Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

The sum of the largest eigenvalues of a symmetric matrix is a nonsmooth convex function of the matrix elements. Max characterizations for this sum are established, giving a concise characterization of the subdifferential in terms of a dual matrix. This leads to a very useful characterization of the generalized gradient of the following convex composite function: the sum of the largest eigenvalues of a smooth symmetric matrix-valued function of a set of real parameters. The dual matrix provides the information required to either verify first-order optimality conditions at a point or to generate a descent direction for the eigenvalue sum from that point, splitting a multiple eigenvalue if necessary. Connections with the classical literature on sums of eigenvalues and eigenvalue perturbation theory are discussed. Sums of the largest eigenvalues in the absolute value sense are also addressed.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.The work of this author was supported by the National Science Foundation under grants CCR-8802408 and CCR-9101640.The work of this author was supported in part during a visit to Argonne National Laboratory by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under contract W-31-109-Eng-38, and in part during a visit to the Courant Institute by the U.S. Department of Energy under Contract DEFG0288ER25053.     

Nonlinear multiple objective optimization: An algorithm and some theory

This paper presents an algorithm and the supporting theory for solving a class of nonlinear multiple criteria optimization problems using Zionts—Wallenius type of interaction. The Zionts—Wallenius method, as extended in this paper, can be used for solving multiple criteria problems with concave objective and (implicit) value functions and convex feasible regions. Modifications of the method to handle nonconvex feasible regions and general nonlinear objective functions are also discussed.This research was supported, in part, by a Faculty Research Development Award and by a Council of 100 Research Grant from Arizona State University (Roy), and by a grant from Y. Jahnsson Foundation, Finland (Wallenius). The research was performed while the second author was a Visiting Professor at Arizona State University.     

Pure adaptive search in global optimization

Pure adaptive seach iteratively constructs a sequence of interior points uniformly distributed within the corresponding sequence of nested improving regions of the feasible space. That is, at any iteration, the next point in the sequence is uniformly distributed over the region of feasible space containing all points that are strictly superior in value to the previous points in the sequence. The complexity of this algorithm is measured by the expected number of iterations required to achieve a given accuracy of solution. We show that for global mathematical programs satisfying the Lipschitz condition, its complexity increases at mostlinearly in the dimension of the problem.This work was supported in part by NATO grant 0119/89.     

On a new stochastic global optimization algorithm based on censored observations

In this paper a new algorithm is proposed for global optimization problems. The main idea is that of modifying a standard clustering approach by sequentially sampling the objective function while adaptively deciding an appropriate sample size. Theoretical as well as computational results are presented.     

Asymptotic analysis of the effect of arrival model uncertainties in some optimal routing problems

We study the effect of arrival model uncertainties on the optimal routing in a system of parallel queues. For exponential service time distributions and Bernoulli routing, the optimal mean system delay generally depends on the interarrival time distribution. Any error in modeling the arriving process will cause a model-based optimal routing algorithm to produce a mean system delay higher than the true optimum. In this paper, we present an asymptotic analysis of the behavior of this error under heavy traffic conditions for a general renewal arrival process. An asymptotic analysis of the error in optimal mean delay due to uncertainties in the service time distribution for Poisson arrivals was reported in Ref. 6, where it was shown that, when the first moment of the service time distribution is known, this error in performance vanishes asymptotically as the traffic load approaches the system capacity. In contrast, this paper establishes the somewhat surprising result that, when only the first moment of the arrival distribution is known, the error in optimal mean delay due to uncertainties in the arrival model is unbounded as the traffic approaches the system capacity. However, when both first and second moments are known, the error vanishes asymptotically. Numerical examples corroborating the theoretical results are also presented.This work was supported by the National Science Foundation under Grants ECS-88-01912 and EID-92-12122 and by NASA under Contract NAG 2-595.The authors wish to thank an anonymous referee for pointing out Ref. 20, thus avoiding the need for an explicit proof of convexity of the cost function considered in the paper.     

Global optimization by random perturbation of the gradient method with a fixed parameter

The paper deals with the global minimization of a differentiable cost function mapping a ball of a finite dimensional Euclidean space into an interval of real numbers. It is established that a suitable random perturbation of the gradient method with a fixed parameter generates a bounded minimizing sequence and leads to a global minimum: the perturbation avoids convergence to local minima. The stated results suggest an algorithm for the numerical approximation of global minima: experiments are performed for the problem of fitting a sum of exponentials to discrete data and to a nonlinear system involving about 5000 variables. The effect of the random perturbation is examined by comparison with the purely deterministic gradient method.     

无约束全局优化的一个可调参数的填充函数

A filled function with adjustable parameters is suggested in this paper for finding a global minimum point of a general class of nonlinear programming problems with a bounded and closed domain. This function has two adjustable parameters. We will discuss the properties of the proposed filled function. Conditions on this function and on the values of parameters are given so that the constructed function has the desired properties of traditional filled function.     

Polynomial-time approximation schemes for packing and piercing fat objects

We consider two problems: given a collection of n fat objects in a fixed dimension, (1) ( packing) find the maximum subcollection of pairwise disjoint objects, and (2) ( piercing) find the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomial-time approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.'s algorithm requires polynomial space. We describe a different algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the first PTAS for that problem.     

集值映射的切导数与向量优化问题的最优性条件

利用集值映射切导数与半可微概念，给出了无约束与具约束的集值向量优化问题局部真有效解与局部强有效解的最优性条件。     

An embedding domain approach for a class of 2-d shape optimization problems: mathematical analysis

This contribution combines a shape optimization approach to free boundary value problems of Bernoulli type with an embedding domain technique. A theoretical framework is developed which allows to prove continuous dependence of the primal and dual variables in the resulting saddle point problems with respect to the domain. This ensures the existence of a solution of a related shape optimization problem in a sufficiently large class of admissible domains.