含有锥约束的多目标变分问题的广义对称对偶性

本文研究了一类含有锥约束多目标变分问题的广义对称对偶性.利用函数的(F,ρ)-不变凸性的条件,得出了多目标变分问题关于有效解的弱对偶定理、强对偶定理和逆对偶定理,将多目标变分问题的对称对偶性理论推广到含有锥约束的广义对称对偶性上来.     

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.     

The minimal cone for conic linear programming

It is known that the minimal cone for the constraint system of a conic linear programming problem is a key component in obtaining strong duality without any constraint qualification. For problems in either primal or dual form, the minimal cone can be written down explicitly in terms of the problem data. However, due to possible lack of closure, explicit expressions for the dual cone of the minimal cone cannot be obtained in general. In the particular case of semidefinite programming, an explicit expression for the dual cone of the minimal cone allows for a dual program of polynomial size that satisfies strong duality. In this paper we develop a recursive procedure to obtain the minimal cone and its dual cone. In particular, for conic problems with so-called nice cones, we obtain explicit expressions for the cones involved in the dual recursive procedure. As an example of this approach, the well-known duals that satisfy strong duality for semidefinite programming problems are obtained. The relation between this approach and a facial reduction algorithm is also discussed.     

Notes on Duality in Second Order and p -Order Cone Optimization

Recently, the so-called second order cone optimization problem has received much attention, because the problem has many applications and the problem can in theory be solved efficiently by interior-point methods. In this note we treat duality for second order cone optimization problems and in particular whether a nonzero duality gap can be obtained when casting a convex quadratically constrained optimization problem as a second order cone optimization problem. Furthermore, we also discuss the p -order cone optimization problem which is a natural generalization of the second order case. Specifically, we suggest a new self-concordant barrier for the p -order cone optimization problem.     

Strong duality in robust semi-definite linear programming under data uncertainty

This article develops the deterministic approach to duality for semi-definite linear programming problems in the face of data uncertainty. We establish strong duality between the robust counterpart of an uncertain semi-definite linear programming model problem and the optimistic counterpart of its uncertain dual. We prove that strong duality between the deterministic counterparts holds under a characteristic cone condition. We also show that the characteristic cone condition is also necessary for the validity of strong duality for every linear objective function of the original model problem. In addition, we derive that a robust Slater condition alone ensures strong duality for uncertain semi-definite linear programs under spectral norm uncertainty and show, in this case, that the optimistic counterpart is also computationally tractable.     

ON CONE D.C. OPTIMIZATION AND CONJUGATE DUALITY

This paper derives first order necessary and sufficient conditions for unconstrained cone d.c. programming problems where the underlined space is partially ordered with respect to a cone. These conditions are given in terms of directional derivatives and subdifferentials of the component functions. Moreover, conjugate duality for cone d.c. optimization is discussed and weak duality theorem is proved in a more general partially ordered linear topological vector space (generalizing the results in [11]).     

Modified regularity conditions for nonlinear programming problems over mixed cone domains

In earlier results by Sposito and David, Kuhn—Tucker duality was established over nondegenerate cone domains (not necessarily polyhedral) without differentiability under a certain natural modification of the Slater condition, in addition to the convexity of a certain auxiliary set. This note extends Kuhn—Tucker duality to optimization problems with both nondegenerate and degenerate cone domains. Moreover, under a different condition than presented in earlier results by the author, this note develops Kuhn—Tucker duality for a certain class of nonlinear problems with linear constraints and an arbitrary objective function.     

Second order duality for multiobjective programming with cone constraints

We focus on second order duality for a class of multiobjective programming problem subject to cone constraints. Four types of second order duality models are formulated. Weak and strong duality theorems are established in terms of the generalized convexity, respectively. Converse duality theorems, essential parts of duality theory, are presented under appropriate assumptions. Moreover, some deficiencies in the work of Ahmad and Agarwal (2010) are discussed.     

锥约束多目标规划的二阶对称对偶性质

给出一对锥约束多目标非线性规划的二阶对称对偶问题,以及二阶F凸函数类的概念.在二阶F凸假设下证明了真有效解的对偶性质———弱对偶性、强对偶性及逆对偶性.     

On Set Containment Characterization and Constraint Qualification for Quasiconvex Programming

Dual characterizations of the containment of a convex set with quasiconvex inequality constraints are investigated. A new Lagrange-type duality and a new closed cone constraint qualification are described, and it is shown that this constraint qualification is the weakest constraint qualification for the duality.     

Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone

We define weakly minimal elements of a set with respect to a convex cone by means of the quasi-interior of the cone and characterize them via linear scalarization, generalizing the classical weakly minimal elements from the literature. Then we attach to a general vector optimization problem, a dual vector optimization problem with respect to (generalized) weakly efficient solutions and establish new duality results. By considering particular cases of the primal vector optimization problem, we derive vector dual problems with respect to weakly efficient solutions for both constrained and unconstrained vector optimization problems and the corresponding weak, strong and converse duality statements.     

Strong duality and minimal representations for cone optimization

The elegant theoretical results for strong duality and strict complementarity for linear programming, LP, lie behind the success of current algorithms. In addition, preprocessing is an essential step for efficiency in both simplex type and interior-point methods. However, the theory and preprocessing techniques can fail for cone programming over nonpolyhedral cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, CQ, and strict complementarity, for linear cone optimization problems in finite dimensions. One theme is the notion of minimal representation of the cone and the constraints. This provides a framework for preprocessing cone optimization problems in order to avoid both the theoretical and numerical difficulties that arise due to the (near) loss of the strong CQ, strict feasibility. We include results and examples on the surprising theoretical connection between duality gaps in the original primal-dual pair and lack of strict complementarity in their homogeneous counterpart. Our emphasis is on results that deal with Semidefinite Programming, SDP.     

向量集值优化超有效解的对偶问题

借助于Contingent切锥和集值映射的上图而引入的有关集值映射的Contingent切导数，对约束集值优化问题的超有效解建立了最优性Kuhn Tucker必要及充分性条件,借此建立了向量集值优化超有效解的Wolfe型和Mond Weir型对偶定理.     

On duality theory of convex semi-infinite programming

In this article we discuss weak and strong duality properties of convex semi-infinite programming problems. We use a unified framework by writing the corresponding constraints in a form of cone inclusions. The consequent analysis is based on the conjugate duality approach of embedding the problem into a parametric family of problems parameterized by a finite-dimensional vector.     

Non-differentiable symmetric duality for multiobjective programming with cone constraints

Two pairs of non-differentiable multiobjective symmetric dual problems with cone constraints over arbitrary cones, which are Wolfe type and Mond–Weir type, are considered. On the basis of weak efficiency with respect to a convex cone, we obtain symmetric duality results for the two pairs of problems under cone-invexity and cone-pseudoinvexity assumptions on the involved functions. Our results extend the results in Khurana [S. Khurana, Symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research 165 (2005) 592–597] to the non-differentiable multiobjective symmetric dual problem.     

Lagrangian Duality and Cone Convexlike Functions

In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called image space approach initiated by Giannessi.     

向量值最优化问题的最优性条件与对偶性

本文我们首先给出一类向量值优化问题（VP）的正切锥真有效解的定义，在锥方向导数的假设下，讨论了一类单目标问题 的最优性必要条件；然后利用正切锥方向导数定义一类正切锥F－凸函数类，并给出了（VP）正切锥真有效解的充分性条件，最后我们亦讨论了（VP）在正切锥真有效解意义下的对偶性质。     

关于向量集值优化的Benson真有效性

对广义锥次数凸向量集值优化问题Benson真有效性解的标量化问题进行了研究,借助于一种新的择一性定理建立了广义锥次类凸向是集值优化问题Benson真有效解的Lagrange乘子型定理并讨论了乘子型对偶问题。     

Higher-order Mond-Weir converse duality in multiobjective programming involving cones

In this work,we established a converse duality theorem for higher-order Mond-Weir type multiobjective programming involving cones.This flls some gap in recently work of Kim et al.[Kim D S,Kang H S,Lee Y J,et al.Higher order duality in multiobjective programming with cone constraints.Optimization,2010,59:29–43].     

Duality and Lipschitzian selections in best approximation from nonconvex cones

Duality relationships in finding a best approximation from a nonconvex cone in a normed linear space in general and in the space of bounded functions in particular, are investigated. The cone and the dual problems are defined in terms of positively homogeneous super-additive functional on the space. Conditions are developed on the cone so that the duality gap between a pair of primal and dual problems does not exist. In addition, Lipschitz continuous selections of the metric projection are identified. The results are specialized to a convex cone. Applications are indicated to approximation problems.