Constraint qualifications for constrained Lipschitz optimization problems and applications to a MPCC

Three constraint qualifications (the weak generalized Robinson constraint qualification, the bounded constraint qualification, and the generalized Abadie constraint qualification), which are weaker than the generalized Robinson constraint qualification (GRCQ) given by Yen (1997) [1], are introduced for constrained Lipschitz optimization problems. Relationships between those constraint qualifications and the calmness of the solution mapping are investigated. It is demonstrated that the weak generalized Robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping, whereas the proposed generalized Abadie constraint qualification, described in terms of graphical derivatives in variational analysis, is weaker than the calmness of the solution mapping. Finally, those constraint qualifications are written for a mathematical program with complementarity constraints (MPCC), and new constraint qualifications ensuring the C-stationary point condition of a MPCC are obtained.     

New Constraint Qualifications and Optimality Conditions for Second Order Cone Programs

We introduce three new constraint qualifications for nonlinear second order cone programming problems that we call constant rank constraint qualification, relaxed constant rank constraint qualification and constant rank of the subspace component condition. Our development is inspired by the corresponding constraint qualifications for nonlinear programming problems. We provide proofs and examples that show the relations of the three new constraint qualifications with other known constraint qualifications. In particular, the new constraint qualifications neither imply nor are implied by Robinson’s constraint qualification, but they are stronger than Abadie’s constraint qualification. First order necessary optimality conditions are shown to hold under the three new constraint qualifications, whereas the second order necessary conditions hold for two of them, the constant rank constraint qualification and the relaxed constant rank constraint qualification.

     

Abadie Constraint Qualifications for Convex Constraint Systems and Applications to Calmness Property

In this paper, we mainly study concepts of Abadie constraint qualification and strong Abadie constraint qualification for a convex constraint system defined by a closed convex multifunction and a closed convex subset. These concepts can cover Abadie constraint qualifications for the feasible region of convex optimization problem and the convex multifunction. Several characterizations for these Abadie constraint qualifications are also provided. As applications, we use these Abadie constraint qualifications to characterize calmness properties of the convex constraint system.     

线性规划的约束条件“滚雪球”算法

为使线性规划的每个约束条件部分或全部地拥有原整个约束条件所包含的信息,将线性规划的约束条件“滚雪球”后得到与原约束条件等价的新约束条件,对新约束条件所构成的线性规划采用目标函数最速递减算法.有一定规模的随机数值算例显示了该算法只需进行m(约束条件数)次迭代即可求得最优解.     

On Several Types of Basic Constraint Qualifications via Coderivatives for Generalized Equations

In this paper, we study several types of basic constraint qualifications in terms of Clarke/Fréchet coderivatives for generalized equations. Several necessary and/or sufficient conditions are given to ensure these constraint qualifications. It is proved that basic constraint qualification and strong basic constraint qualification for convex generalized equations can be obtained by these constraint qualifications, and the existing results on constraint qualifications for the inequality system can be deduced from the given conditions in this paper. The main work of this paper is an extension of the study on constraint qualifications from inequality systems to generalized equations.     

均衡约束数学规划的约束规格和最优性条件综述

约束规格在约束优化问题的最优性条件中起着重要的作用,介绍了近几年国际上关于均衡约束数学规划(简记为MPEC)的约束规格以及最优性条件的研究成果, 包括以下主要内容: (1) MPEC常用的约束规格(如线性无关约束规格 (MPEC-LICQ)、Mangasarian-Fromovitz约束规格 (MPEC-MFCQ)等)和新的约束规格(如恒秩约束规格、常数正线性相关约束规格等), 以及它们之间的关系; (2) MPEC常用的稳定点; (3) MPEC的最优性条件. 最后还对MPEC的约束规格和最优性条件的研究前景进行了探讨.     

Necessary and Sufficient Constraint Qualification for Surrogate Duality

In mathematical programming, constraint qualifications are essential elements for duality theory. Recently, necessary and sufficient constraint qualifications for Lagrange duality results have been investigated. Also, surrogate duality enables one to replace the problem by a simpler one in which the constraint function is a scalar one. However, as far as we know, a necessary and sufficient constraint qualification for surrogate duality has not been proposed yet. In this paper, we propose necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, which are closely related with ones for Lagrange duality.     

Linear and nonlinear constructions of DNA codes with Hamming distance d,constant GC-content and a reverse-complement constraint

In a previous paper, the authors used cyclic and extended cyclic constructions to obtain codes over an alphabet {A,C,G,T} satisfying a Hamming distance constraint and a GC-content constraint. These codes are applicable to the design of synthetic DNA strands used in DNA microarrays, as DNA tags in chemical libraries and in DNA computing. The GC-content constraint specifies that a fixed number of positions are G or C in each codeword, which ensures uniform melting temperatures. The Hamming distance constraint is a step towards avoiding unwanted hybridizations. This approach extended the pioneering work of Gaborit and King. In the current paper, another constraint known as a reverse-complement constraint is added to further prevent unwanted hybridizations.Many new best codes are obtained, and are reproducible from the information presented here. The reverse-complement constraint is handled by searching for an involution with 0 or 1 fixed points, as first done by Gaborit and King. Linear codes and additive codes over GF(4) and their cosets are considered, as well as shortenings of these codes. In the additive case, codes obtained from two different mappings from GF(4) to {A,C,G,T} are considered.     

低碳约束下湖南省产业结构优化研究

在湖南省"十二五"规划工业产业节能减排的背景下,如何在碳减排目标的基础上,实现产业结构优化,成为湖南省面临的重大挑战.本文构建低碳约束下湖南省主导产业选择指标体系,计算并选取湖南省无低碳约束和有低碳约束下的主导产业.进一步,本文设计无低碳约束、弱低碳约束和强低碳约束三种情景模拟,分析不同的低碳约束条件对湖南省主导产业选择的影响.本文结论如下:1)无低碳约束条件下,湖南省主导产业包括有色金属冶炼及压延加工业等碳生产力较低的行业,低碳约束条件下,湖南省主导产业包括食品制造业等碳生产力较高的行业.这表明,考虑低碳约束条件,湖南省主导产业由碳生产力较低的行业向碳生产力较高的行业转变.2)情景模拟结果表明,一些传统支柱型产业仍是湖南省产业结构优化中需重点发展的产业,且随着低碳政策的深化和节能减排目标的扩大,战略性产业如医药制造业成为主导产业,在产业结构优化中发挥中坚作用.     

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.     

On Constraint Qualification in Multiobjective Optimization Problems: Semidifferentiable Case

Some versions of constraint qualifications in the semidifferentiable case are considered for a multiobjective optimization problem with inequality constraints. A Maeda-type constraint qualification is given and Kuhn–Tucker-type necessary conditions for efficiency are obtained. In addition, some conditions that ensure the Maeda-type constraint qualification are stated.     

Computational Discretization Algorithms for Functional Inequality Constrained Optimization

In this paper, a functional inequality constrained optimization problem is studied using a discretization method and an adaptive scheme. The problem is discretized by partitioning the interval of the independent parameter. Two methods are investigated as to how to treat the discretized optimization problem. The discretization problem is firstly converted into an optimization problem with a single nonsmooth equality constraint. Since the obtained equality constraint is nonsmooth and does not satisfy the usual constraint qualification condition, relaxation and smoothing techniques are used to approximate the equality constraint via a smooth inequality constraint. This leads to a sequence of approximate smooth optimization problems with one constraint. An adaptive scheme is incorporated into the method to facilitate the computation of the sum in the inequality constraint. The second method is to apply an adaptive scheme directly to the discretization problem. Thus a sequence of optimization problems with a small number of inequality constraints are obtained. Convergence analysis for both methods is established. Numerical examples show that each of the two proposed methods has its own advantages and disadvantages over the other.     

Existence and Boundedness of the Kuhn-Tucker Multipliers in Nonsmooth Multiobjective Optimization

Using the idea of upper convexificators, we propose constraint qualifications and study existence and boundedness of the Kuhn-Tucker multipliers for a nonsmooth multiobjective optimization problem with inequality constraints and an arbitrary set constraint. We show that, at locally weak efficient solutions where the objective and constraint functions are locally Lipschitz, the constraint qualifications are necessary and sufficient conditions for the Kuhn-Tucker multiplier sets to be nonempty and bounded under certain semiregularity assumptions on the upper convexificators of the functions.     

Constraint satisfaction problems: Algorithms and applications

A constraint satisfaction problem (CSP) requires a value, selected from a given finite domain, to be assigned to each variable in the problem, so that all constraints relating the variables are satisfied. Many combinatorial problems in operational research, such as scheduling and timetabling, can be formulated as CSPs. Researchers in artificial intelligence (AI) usually adopt a constraint satisfaction approach as their preferred method when tackling such problems. However, constraint satisfaction approaches are not widely known amongst operational researchers. The aim of this paper is to introduce constraint satisfaction to the operational researcher. We start by defining CSPs, and describing the basic techniques for solving them. We then show how various combinatorial optimization problems are solved using a constraint satisfaction approach. Based on computational experience in the literature, constraint satisfaction approaches are compared with well-known operational research (OR) techniques such as integer programming, branch and bound, and simulated annealing.     

Locally Farkas–Minkowski Systems in Convex Semi-Infinite Programming

A pair of constraint qualifications in convex semi-infinite programming, namely the locally Farkas–Minkowski constraint qualification and generalized Slater constraint qualification, are studied in the paper. We analyze the relationship between them, as well as the behavior of the so-called active and sup-active mappings, accounting for the tightness of the constraint system at each point of the variables space. The generalized Slater constraint qualification guarantees a regular behavior of the supremum function (defined as supremum of the infinitely many functions involved in the constraint system), giving rise to the well-known Valadier formula.     

Second-Order Conditions for Efficiency in Nonsmooth Multiobjective Optimization Problems

In this paper, we are concerned with a nonsmooth multiobjective optimization problem with inequality constraints. We introduce a second-order constraint qualification, which is a generalization of the Abadie constraint qualification and derive second-order Kuhn-Tucker type necessary conditions for efficiency under the constraint qualification. Moreover, we give some conditions which ensure the constraint qualification holds.     

Bounded knapsack sharing

A bounded knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more linear inequality constraints, an objective function composed of single variable continuous functions called tradeoff functions, and lower and upper bounds on the variables. A single constraint problem which can have negative or positive constraint coefficients and any type of continuous tradeoff functions (including multi-modal, multiple-valued and staircase functions) is considered first. Limiting conditions where the optimal value of a variable may be plus or minus infinity are explicitly considered. A preprocessor procedure to transform any single constraint problem to a finite form problem (an optimal feasible solution exists with finite variable values) is developed. Optimality conditions and three algorithms are then developed for the finite form problem. For piecewise linear tradeoff functions, the preprocessor and algorithms are polynomially bounded. The preprocessor is then modified to handle bounded knapsack sharing problems with multiple constraints. An optimality condition and algorithm is developed for the multiple constraint finite form problem. For multiple constraints, the time needed for the multiple constraint finite form algorithm is the time needed to solve a single constraint finite form problem multiplied by the number of constraints. Some multiple constraint problems cannot be transformed to multiple constraint finite form problems.     

The Tangent cones on constraint qualifications in optimization problems

This article proposes a few tangent cones,which are relative to the constraint qualifications of optimization problems.With the upper and lower directional derivatives of an objective function,the characteristics of cones on the constraint qualifications are presented.The interrelations among the constraint qualifications,a few cones involved, and level sets of upper and lower directional derivatives are derived.     

On Error Bounds for Quasinormal Programs

The Mangasarian-Fromovitz constraint qualification is a central concept within the theory of constraint qualifications in nonlinear optimization. Nevertheless there are problems where this condition does not hold though other constraint qualifications can be fulfilled. One of such constraint qualifications is the so-called quasinormality by Hestenes. The well known error bound property (R-regularity) can also play the role of a general constraint qualification providing the existence of Lagrange multipliers. In this note we investigate the relation between some constraint qualifications and prove that quasinormality implies the error bound property, while the reciprocal is not true.     

约束半环诱导的赋值代数的轮廓解及其算法

本文研究了约束半环所诱导的赋值代数的轮廓解及其算法的问题.利用约束半环的性质,以及基于记忆约束半环赋值的方法,获得了约束半环所诱导的赋值代数的轮廓解的概念,性质以及算法的相关结论,推广了文献[2]关于全序幂等半环诱导的赋值代数的轮廓解的结果.