An algorithm for global optimization of Lipschitz continuous functions

An algorithm is presented which locates the global minimum or maximum of a function satisfying a Lipschitz condition. The algorithm uses lower bound functions defined on a partitioned domain to generate a sequence of lower bounds for the global minimum. Convergence is proved, and some numerical results are presented.     

Efficient domain partitioning algorithms for global optimization of rational and Lipschitz continuous functions

A domain partitioning algorithm for minimizing or maximizing a Lipschitz continuous function is enhanced to yield two new, more efficient algorithms. The use of interval arithmetic in the case of rational functions and the estimates of Lipschitz constants valid in subsets of the domain in the case of others and the addition of local optimization have resulted in an algorithm which, in tests on standard functions, performs well.     

A class of continuous nonlinear complementarity problems

The concept of continuous nonlinear complementarity is defined. Basic properties and existence theorem are proven. Applications to continuous linear and nonlinear programming are presented. Kuhn-Tucker type conditions are established.This work was supported in part by the National Institute of General Medical Sciences under Training Grant No. 5-TO1-GM00913.     

On a subclass of quasi-convex mappings

A class of biholomorphic mappings named “quasi-convex mappings of type D” is introduced in the unit ball of a complex Banach space, which is the natural generalization of the convex functions in one variable. It is proved that this class of mappings is a subset of the class of “quasi-convex mappings of type A” introduced by K.A. Roper and T.J. Suffridge.     

Lipschitzian inverse functions,directional derivatives,and applications inC 1,1 optimization

The paper shows that Thibault's limit sets allow an iff-characterization of local Lipschitzian invertibility in finite dimension. We consider these sets as directional derivatives and extend the calculus in a way that can be used to clarify whether critical points are strongly stable inC ^1,1 optimization problems.Many fruitful discussions with colleagues D. Klatte and K. Tammer as well as with H. Th. Jongen and F. Nozicka have influenced the present investigations in a very constructive manner. For the original papers concerning the sets f(x; u), the author is indebted to Prof. L. Thibault.     

On lengths,areas and Lipschitz continuity of polyharmonic mappings

In this paper, we continue our investigation of polyharmonic mappings in the complex plane. First, we establish two Landau type theorems. We also show a three circles type theorem and an area version of the Schwarz lemma. Finally, we study Lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric.     

Lipschitz mappings,contingents, and differentiability

The main purpose of the paper is to show that, for each real normed space Y of infinite dimension, each number L > 0, and each at most countable set Q ? ?, there exists a Lipschitz mapping ?: ? → Y, with constant L, whose graph has a tangent everywhere, whereas ? is not differentiable at any point of Q.     

Lipschitz properties of solutions in mathematical programming

This paper deals with the dependence of the solutions and the associated multipliers of a nonlinear programming problem when the data of the problem are subjected to small perturbations. Sufficient conditions are given which imply that the solutions and the multipliers of a perturbed nonlinear programming problem are Lipschitzian with respect to the perturbations.The authors wish to thank J. Drèze and J. P. Vial for many helpful discussions and J. B. Hiriart-Urruty for comments on a previous version of the paper.     

有限个李普希兹伪压缩映射近迫点序列的收敛性

研究了Banach空间中有限个李普希兹伪压缩映射近迫点序列的收敛性问题,此结果推广了以前的结论.     

On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings

In this note, we speed up the convergence of the Picard sequence of iterations for strongly accretive and strongly pseudo-contractive mappings. Our results improve the results of Chidume [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint no. IC2000098; C.E. Chidume, Iterative Algorithms for Non-expansive Mappings and Some of Their Generalizations, in: Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, vol. 1, 2, Kluwer Acad. Publ, Dordrecht, 2003, pp. 383–429], and Liu [L. Liu, Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. Amer. Math. Soc. 125 (2) (1997) 1363–1366], and some other known results. The technique of the proof, presented in this paper, is different from the technique used by Chidume.     

Increasing Lipschitz continuous maximizers of some dynamic programs

Conditions are presented for the existence of increasing and Lipschitz continuous maximizers in a general one-stage optimization problem. This property results in substantial numerical savings in case of a discrete parameter space. The one-stage result and properties of concave functions lead to simple conditions for the existence of optimal policies, composed of increasing and Lipschitz continuous decision rules, for several dynamic programs with discrete state and action space, in which case discrete concavity plays a dominant role. One of the examples, a general multi-stage allocation problem, is considered in detail. Finally, some known results in the case of a continuous state and action space are generalized.     

Higher-order necessary conditions for infinite and semi-infinite optimization

In this paper, we present a unified theory of first-order and higher-order necessary optimality conditions for abstract vector optimization problems in normed linear spaces. We prove general multiplier rules, from which nearly all known first-order, second-order, and higher-order necessary conditions can be derived. In the last section, we prove higher-order necessary conditions for semi-infinite programming problems.This work was developed within the Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, Bremen, West Germany.The author wishes to thank Prof. Dr. D. Hinrichsen for his helpful remarks and discussions during the preparation of this work.     

Preparation of papers

Recent studies are concerned with two types of questions in nonconvex optimization: (a) conditions for having bounded Lagrange multipliers, Refs. 1–2; (b) a priori bounds for such Lagrange multipliers, Ref. 3. Such topics have been investigated under suitable regularity assumptions. The purpose of this paper is to study the same problems for the generalized Lagrange multipliers of a locally Lipschitz programming.The author thanks the referees for helpful suggestions     

Lipschitz behavior of convex semi-infinite optimization problems: a variational approach

In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103–162; Engligh translation Math. Surveys 55, 501–558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543–554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.      

General Strongly Nonlinear Quasivariational Inequalities with Relaxed Lipschitz and Relaxed Monotone Mappings

In this paper, we introduce and study a new class of general strongly nonlinear quasivariational inequalities and construct a general iterative algorithm by using the projection method. We establish the existence of a unique solution for general strongly nonlinear quasivariational inequalities involving relaxed Lipschitz, relaxed monotone, and strongly monotone mappings; we obtain the convergence and stability of the iterative sequences generated by the algorithm. Our results extend, improve, and unify many known results due to Bose, Noor, Siddiqi-Ansari, Verma, Yao, Zeng, and others.     

关于实李普希兹映射的lshikawa迭代的收敛率和加速收敛问题

本文研究使用Ｉｓｈｉｋａｗａ迭代格式求实李普希兹映射的不动点，指出该迭代格式仅具线性收敛率；对于参变量序列｛αｎ｝、｛βｎ｝所取的不同的值，比较了迭代格式的收敛速度．在给出加速因子定义的基础上，本文给出了加速收敛的一个充分条件．     

Increasing and Lipschitz continuous minimizers in one-dimensional linear-convex systems without constraints: The continuous and the discrete case

We consider a stochastic control model with linear transition law and arbitrary convex cost functions, a far-reaching generalization of the familiar linear quadratic model. Firstly conditions are given under which the continuous state version has minimizersf _n at each stagen which are increasing and in addition either right continuous or continuous or Lipschitz continuous with explicitly given Lipschitz constant. For the computationally important discrete version we verify some analogous properties under stronger assumptions.     

Global analysis of continuous analogues of the Levenberg-Marquardt and Newton-Raphson methods for solving nonlinear equations

Summary Global analyses are given to continuous analogues of the Levenberg-Marquardt methoddx/dt=−(J ^t(x)J(x)+δI)⁻¹J^t(x)g(x), and the Newton-Raphson-Ben-Israel methoddx/dt=−J ⁺(x)g(x), for solving an over- and under-determined systemg(x)=0 of nonlinear equations. The characteristics of both methods are compared. Erros in some literature which dealt with related continous analogue methods are pointed out. The Institute of Statistical Mathematics