Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations

Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations and the conjugacy of primal and dual value functions are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points.

     

向量映射的鞍点和Lagrange对偶问题

本文研究拓扑向量空间广义锥-次类凸映射向量优化问题的鞍点最优性条件和Lagrange对偶问题,建立向量优化问题的Fritz John鞍点和Kuhn-Tucker鞍点的最优性条件及其与向量优化问题的有效解和弱有效解之间的联系。通过对偶问题和向量优化问题的标量化刻画各解之间的关系,给出目标映射是广义锥-次类凸的向量优化问题在其约束映射满足广义Slater约束规格的条件下的对偶定理。     

Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.     

Efficient algorithms for buffer space allocation

This paper describes efficient algorithms for determining how buffer space should be allocated in a flow line. We analyze two problems: a primal problem, which minimizes total buffer space subject to a production rate constraint; and a dual problem, which maximizes production rate subject to a total buffer space constraint. The dual problem is solved by means of a gradient method, and the primal problem is solved using the dual solution. Numerical results are presented. Profit optimization problems are natural generalizations of the primal and dual problems, and we show how they can be solved using essentially the same algorithms.     

Projection and restriction methods in geometric programming and related problems

Mathematical programming problems with unattained infima or unbounded optimal solution sets are dual to problems which lackinterior points, e.g., problems for which the Slater condition fails to hold or for which the hypothesis of Fenchel's theorem fails to hold. In such cases, it is possible to project the unbounded problem onto a subspace and to restrict the dual problem to an affine set so that the infima are not altered. After a finite sequence of such projections and restrictions, dual problems are obtained which have bounded optimal solution sets andinterior points. Although results of this kind have occasionally been used in other contexts, it is in geometric programming (both in the original psynomial form and the generalized form) where such methods appear most useful. In this paper, we present a treatment of dual projection and restriction methods developed in terms of dual generalized geometric programming problems. Analogous results are given for Fenchel and ordinary dual problems.This research was supported in part by Grant No. AFOSR-73-2516 from the Air Force Office of Scientific Research and by Grant No. NSF-ENG-76-10260 from the National Science Foundation.The authors wish to express their appreciation to the referees for several helpful comments.     

Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.     

Understanding linear semi-infinite programming via linear programming over cones

In this article, the standard primal and dual linear semi-infinite programming (DLSIP) problems are reformulated as linear programming (LP) problems over cones. Therefore, the dual formulation via the minimal cone approach, which results in zero duality gap for the primal–dual pair for LP problems over cones, can be applied to linear semi-infinite programming (LSIP) problems. Results on the geometry of the set of the feasible solutions for the primal LSIP problem and the optimality criteria for the DLSIP problem are also discussed.     

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     

Strong Duality with Proper Efficiency in Multiobjective Optimization Involving Nonconvex Set-Valued Maps

In this paper, we consider some dual problems of a primal multiobjective problem involving nonconvex set-valued maps. For each dual problem, we give conditions under which strong duality between the primal and dual problems holds in the sense that, starting from a Benson properly efficient solution of the primal problem, we can construct a Benson properly efficient solution of the dual problem such that the corresponding objective values of both problems are equal. The notion of generalized convexity of set-valued maps we use in this paper is that of near-subconvexlikeness.     

The use of Hestenes' method of multipliers to resolve dual gaps in engineering system optimization

For nonconvex problems, the saddle point equivalence of the Lagrangian approach need not hold. The nonexistence of a saddle point causes the generation of a dual gap at the solution point, and the Lagrangian approach then fails to give the solution to the original problem. Unfortunately, dual gaps are a fairly common phenomenon for engineering system design problems. Methods which are available to resolve the dual gaps destroy the separability of separable systems. The present work employs the method of multipliers by Hestenes to resolve the dual gaps of engineering system design problems; it then develops an algorithmic procedure which preserves the separability characteristics of the system. The theoretical foundations of the proposed algorithm are developed, and examples are provided to clarify the approach taken.     

Dual gradient method for linearly constrained,strongly convex,separable mathematical programming problems

A new dual gradient method is given to solve linearly constrained, strongly convex, separable mathematical programming problems. The dual problem can be decomposed into one-dimensional problems whose solutions can be computed extremely easily. The dual objective function is shown to have a Lipschitz continuous gradient, and therefore a gradient-type algorithm can be used for solving the dual problem. The primal optimal solution can be obtained from the dual optimal solution in a straightforward way. Convergence proofs and computational results are given.     

Dual interior point methods for linear semidefinite programming problems

Dual interior point methods for solving linear semidefinite programming problems are proposed. These methods are an extension of dual barrier-projection methods for linear programs. It is shown that the proposed methods converge locally at a linear rate provided that the solutions to the primal and dual problems are nondegenerate.     

具广义V-不变凸多目标变分的混合对偶性

在函数广义V-不变凸性的条件下,建立了多目标变分关于有效解的混合对偶理论.     

Wolfe Duality for Interval-Valued Optimization

Weak and strong duality theorems in interval-valued optimization problem based on the formulation of the Wolfe primal and dual problems are derived. The solution concepts of the primal and dual problems are based on the concept of nondominated solution employed in vector optimization problems. The concepts of no duality gap in the weak and strong sense are also introduced, and strong duality theorems in the weak and strong sense are then derived.     

Conjugate dual problems in constrained set-valued optimization and applications

In this paper, two conjugate dual problems are proposed by considering the different perturbations to a set-valued vector optimization problem with explicit constraints. The weak duality, inclusion relations between the image sets of dual problems, strong duality and stability criteria are investigated. Some applications to so-called variational principles for a generalized vector equilibrium problem are shown.     

Experiments with primal - dual decomposition and subgradient methods for the uncapacitatied facility location problem

For optimization problems that are structured both with respect to the constraints and with respect to the variables, it is possible to use primal–dual solution approaches, based on decomposition principles. One can construct a primal subproblem, by fixing some variables, and a dual subproblem, by relaxing some constraints and king their Lagrange multipliers, so that both these problems are much easier to solve than the original problem. We study methods based on these subproblems, that do not include the difficult Benders or Dantzig-Wolfe master problems, namely primal–dual subgradient optimization methods, mean value cross decomposition, and several comtbinations of the different techniques. In this paper, these solution approaches are applied to the well-known uncapacitated facility location problem. Computational tests show that some combination methods yield near-optimal solutions quicker than the classical dual ascent method of Erlenkotter     

A dual view of equilibrium problems

The aim of this paper is to introduce and study a dual problem associated to a generalized equilibrium problem (GEP). We show that the solutions of (GEP) and its dual are strictly related to the saddle points of an associated Lagrangian function, and, under some suitable conditions, to the solutions of a family of parametric optimization problems and their dual problems. Our results allow us to show that well-known concepts and results from duality theory of some important particular cases of (GEP) like variational inequalities and optimization problems can be recovered.     

非凸集值优化问题严有效解的强对偶定理

本文研究了非凸集值向量优化的严有效解在两种对偶模型的强对偶问题.利用Lagrange对偶和Mond-Weir对偶原理,获得了如下结果：原集值优化问题的严有效解,在一些条件下是对偶问题的强有效解,并且原问题和对偶问题的目标函数值相等;推广了集值优化对偶理论在锥-凸假设下的相应结果.     

Duality for Equilibrium Problems under Generalized Monotonicity

Duality is studied for an abstract equilibrium problem which includes, among others, optimization problems and variational inequality problems. Following different schemes, various duals are proposed and primal–dual relationships are established under certain generalized convexity and generalized monotonicity assumptions. In a primal–dual setting, existence results for a solution are derived for different generalized monotone equilibrium problems within each duality scheme.     

Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for primal problems which are perturbed by continuous linear functionals and their respective dual problems, which is named stable strong duality, are established. In these conditions, the fact that the perturbation function is evenly convex will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal–dual problems, obtaining a characterization for Fenchel case.