Proving Strong Duality for Geometric Optimization Using a Conic Formulation

Geometric optimization¹ is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the well-known associated duality theory, using conic optimization. After introducing suitable convex cones and studying their properties, we model geometric optimization problems with a conic formulation, which allows us to apply the powerful duality theory of conic optimization and derive the duality results valid for geometric optimization.     

Surrogate duality for robust optimization

Robust optimization problems, which have uncertain data, are considered. We prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min–max duality theorems for robust convex optimization problems. We give necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, and show some examples at which such duality results are used effectively. Moreover, we obtain a surrogate duality theorem and a surrogate min–max duality theorem for semi-definite optimization problems in the face of data uncertainty.     

A generalized duality and applications

The aim of this paper is to present a nonconvex duality with a zero gap and its connection with convex duality. Since a convex program can be regarded as a particular case of convex maximization over a convex set, a nonconvex duality can be regarded as a generalization of convex duality. The generalized duality can be obtained on the basis of convex duality and minimax theorems. The duality with a zero gap can be extended to a more general nonconvex problems such as a quasiconvex maximization over a general nonconvex set or a general minimization over the complement of a convex set. Several applications are given.On leave from the Institute of Mathematics, Hanoi, Vietnam.     

Duality for Multiobjective Optimization via Nonlinear Lagrangian Functions

In this paper, a strong nonlinear Lagrangian duality result is established for an inequality constrained multiobjective optimization problem. This duality result improves and unifies existing strong nonlinear Lagrangian duality results in the literature. As a direct consequence, a strong nonlinear Lagrangian duality result for an inequality constrained scalar optimization problem is obtained. Also, a variant set of conditions is used to derive another version of the strong duality result via nonlinear Lagrangian for an inequality constrained multiobjective optimization problem.     

Duality theory for preferences in multiobjective decisionmaking

A duality theory for convex multiobjective decisionmaking is developed. This duality theory is designed for a decisionmaker determining preferred solutions, while duality theories presented earlier have focused on how to generate the whole Pareto optimal set.     

Inverse conic programming with applications

Linear programming duality yields efficient algorithms for solving inverse linear programs. We show that special classes of conic programs admit a similar duality and, as a consequence, establish that the corresponding inverse programs are efficiently solvable. We discuss applications of inverse conic programming in portfolio optimization and utility function identification.     

群体多目标最优化的Lagrange对偶性

本文建立了目标和约束为不对称的群体多目标最优化问题的Lagrange对偶规划，在问题的联合弱有效解意义下，得到群体多目标最优化Lagrange型的弱对偶定理、基本对偶定理、直接对偶定理和逆对偶定理。     

Lagrange duality,stability and subdifferentials in vector optimization

Abstract

Lagrange duality theorems for vector and set optimization problems which are based on a consequent usage of infimum and supremum (in the sense of greatest lower and least upper bounds with respect to a partial ordering) have been recently proven. In this note, we provide an alternative proof of strong duality for such problems via suitable stability and subdifferential notions. In contrast to most of the related results in the literature, the space of dual variables is the same as in the scalar case, i.e. a dual variable is a vector rather than an operator. We point out that duality with operators is an easy consequence of duality with vectors as dual variables.     

向量最优化问题的Lagrange对偶与择一定理

本文讨论无限维向量最优化问题的Lagrange对偶与弱对偶，建立了若干鞍点定理与弱鞍点定理．作为研究对偶问题的工具，建立了一个新的择一定理．     

非凸优化问题的一个对偶结论

本文研究了单约束条件的非凸极小问题的对偶形式,我们的结论是通过变换,可以化成无缝对偶情形,同时我们研究了多约束条件的同类问题的处理方法。     

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.     

A necessary and sufficient condition for duality in control problems: KT-invex functionals

In this article, we introduce a new condition on functionals of a control problem, and for that purpose we define the KT-invex functionals. We extend recent optimality control works to the study of duality. In this way we establish weak, strong and converse duality results under KT-invexity. Furthermore, we prove that KT-invexity is not only a sufficient condition for establishing duality, but it is necessary.     

一类非光滑优化问题的最优性与对偶（英文）

本文研究了一类带等式和不等式约束的非光滑多目标优化问题,给出了该类问题的Karush-Kuhn-Tucker最优性必要条件和充分条件,建立了该类规划问题的一类混合对偶模型的弱对偶定理、强对偶定理、逆对偶定理、严格逆对偶定理和限制逆对偶定理.     

一类非光滑优化问题的最优性与对偶

本文研究了一类带等式和不等式约束的非光滑多目标优化问题,给出了该类问题的Karush-Kuhn-Tucker最优性必要条件和充分条件,建立了该类规划问题的一类混合对偶模型的弱对偶定理、强对偶定理、逆对偶定理、严格逆对偶定理和限制逆对偶定理.     

Lower level duality and the global solution of generalized semi-infinite programs

The reformulation of generalized semi-infinite programs (GSIP) to simpler problems is considered. These reformulations are achieved under the assumption that a duality property holds for the lower level program (LLP). Lagrangian duality is used in the general case to establish the relationship between the GSIP and a related semi-infinite program (SIP). Practical aspects of this reformulation, including how to bound the duality multipliers, are also considered. This SIP reformulation result is then combined with recent advances for the global, feasible solution of SIP to develop a global, feasible point method for the solution of GSIP. Reformulations to finite nonlinear programs, and the practical aspects of solving these reformulations globally, are also discussed. When the LLP is a linear program or second-order cone program, specific duality results can be used that lead to stronger results. Numerical examples demonstrate that the global solution of GSIP is computationally practical via the solution of these duality-based reformulations.     

On Duality Theorems for Nonsmooth Lipschitz Optimization Problems

A nonsmooth Lipschitz vector optimization problem (VP) is considered. Using the Fritz John type necessary optimality conditions for (VP), we formulate the Mond–Weir dual problem (VD) and establish duality theorems for (VP) and (VD) under (strict) pseudoinvexity assumptions on the functions. Our duality theorems do not require a constraint qualification.     

Zero duality gap in surrogate constraint optimization: A concise review of models

Surrogate constraint relaxation was proposed in the 1960s as an alternative to the Lagrangian relaxation for solving difficult optimization problems. The duality gap in the surrogate relaxation is always as good as the duality gap in the Lagrangian relaxation. Over the years researchers have proposed procedures to reduce the gap in the surrogate constraint. Our aim is to review models that close the surrogate duality gap. Five research streams that provide procedures with zero duality gap are identified and discussed. In each research stream, we will review major results, discuss limitations, and suggest possible future research opportunities. In addition, relationships between models if they exist, are also discussed.     

Scalarization and lagrange duality in multiobjective optimization

In this paper, we are concerned with scalarization and the Lagrange duality in multiobjective optimization. After exposing a property of a cone-subconvexlike function, we prove two theorems on scalarization and three theorems of the Lagrange duality.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments