Existence and Convergence of Pareto Minima

In the context of vector optimization for functions with values in an ordered topological vector space, we give a result for the existence of global minima. Moreover, we find a set of conditions ensuring the convergence of minimal points and minimal values. More general assumptions are excluded by several counterexamples.The authors thank Professor F. Giannessi for useful suggestions.Communicated by F. Giannessi     

New Modified Function Method for Global Optimization

In this paper, a class of global optimization problems is considered. Corresponding to each local minimizer obtained, we introduced a new modified function and construct a corresponding optimization subproblem with one constraint. Then, by applying a local search method to the one-constraint optimization subproblem and using the local minimizer as the starting point, we obtain a better local optimal solution. This process is continued iteratively. A termination rule is obtained which can serve as stopping criterion for the iterating process. To demonstrate the efficiency of the proposed approach, numerical examples are solved.This research was partially supported by the National Science Foundation of China, Grant 10271073.     

On functions whose stationary points are global minima

In this paper, a characterization of functions whose stationary points are global minima is studied. By considering the level sets of a real function as a point-to-set mapping, and by examining its semicontinuity properties, we obtain the result that a real function, defined on a subset ofR ⁿ and satisfying some mild regularity conditions, belongs to the above family iff the point-to-set mapping of its level sets is strictly lower semicontinuous. Mathematical programming applications are also mentioned.The authors are thankful to an anonymous referee of an earlier version of this paper for his valuable comments. This research was partially supported by the Office of Naval Research, Contract No. N-00014-75-C-0267, by the National Science Foundation, Grant No. MPS-71-03341-A03, and by the US Energy Research and Development Administration, Contract No. E(04-3)-326 PA-18. An earlier version of this paper appeared as CORE Discussion Paper No. 7502. Part of this research was carried out while the first author was at CORE, Louvain, Belgium.     

Generalized weak sharp minima in cone-constrained convex optimization on Hadamard manifolds

In this paper, a convex optimization problem with cone constraint (for short, CPC) is introduced and studied on Hadamard manifolds. Some criteria and characterizations for the solution set to be a set of generalized global weak sharp minima, generalized local weak sharp minima and generalized bounded weak sharp minima for (CPC) are derived on Hadamard manifolds.     

Generalized Connected Functions with Respect to Cones

In this paper, generalized connected functions with respect to cones such as quasi cone-connected, pseudo cone-connected, strongly pseudo cone-connected, and strictly pseudo cone-connected functions are introduced; necessary and sufficient optimality conditions are obtained for a weak minimum, a minimum, and a strong minimum of a vector-valued minimization problem. A Mond–Weir type dual is associated, and weak and strong duality results are established.     

格中依次最短无关组与Minkowski约化基等价的充分条件

本文讨论了格中基子集、依次最短无关组及Minkowski约化基之间的向量长度关系,利用无关组与基之间的一些制约性质,给出了Minkowski约化基达到依次最短长度,以及依次最短无关组成为Minkowski约化基的一些充分条件.     

Optimal Bang-Bang Controls for a Two-Compartment Model in Cancer Chemotherapy

A class of mathematical models for cancer chemotherapy which have been described in the literature take the form of an optimal control problem over a finite horizon with control constraints and dynamics given by a bilinear system. In this paper, we analyze a two-dimensional model in which the cell cycle is broken into two compartments. The cytostatic agent used as control to kill the cancer cells is active only in the second compartment where cell division occurs and the cumulative effect of the drug is used to model the negative effect of the treatment on healthy cells. It is shown that singular controls are not optimal for this model and the optimality properties of bang-bang controls are established. Specifically, transversality conditions at the switching surfaces are derived. In a nondegenerate setting, these conditions guarantee the local optimality of the flow if satisfied, while trajectories will be no longer optimal if they are violated.     

On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian

Mangasarian and Solodov (Ref. 1) proposed to solve nonlinear complementarity problems by seeking the unconstrained global minima of a new merit function, which they called implicit Lagrangian. A crucial point in such an approach is to determine conditions which guarantee that every unconstrained stationary point of the implicit Lagrangian is a global solution, since standard unconstrained minimization techniques are only able to locate stationary points. Some authors partially answered this question by giving sufficient conditions which guarantee this key property. In this paper, we settle the issue by giving a necessary and sufficient condition for a stationary point of the implicit Lagrangian to be a global solution and, hence, a solution of the nonlinear complementarity problem. We show that this new condition easily allows us to recover all previous results and to establish new sufficient conditions. We then consider a constrained reformulation based on the implicit Lagrangian in which nonnegative constraints on the variables are added to the original unconstrained reformulation. This is motivated by the fact that often, in applications, the function which defines the complementarity problem is defined only on the nonnegative orthant. We consider the KKT-points of this new reformulation and show that the same necessary and sufficient condition which guarantees, in the unconstrained case, that every unconstrained stationary point is a global solution, also guarantees that every KKT-point of the new problem is a global solution.     

组合优化算法中摆脱局部极小值的几种策略

局部搜索算法是一种非常有效的求解组合优化问题的算法 ,它具有通用、灵活等特点 .但是 ,由于搜索空间和目标函数的复杂性 ,目标函数在搜索空间中有许多局部极小值点 ,使算法在这些局部极小值点处被“卡住”,大大影响算法的效果 .对于此问题 ,笔者查阅了大量文献资料 ,结合自己的研究实践 ,总结出几种跳出局部极小“陷井”的策略 .使用这些策略 ,有望使算法更加完善 ,在求解组合优化问题过程中更能发挥其作用 .     

New Uniform Parametric Error Bounds

A uniform parametric error bound is a uniform error estimate for feasible solutions of a family of parametric mathematical programming problems. It has been proven useful in exact penalty formulation for bilevel programming problems. In this paper, we derive new sufficient conditions for the existence of uniform parametric error bounds.     

Weak sharp minima revisited, part II: application to linear regularity and error bounds

The notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. It has been studied extensively by several authors. This paper is the second of a series on this subject where the basic results on weak sharp minima in Part I are applied to a number of important problems in convex programming. In Part II we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming. We obtain both new results and reproduce several existing results from a fresh perspective. We dedicate this paper to our friend and mentor Terry Rockafellar on the occasion of his 70th birthday. He has been our guide in mathematics as well as in the backcountry and waterways of the Olympic and Cascade mountains. Research supported in part by the National Science Foundation Grant DMS-0203175.     

欧氏空间中的标准格

给出了欧氏空间中标准格的定义,证明了在1维,2维欧氏空间中所有格都是标准格,并构造了当维数大于4时非标准格的例子.还给出了标准格的若干性质.     

Successive minima and slopes of hermitian vector bundles over number fields

The purpose of this paper is to clarify the relationship between the successive minima and the slopes of a hermitian vector bundle on the spectrum of the ring of integers of an algebraic number field. The main result is a lower and an upper bound for each successive minimum in terms of the corresponding slope.     

Derivative,maxima and minima in a graphical context

A deeper learning of the properties and applications of the derivative for the study of functions may be achieved when teachers present lessons within a highly graphic context, linking the geometric illustrations to formal proofs. Each concept is better understood and more easily retained when it is presented and explained visually using graphs. In this article, we explore the conditions of necessity or sufficiency of the criteria for determining the maxima and minima of a function. The implications for the teaching of derivatives and functions in undergraduate courses are discussed in light of our analysis of textbooks.     

Uniqueness conditions for Kuhn-Tucker points on a disk

Consider minimizingf onD which is diffeomorphic to a disk. Under a genericity assumption, the number of points onD satisfying the Kuhn-Tucker necessary conditions for minimum is odd. We give conditions which imply that a local minimum is global and a necessary and sufficient condition that a Kuhn-Tucker point is the solution. Convex transformable problems satisfy the latter condition.D may be of full dimension or be embedded on a manifold or it may be given by a system of concave inequalities.The work on which this paper is based was done while the author was visiting the Group for the Application of Mathematics and Statistics to Economics at the University of California, Berkeley, Spring 1981. The author would like to thank the Group, and Professor G. Debreu in particular, for the hospitality. He benefited from discussions with Professors S. Smale, A. Mas-Colell, K. Nishimura, and L. Chenault, among others. He also wishes to thank two referees of this journal for helpful comments.     

泛函极小与非线性椭圆方程解的局部正则性

利用Giaquinta和Giusti的嵌入不等式和Sobolev空间方法,证明了在更一般条件下的变分问题中的泛函极小与非线性椭圆方程弱解的局部正则性.     

Objective Function Features Providing Barriers to Rapid Global Optimization

The purposes of this discussion paper are twofold. First, features of an objective function landscape which provide barriers to rapid finding of the global optimum are described. Second, stochastic algorithms are discussed and their performance examined, both theoretically and computationally, as the features change. The paper lays a foundation for the later findings paper.     

On the role of continuously differentiable exact penalty functions in constrained global optimization

The aim of this paper is to show that the new continuously differentiable exact penalty functions recently proposed in literature can play an important role in the field of constrained global optimization. In fact they allow us to transfer ideas and results proposed in unconstrained global optimization to the constrained case.First, by drawing our inspiration from the unconstrained case and by using the strong exactness properties of a particular continuously differentiable penalty function, we propose a sufficient condition for a local constrained minimum point to be global.Then we show that every constrained local minimum point satisfying the second order sufficient conditions is an attraction point for a particular implementable minimization algorithm based on the considered penalty function. This result can be used to define new classes of global algorithms for the solution of general constrained global minimization problems. As an example, in this paper we describe a simulated annealing algorithm which produces a sequence of points converging in probability to a global minimum of the original constrained problem.     

Inexact subgradient methods for quasi-convex optimization problems

In this paper, we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi-convex constrained optimization problem. The inexactness stems from computation errors and noise, which come from practical considerations and applications. Assuming that the computational errors and noise are deterministic and bounded, we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima. In both cases, using the constant and diminishing stepsize rules, we describe convergence results in both objective values and iterates, and finite convergence to approximate optimality. We also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the Cobb–Douglas production efficiency problem. The numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm, especially for the large-scale problems.     

Note on generalized convex functions

In this note, an important class of generalized convex functions, called invex functions, is defined under a general framework, and some properties of the functions in this class are derived. It is also shown that a function is (generalized) pseudoconvex if and only if it is quasiconvex and invex.