Strong Duality for Generalized Convex Optimization Problems

In this paper, strong duality for nearly-convex optimization problems is established. Three kinds of conjugate dual problems are associated to the primal optimization problem: the Lagrange dual, Fenchel dual, and Fenchel-Lagrange dual problems. The main result shows that, under suitable conditions, the optimal objective values of these four problems coincide. The first author was supported in part by Gottlieb Daimler and Karl Benz Stiftung 02-48/99. This research has been performed while the second author visited Chemnitz University of Technology under DAAD (Deutscher Akademischer Austauschdienst) Grant A/02/12866. Communicated by T. Rapcsák     

Duality for Equilibrium Problems

We consider equilibrium problems in the framework of the formulation proposed by Blum and Oettli. We establish a new dual formulation for this equilibrium problem using the classical Fenchel conjugation, thus generalizing the classical convex duality theory for optimization problems. This work was begun when the first author was visiting the Instituto de Matemática y Ciencias Afines (Lima-Peru) in July 2002 and was finished when the second author was visiting the Centre de Recerca Matemàtica (Bellaterra-Spain) in September 2002.     

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.     

Optimality Conditions for Vector Optimization Problems of a Difference of Convex Mappings

In this paper, we study optimality conditions for vector optimization problems of a difference of convex mappings

Fenchel’s Duality Theorem for Nearly Convex Functions

We present an extension of Fenchel’s duality theorem by weakening the convexity assumptions to near convexity. These weak hypotheses are automatically fulfilled in the convex case. Moreover, we show by a counterexample that a further extension to closely convex functions is not possible under these hypotheses. The authors are grateful to the Associate Editor for helpful suggestions and remarks which improved the quality of the paper. The second author was supported by DFG (German Research Foundation), project WA 922/1.     

Duality of Unary Convex Functions Programming

Most nonliner programming problems consist of functions which are sums of unary,convexfunctions of linear fuctions.In this paper.we derive the duality forms of the unary oonvex optimization,and these technuqucs are applied to the geometric programming and minimum discrimination informationproblems.     

复合凸优化问题的Fenchel-Lagrange强对偶之研究

利用共轭函数的上图性质,引入新的约束规范条件,等价刻画了目标函数为凸函数与凸复合函数之和的复合优化问题及其Fenchel-Lagrange对偶问题之间的强对偶与稳定强对偶.     

Dynamic Optimization for Reachability Problems

This paper uses dynamic programming techniques to describe reach sets andrelated problems of forward and backward reachability. The original problemsdo not involve optimization criteria and are reformulated in terms ofoptimization problems solved through the Hamilton–Jacobi–Bellmanequations. The reach sets are the level sets of the value function solutionsto these equations. Explicit solutions for linear systems with hard boundsare obtained. Approximate solutions are introduced and illustrated forlinear systems and for a nonlinear system similar to that of theLotka–Volterra type.     

Optimality Conditions for C1,1 Vector Optimization Problems

The main purpose of this paper is to make use of the second-order subdifferential of vector functions to establish necessary and sufficient optimality conditions for vector optimization problems.     

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.     

Proof of a Conjecture by Deutsch, Li, and Swetits on Duality of Optimization Problems

A conjecture of Deutsch, Li, and Swetits on duality relationships among three optimization problems is shown to hold true. The proof relies on a reformulation of one of the problems in a suitable product space, to which then a version of the classical Fenchel duality theorem applies.     

An Application of the Duality Method to the Regularity of Membrane Problems with Friction

Duality is applied to study the regularity of solutions to membrane problems with friction. The method consists of the characterization of the regularity of the subdifferentials of the friction functional. Then the regularity of a solution reduces to the regularity of a solution to a related elliptic boundary value problem or to that of an obstacle problem without friction.     

Approximate Efficiency and Scalar Stationarity in Unbounded Nonsmooth Convex Vector Optimization Problems

We deal with extended-valued nonsmooth convex vector optimization problems in infinite-dimensional spaces where the solution set (the weakly efficient set) may be empty. We characterize the class of convex vector functions having the property that every scalarly stationary sequence is a weakly-efficient sequence. We generalize the results obained in the scalar case by Auslender and Crouzeix about asymptotically well-behaved convex functions and improve substantially the few results known in the vector case.     

Fast Moreau envelope computation I: numerical algorithms

The present article summarizes the state of the art algorithms to compute the discrete Moreau envelope, and presents a new linear-time algorithm, named NEP for NonExpansive Proximal mapping. Numerical comparisons between the NEP and two existing algorithms: The Linear-time Legendre Transform (LLT) and the Parabolic Envelope (PE) algorithms are performed. Worst-case time complexity, convergence results, and examples are included. The fast Moreau envelope algorithms first factor the Moreau envelope as several one-dimensional transforms and then reduce the brute force quadratic worst-case time complexity to linear time by using either the equivalence with Fast Legendre Transform algorithms, the computation of a lower envelope of parabolas, or, in the convex case, the non expansiveness of the proximal mapping.      

Optimality Conditions for the Difference of Convex Set-Valued Mappings

Using the concept of subdifferential of cone-convex set valued mappings recently introduced by Baier and Jahn J. Optimiz. Theory Appl. 100 (1999), 233–240, we give necessary optimality conditions for nonconvex multiobjective optimization problems. An example illustrating the usefulness of our results is also given. Mathematics Subject classification: Primary 90C29, 90C26; Secondary 49K99.     

Optimality Conditions for Vector Optimization Problems with Difference of Convex Maps

In this paper, by using the notion of strong subdifferential and epsilon-subdifferential, necessary optimality conditions are established firstly for an epsilon-weak Pareto minimal point and an epsilon-proper Pareto minimal point of a vector optimization problem, where its objective function and constraint set are denoted by using differences of two vector-valued maps, respectively. Then, by using the concept of approximate pseudo-dissipativity, sufficient optimality conditions are obtained. As an application of these results, sufficient and necessary optimality conditions are also given for an epsilon-weak Pareto minimal point and an epsilon-proper Pareto minimal point of a vector fractional mathematical programming.     

New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems

The approach to optimal control problems based on a purely variational reformulation may lead to new existence results by using fine, general existence theorems for variational problems without convexity assumptions. We illustrate this perspective here for autonomous one-dimensional problems and defer the study of more complex situations to later work. This research was supported by Project MTM2004-07114 from Ministerio de Educación y Ciencia (Spain) and by Grant PAI05-029 from JCCM (Castilla-La Mancha).     

Globally Convergent BFGS Method for Nonsmooth Convex Optimization1

We propose an implementable BFGS method for solving a nonsmooth convex optimization problem by converting the original objective function into a once continuously differentiable function by way of the Moreau–Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. We prove the global convergence of the proposed method under the assumption of strong convexity of the objective function.     

Faster than the Fast Legendre Transform,the Linear-time Legendre Transform

A new algorithm to compute the Legendre–Fenchel transform is proposed and investigated. The so-called Linear-time Legendre Transform (LLT) improves all previously known Fast Legendre Transform algorithms by reducing their log-linear worst-case time complexity to linear. Since the algorithm amounts to computing several convex hulls and sorting, any convex hull algorithm well-suited for a particular problem gives a corresponding LLT algorithm. After justifying the convergence of the Discrete Legendre Transform to the Legendre–Fenchel transform, an extended computation time complexity analysis is given and confirmed by numerical tests. Finally, the LLT is illustrated with several examples and a LLT MATLAB package is described. This revised version was published online in June 2006 with corrections to the Cover Date.     

集函数多目标分式规划的最优性和广义凸对偶性

本文讨论了集函数多目标（分母不同）分式规划，给出了Ｇｅｏｆｆｒｉｏｎ正常有效解的必要和充分条件，并讨论了关于有效解的广义凸对偶理论．