions for a weakly efficient solution of the convex composite multiobjective optimization problem are established via scalarization. These conditions are then extended to derive second-order optimality conditions.     

Characterizations of the Nonemptiness and Compactness of Solution Sets in Convex Vector Optimization

In this paper, various necessary and sufficient conditions are given for the nonemptiness and compactness of the weakly efficient solution set of a convex vector optimization problem.     

Variational Methods in Convex Analysis

We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth subdifferential theory. Research was supported by NSERC and by the Canada Research Chair Program and National Science Foundation under grant DMS 0102496.     

带多面体控制锥的锥约束凸向量优化问题的有效解集的非空有界性的刻画

本文刻画了控制锥为多面凸锥的锥约束凸向量优化问题有效解集的非空有界性．然后将其中的一个重要条件应用于一类罚函数方法收敛性的研究．     

Computable bounds on parametric solutions of convex problems

We present a new method for computing bounds on parametric solutions of convex problems. The approach is based on a uniform quadratic underestimation of the objective function and a simple technique for the calculation of bounds on the optimal value function.Research supported by Grant ECS-8619859, National Science Foundation and Contract N00017-86-K-0052, Office of Naval Research.     

Convergence results for a self-dual regularization of convex problems

We study a one-parameter regularization technique for convex optimization problems whose main feature is self-duality with respect to the Legendre–Fenchel conjugation. The self-dual technique, introduced by Goebel, can be defined for both convex and saddle functions. When applied to the latter, we show that if a saddle function has at least one saddle point, then the sequence of saddle points of the regularized saddle functions converges to the saddle point of minimal norm of the original one. For convex problems with inequality and state constraints, we apply the regularization directly on the objective and constraint functions, and show that, under suitable conditions, the associated Lagrangians of the regularized problem hypo/epi-converge to the original Lagrangian, and that the associated value functions also epi-converge to the original one. Finally, we find explicit conditions ensuring that the regularized sequence satisfies Slater's condition.     

Unifying Optimal Partition Approach to Sensitivity Analysis in Conic Optimization

We study convex conic optimization problems in which the right-hand side and the cost vectors vary linearly as functions 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 optimal partition approach to sensitivity analysis in LP and SDP, the range of perturbations for which the optimal partition remains constant can be computed by solving two conic optimization problems. Under a weaker notion of nondegeneracy, this range is simply given by a minimum ratio test. We discuss briefly the properties of the optimal value function under such perturbations.     

An Algorithm for Approximate Multiparametric Convex Programming

For multiparametric convex nonlinear programming problems we propose a recursive algorithm for approximating, within a given suboptimality tolerance, the value function and an optimizer as functions of the parameters. The approximate solution is expressed as a piecewise affine function over a simplicial partition of a subset of the feasible parameters, and it is organized over a tree structure for efficiency of evaluation. Adaptations of the algorithm to deal with multiparametric semidefinite programming and multiparametric geometric programming are provided and exemplified. The approach is relevant for real-time implementation of several optimization-based feedback control strategies.     

Connectedness of Cone-Efficient Solution Set for Cone-Quasiconvex Multiobjective Programming in Locally Convex Spaces

This paper deals with the connectedness of the cone-efficient solution set for vector optimization inlocally convex Hausdorff topological vector spaces.The connectedness of the cone-efficient solution set is provedfor multiobjective programming defined by a continuous cone-quasiconvex mapping on a compact convex set ofalternatives.The generalized saddle theorem plays a key role in the proof.     

鲁棒凸优化问题拟近似解的刻划

本文研究鲁棒凸优化问题拟近似解的最优性条件和对偶理论.首先利用鲁棒优化方法,在由约束函数的共轭函数的上图给出的闭凸锥约束规格条件下,建立了拟近似解的最优性充要条件.其次给出了鲁棒凸优化问题拟近似解在Wolf型和Mond-weir型对偶模型下的强(弱)对偶定理.最后给出具体实例验证了本文获得的结果.     

Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

We characterize the class of those closed convex sets which have a barrier cone with a nonempty interior. As a consequence, we describe the set of those proper extended-real-valued functionals for which the domain of their Fenchel conjugate has a nonempty interior. As an application, we study the stability of the solution set of a semi-coercive variational inequality.     

自反Banach空间中的约束凸向量优化问题的弱有效解集的非空有界性的刻画

本文首先研究无限维自反Banach空间中的锥约束凸向量优化问题的弱有效解集的非空有界性的各种刻画.然后将获得的结果用于研究一类罚函数方法的收敛性.     

Fenchel-Lagrange Duality Versus Geometric Duality in Convex Optimization

We present a new duality theory to treat convex optimization problems and we prove that the geometric duality used by Scott and Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions to achieve strong duality are considered and optimality conditions are derived. Next, we apply our approach to some problems considered by Scott and Jefferson, determining their duals. We give weaker sufficient conditions to achieve strong duality and the corresponding optimality conditions. Finally, posynomial geometric programming is viewed also as a particular case of the duality approach that we present. Communicated by V. F. Demyanov The first author was supported in part by Gottlieb Daimler and Karl Benz Stiftung 02-48/99. The second author was supported in part by Karl und Ruth Mayer Stiftung.     

Value Estimation Approach to the Iri-Imai Method for Constrained Convex Optimization

In this paper we propose an extension of the so-called Iri-Imai method to solve constrained convex programming problems. The original Iri-Imai method is designed for linear programs and assumes that the optimal objective value of the optimization problem is known in advance. Zhang (Ref. 9) extends the method for constrained convex optimization but the optimum value is still assumed to be known in advance. In our new extension this last requirement on the optimal value is relaxed; instead only a lower bound of the optimal value is needed. Our approach uses a multiplicative barrier function for the problem with a univariate parameter that represents an estimated optimum value of the original optimization problem. An optimal solution to the original problem can be traced down by minimizing the multiplicative barrier function. Due to the convexity of this barrier function the optimal objective value as well as the optimal solution of the original problem are sought iteratively by applying Newtons method to the multiplicative barrier function. A new formulation of the multiplicative barrier function is further developed to acquire computational tractability and efficiency. Numerical results are presented to show the efficiency of the new method.His research supported by Hong Kong RGC Earmarked Grant CUHK4233/01E.Communicated by Z. Q. Luo     

不等式约束的向量优化中值映射的余切上图导数

考虑如下的参数向量优化问题minK{f(w,x)|x∈X,g(w,x)∈C},这里f:W×X→Y是从赋范空间W和X的积到另一个赋范空间Y的Hadamard可微的单值映射,K Y是一个尖闭凸锥,C是Banach空间Z中的一个尖闭凸锥,g:W×X→Z是一个Fréchet可微的映射.借助目标函数的导数、约束映射的余切导数及拉格朗日映射给出了值映射的余切上图导数的两个表示.