On constraint qualifications for second-order optimality conditions depending on a single Lagrange multiplier

We present new constraint qualifications (CQs) to ensure the validity of some well-known second-order optimality conditions. Our main interest is on second-order conditions that can be associated with numerical methods for solving constrained optimization problems. Such conditions depend on a single Lagrange multiplier, instead of the whole set of Lagrange multipliers. For each condition, we characterize the weakest CQ that guarantees its fulfillment at local minimizers, while proposing new weak conditions implying them. Relations with other CQs are discussed.     

On alternative theorems and necessary conditions for efficiency

In this article, we establish theorems of the alternative for a system described by inequalities, equalities and a set inclusion, which are generalizations of Tucker's classical theorem of the alternative, and develop Kuhn–Tucker necessary conditions for efficiency to mathematical programs in normed linear spaces involving inequality, equality and set constraints with positive Lagrange multipliers of all the components of objective functions.     

Necessary and sufficient conditions for regularity of constraints in convex programming

This paper derives some necessary and sufficient conditions for (Lagrangian) regularity of the nondifferentiable convex programming problem. Furthermore, some weakest constraint qualifications are presented using the supporting functions and their derivatives, the outer normal cones, the single constraint function and its directional derivatives and epigraph and the projections of the outer normal cones     

基于蕴涵算子族L-λ-R0的模糊推理反向α-三I约束算法

提出了基于蕴涵算子族L-λ-R0的模糊推理的思想,这将有助于提高推理结果的可靠性.针对蕴涵算子族L-λ-R0给出了模糊推理的FMP模型及FMT模型的反向α-三I约束算法.     

国家教师资格考试制度下化学教师教育课程优化探讨

以国家教师资格考试大纲和《教师教育课程标准》为本,针对参加化学教师资格考试人群提出备考建议。以样本院校化学专业职前教师培养方案为基础,提出该学科调整课程结构,调整学科教育类课程比例,调整课程开设时间,突出理论与实践相结合;通过开设不同层次、不同内容和形式的学科教育课程,促进化学专业职前教师教育课程能与“国考”有效对接,以适应制度创新带来的新变化,提高化学教师培养的适应性和针对性。     

Some theoretical limitations of second-order algorithms for smooth constrained optimization

In second-order algorithms, we investigate the relevance of the constant rank of the full set of active constraints in ensuring global convergence to a second-order stationary point under a constraint qualification. We show that second-order stationarity is not expected in the non-constant rank case if the growth of so-called tangent AKKT2 sequences is not controlled. Since no algorithm controls their growth, we argue that there is a theoretical limitation of algorithms in finding second-order stationary points without constant rank assumptions.     

Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators

The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function.     

Principles for the Design of Large Neighborhood Search

Large neighborhood search (LNS) is a combination of constraint programming (CP) and local search (LS) that has proved to be a very effective tool for solving complex optimization problems. However, the practice of applying LNS to real world problems remains an art which requires a great deal of expertise. In this paper, we show how adaptive techniques can be used to create algorithms that adjust their behavior to suit the problem instance being solved. We present three design principles towards this goal: cost-based neighborhood heuristics, growing neighborhood sizes, and the application of learning algorithms to combine portfolios of neighborhood heuristics. Our results show that the application of these principles gives strong performance on a challenging set of job shop scheduling problems. More importantly, we are able to achieve robust solving performance across problem sets and time limits. This material is based upon works supported by the Science Foundation Ireland under Grant No. 00/PI.1/C075, the Embark Initiative of the Irish Research Council of Science Engineering and Technology under Grant PD2002/21, and ILOG, S.A.     

解互补约束优化问题的一种新的光滑化近似方法

互补约束优化问题应用十分广泛.利用Sigmoid函数的积分函数提出了一种新的光滑化近似算法,将互补约束优化问题转化为一般的非线性规划近似问题,通过求解近似问题的一系列光滑子问题得到原问题的近似解.在线性独立约束规范和其他一些较弱的假设条件下：无须上水平严格互补和渐进弱非退化,证明了光滑近似问题的KKT稳定点序列收敛于原问题的C-稳定点.进而考虑弱二阶必要条件,证明了上述KKT稳定点序列收敛于原问题的S-稳定点.最后,设计了相应算法,并对MacMPEC测试题库中的一些算例进行了数值实验,将得到的结果与其他算法的结果进行比较,显示本方法是有效的.     

Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification

A global error bound is given on the distance between an arbitrary point in then-dimensional real spaceR ⁿ and its projection on a nonempty convex set determined bym convex, possibly nondifferentiable, inequalities. The bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong Slater constraint qualification (CQ) which implies the ordinary Slater CQ. A very simple bound on the distance to the projection relative to the distance to a point satisfying the ordinary Slater CQ is given first and then used to derive the principal global error bound. This material is based on research supported by National Science Foundation Grant CCR-9322479 and Air Force Office of Scientific Research grant F49620-97-1-0326.