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.     

B-(p,γ)-不变凸规划的最优性条件及混合型对偶

函数的广义凸性在数学规划及数学规划的对偶理论中起着非常重要的作用.在一种函数的广义凸性-关于n和b的B-(p,γ)-不变凸性的假设下,讨论了一类含有无穷多分式函数的约束广义分式规划及其对偶的某些问题:首先,给出并证明了这类约束广义分式规划的一个最优性充分条件,接着,针对这一类广义分式规划,提出了它的一个混合型对偶,然后又在适当的条件下,进一步给出并证明了相应的弱对偶定理,强对偶定理以及严格逆对偶定理.     

广义凸多目标规划的Mond─Weir型对偶性

本文研究带不等式和等式约束的多目标规划的Ｍｏｎｄ－Ｗｅｉｒ型对偶性理论。在目标和约束是广义凸的假设下，证明了弱对偶定理、直接对偶定理以及逆对偶定理     

B-(p,r)-不变凸规划的最优性条件及混合型对偶

函数的广义凸性在数学规划及数学规划的对偶理论中起着非常重要的作用.在一种函数的广义凸性—关于η和b的B-(p,r)-不变凸性的假设下,讨论了一类含有无穷多分式函数的约束广义分式规划及其对偶的某些问题:首先,给出并证明了这类约束广义分式规划的一个最优性充分条件,接着,针对这一类广义分式规划,提出了它的一个混合型对偶,然后又在适当的条件下,进一步给出并证明了相应的弱对偶定理,强对偶定理以及严格逆对偶定理.     

广义凸多目标规划的Mond—Weir型对偶性

本文研究带不等式和等式约束的多目标规划的Ｍｏｎｄ－Ｗｅｉｒ型对偶性理论。在目标和约束是广义凸的假设下，证明了弱对偶定理、直接对偶定理以及逆对偶定理。     

具V-不变凸性的一类多目标控制问题的混合对偶性

本文研究了一类多目标控制问题的混合对偶性.利用函数的广义V-不变凸性条件,得出了关于有效解的弱对偶定理、强对偶定理和严格逆对偶定理,推广了多目标控制问题的对偶性结论.     

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.     

均衡约束为二阶锥约束广义方程的数学规划问题的二阶充分条件(英)

本文考虑一类均衡约束为二阶锥约束广义方程的数学规划问题. 我们通过一个非光滑映射的方向导数, 给出了临界锥的定义, 并建立它在可行点处的等价形式. 基于此临界锥, 我们提出了均衡约束为二阶锥约束广义方程的数学规划问题的二阶充分性条件, 并且验证了在适当的条件下, M-稳定点处的二阶充分性条件是二阶增长条件成立的充分条件.     

群体多目标规划的联合Mond-Weir对偶

对于目标和约束均为不对称的群体多目标规划问题,本文研究它的联合有效解类 的Mond—Weir型对偶性,得到了相应的弱对偶定理、直接对偶定理和逆对偶定理.     

非可微凸规划的对偶问题

文章建立关于非可微凸规划的一个新的对偶问题,它不同于已知的对偶问题,文中证明了弱对偶性及强对偶性。并用Ｌａｇｒａｎｇｅ正则性证明了强对偶性的充要条件。最后,讨论了等式约束的情况。     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

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.     

Canonical duality for solving general nonconvex constrained problems

This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided.While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints. Some fundamental concepts such as the objectivity and Lagrangian in nonlinear programming are addressed.     

Equilibrium constrained optimization problems

We consider equilibrium constrained optimization problems, which have a general formulation that encompasses well-known models such as mathematical programs with equilibrium constraints, bilevel programs, and generalized semi-infinite programming problems. Based on the celebrated KKM lemma, we prove the existence of feasible points for the equilibrium constraints. Moreover, we analyze the topological and analytical structure of the feasible set. Alternative formulations of an equilibrium constrained optimization problem (ECOP) that are suitable for numerical purposes are also given. As an important first step for developing efficient algorithms, we provide a genericity analysis for the feasible set of a particular ECOP, for which all the functions are assumed to be linear.     

求解带均衡约束数学规划问题的一个连续化方法

In this paper, a continuation method for mathematical programs with equilibrium constraints (MPEC) is proposed. By using the KKT conditions for the variational inequality constraints, the MPEC is firstly reformulated as a nonsmooth constrained optimization problem, then we solve a sequence of smooth perturbation problems, which progressively approximate the nonsmooth problem, and study the convergence of the proposed method. Numerical results showing feasibility of the approach are given.     

Nonlinear Lagrange Duality Theorems and Penalty Function Methods In Continuous Optimization

We propose a general dual program for a constrained optimization problem via generalized nonlinear Lagrangian functions. Our dual program includes a class of general dual programs with explicit structures as special cases. Duality theorems with the zero duality gap are proved under very general assumptions and several important corollaries which include some known results are given. Using dual functions as penalty functions, we also establish that a sequence of approximate optimal solutions of the penalty function converges to the optimal solution of the original optimization problem.     

基于广义几何规划的最小广义信息密度区分问题及其对偶理论

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｑｕａｄｒａｔｉｃａｌｌｙｃｏｎｓｔｒａｉｎｅｄａｎｄｅｎｔｒｏｐｙｄｅｎｓｉｔｙｃｏｎｓｔｒａｉｎｅｄｑｕａｄｒａｔｉｃｐｒｏｇｒａｍｔｈａｔｉｓｇｏｉｎｇｔｏｂｅｓｔｕｄｉｅｄｉｎｔｈｉｓｐａｐｅｒｉｓｃｈａｒａｃｔｅｒｉｚｅｄａｓｔｈｅｆｏｌｌｏｗｉｎｇｆｏｒｍ :Ｐｒｏｇｒａｍ (Ｑ)(Ｑ)　　ｍｉｎ　Ｑ0 (ｚ)ｓ .ｔ .　Ｐｊ(ｚ)≤ 0 ,　ｊ =1 ,2 ,...,ｌ,Ｑｉ(ｚ) ≤ 0 ,　ｉ =1 ,2 ,...,ｒ ,ｚ=(ｚ1,...,ｚｎ) Ｔ ≥ 0 ,ｗｈｅｒｅＰｊ(ｚ) = ｎｋ =1ｚｋｌｏｇ ｚｋｅ…     

Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems

In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures regularized Lagrangian duality.     

求解含平衡约束数学规划的熵函数法

提出求解含平衡约束数学规划问题(简记为MPEC问题)的熵函数法，在将原问题等价改写为单层非光滑优化问题的基础上，通过熵函数逼近，给出求解MPEC问题的序列光滑优化方法，证明了熵函数逼近问题解的存在性和算法的全局收敛性，数值算例表明了算法的有效性。     

A closedness condition and its applications to DC programs with convex constraints

This paper concerns a closedness condition called (CC) involving a convex function and a convex constrained system. This type of condition has played an important role in the study of convex optimization problems. Our aim is to establish several characterizations of this condition and to apply them to study problems of minimizing a DC function under a cone-convex constraint and a set constraint. First, we establish several so-called ‘Toland–Fenchel–Lagrange’ duality theorems. As consequences, various versions of generalized Farkas lemmas in dual forms for systems involving convex and DC functions are derived. Then, we establish optimality conditions for DC problem under convex constraints. Optimality conditions for convex problems and problems of maximizing a convex function under convex constraints are given as well. Most of the results are established under the (CC) condition. This article serves as a link between several corresponding known ones published recently for DC programs and for convex programs.