半无限规划的一个非线性Lagrangian对偶模型

本文给出半无限规划的一个对偶罚函数模型,该模型能处理目标函数不是凸函数的情形,从而凸(SIP)对偶为该模型的一个特例.并且,作为罚函数,本模型的罚因子比l1-罚函数要小,这使得算法更可行,最后,给出零对偶间隙证明.     

Perfect duality for convexlike programs

The minimizing problem for a convex program has a dual problem, that is, the maximizing problem of the Lagrangian. Although these problems have a duality gap in general, the duality gap can be eliminated by relaxing the constraint of the minimizing problem, so that the constraint is enforced only in the limit. We extend this result to the convexlike case. Moreover, we obtain a necessary condition for optimality for minimizing problems whose objective function and constraint mapping have convex Gateaux derivative.The authors are indebted to Professor W. Takahashi of Tokyo Institute of Technology for his valuable comments, and also to the referees for their helpful suggestions.     

一类稀疏约束非线性规划的约束规格

研究一类带有闭凸集约束的稀疏约束非线性规划问题,这类问题在变量选择、模式识别、投资组合等领域具有广泛的应用.首先引进了限制性Slater约束规格的概念,证明了该约束规格强于限制性M-F约束规格,然后在此约束规格成立的条件下,分析了其局部最优解成立的充分和必要条件.最后,对约束集合的两种具体形式,指出限制性Slater约束规格必满足,并给出了一阶必要性条件的具体表达形式.     

Duality and Penalization in Optimization via an Augmented Lagrangian Function with Applications

This paper aims to establish duality and exact penalization results for the primal problem of minimizing an extended real-valued function in a reflexive Banach space in terms of a valley-at-0 augmented Lagrangian function. It is shown that every weak limit point of a sequence of optimal solutions generated by the valley-at-0 augmented Lagrangian problems is a solution of the original problem. A zero duality gap property and an exact penalization representation between the primal problem and the valley-at-0 augmented Lagrangian dual problem are obtained. These results are then applied to an inequality and equality constrained optimization problem in infinite-dimensional spaces and variational problems in Sobolev spaces, respectively. The first author was supported by the Research Committee of Hong Kong Polytechnic University, by Grant 10571174 from the National Natural Science Foundation of China and Grant 08KJB11009 from the Jiangsu Education Committee of China. The second author was supported by Grant BQ771 from the Research Grants Council of Hong Kong. We are grateful to the referees for useful suggestions which have contributed to the final presentation of the paper.     

Augmented Lagrangian function, non-quadratic growth condition and exact penalization

In this paper, under the assumption that the perturbation function satisfies a growth condition, necessary and sufficient conditions for an exact penalty representation and a zero duality gap property between the primal problem and its augmented Lagrangian dual problem are established.     

Optimality conditions for nonsmooth mathematical programs with equilibrium constraints,using convexificators

In this paper, we deal with constraint qualifications, stationary concepts and optimality conditions for a nonsmooth mathematical program with equilibrium constraints (MPEC). The main tool in our study is the notion of convexificator. Using this notion, standard and MPEC Abadie and several other constraint qualifications are proposed and a comparison between them is presented. We also define nonsmooth stationary conditions based on the convexificators. In particular, we show that GS-stationary is the first-order optimality condition under generalized standard Abadie constraint qualification. Finally, sufficient conditions for global or local optimality are derived under some MPEC generalized convexity assumptions.     

极小扰动约束非线性规划的最优性条件

考虑当目标函数在约束条件下的最优值作扰动时，使各约束作极小扰动的非线性规划问题．文中引进了极小扰动约束规划的极小扰动有效解概念．利用把问题归为一个相应的多目标规划问题，给出了极小扰动约束有效解的最优性条件．     

Partial Augmented Lagrangian Method and Mathematical Programs with Complementarity Constraints

In this paper, we apply a partial augmented Lagrangian method to mathematical programs with complementarity constraints (MPCC). Specifically, only the complementarity constraints are incorporated into the objective function of the augmented Lagrangian problem while the other constraints of the original MPCC are retained as constraints in the augmented Lagrangian problem. We show that the limit point of a sequence of points that satisfy second-order necessary conditions of the partial augmented Lagrangian problems is a strongly stationary point (hence a B-stationary point) of the original MPCC if the limit point is feasible to MPCC, the linear independence constraint qualification for MPCC and the upper level strict complementarity condition hold at the limit point. Furthermore, this limit point also satisfies a second-order necessary optimality condition of MPCC. Numerical experiments are done to test the computational performances of several methods for MPCC proposed in the literature. This research was partially supported by the Research Grants Council (BQ654) of Hong Kong and the Postdoctoral Fellowship of The Hong Kong Polytechnic University. Dedicated to Alex Rubinov on the occassion of his 65th birthday.     

Duality and Exact Penalization for Vector Optimization via Augmented Lagrangian

In this paper, we introduce an augmented Lagrangian function for a multiobjective optimization problem with an extended vector-valued function. On the basis of this augmented Lagrangian, set-valued dual maps and dual optimization problems are constructed. Weak and strong duality results are obtained. Necessary and sufficient conditions for uniformly exact penalization and exact penalization are established. Finally, comparisons of saddle-point properties are made between a class of augmented Lagrangian functions and nonlinear Lagrangian functions for a constrained multiobjective optimization problem.     

非线性规划问题的精确增广Lagrange函数

对于一般的非线性规划给出一种精确增广Lagrange函数,并讨论其性质.无需假设严格互补条件成立,给出了原问题的局部极小点与增广Lagrange函数在原问题的变量空间上的局部极小的关系.进一步,在适当的假设条件下,建立了两者的全局最优解之间的关系.     

The strong conical hull intersection property for convex programming

The strong conical hull intersection property (CHIP) is a geometric property of a collection of finitely many closed convex intersecting sets. This basic property, which was introduced by Deutsch et al. in 1997, is one of the central ingredients in the study of constrained interpolation and best approximation. In this paper we establish that the strong CHIP of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems. We first show that a sharpened strong CHIP is necessary and sufficient for a complete Lagrange multiplier characterization of optimality for the convex programming model problem where C is a closed convex subset of a Banach space X, S is a closed convex cone which does not necessarily have non-empty interior, Y is a Banach space, is a continuous convex function and g:X→Y is a continuous S-convex function. We also show that the strong CHIP completely characterizes the strong duality for partially finite convex programs, where Y is finite dimensional and g(x)=−Ax+b and S is a polyhedral convex cone. Global sufficient conditions which are strictly weaker than the Slater type conditions are given for the strong CHIP and for the sharpened strong CHIP. The author is grateful to the referees for their constructive comments and valuable suggestions which have contributed to the final preparation of the paper.     

E-凸多目标规划的最优性及Wolfe型对偶

先引入了一类带不等式和等式约束的E-凸多目标优化问题(MP),给出了该类问题的一个最优性充分条件;其次,建立了该类规划问题(MP)的一类Wolfe型对偶模型(WD),并在E-凸条件下,获得了弱对偶定理,强对偶定理和逆对偶定理.     

Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints

Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper, we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ, discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ.Communicated by Z. Q. Luo     

一类Lagrangian对偶问题的零对偶间隙性质及其最优路径的收敛性

针对一般的非线性规划问题,利用某些Lagrange型函数给出了一类Lagrangian对偶问题的一般模型,并证明它与原问题之间存在零对偶间隙．针对具体的一类增广La- grangian对偶问题以及几类由非线性卷积函数构成的Lagrangian对偶问题,详细讨论了零对偶间隙的存在性．进一步,讨论了在最优路径存在的前提下,最优路径的收敛性质．     

Nonlinear Lagrangian Theory for Nonconvex Optimization

The Lagrangian function in the conventional theory for solving constrained optimization problems is a linear combination of the cost and constraint functions. Typically, the optimality conditions based on linear Lagrangian theory are either necessary or sufficient, but not both unless the underlying cost and constraint functions are also convex.We propose a somewhat different approach for solving a nonconvex inequality constrained optimization problem based on a nonlinear Lagrangian function. This leads to optimality conditions which are both sufficient and necessary, without any convexity assumption. Subsequently, under appropriate assumptions, the optimality conditions derived from the new nonlinear Lagrangian approach are used to obtain an equivalent root-finding problem. By appropriately defining a dual optimization problem and an alternative dual problem, we show that zero duality gap will hold always regardless of convexity, contrary to the case of linear Lagrangian duality.     

Convexificators and boundedness of the Kuhn–Tucker multipliers set

In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded.     

Efficiency and Solution Approaches to Bicriteria Nonconvex Programs

A new nonlinear scalarization specially designed for bicriteria nonconvexprogramming problems is presented. The scalarization is based on generalizedLagrangian duality theory and uses an augmented Lagrange function. The newconcepts, q _i-approachable points and augmented duality gap, are introducedin order to determine the location of nondominated solutions with respect to aduality gap as well as the connectedness of the nondominated set.     

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.