On the Stability of Globally Projected Dynamical Systems

Two types of projected dynamical systems, whose equilibrium states solve the corresponding variational inequality problems, were proposed recently by Dupuis and Nagurney (Ref. 1) and by Friesz et al. (Ref. 2). The stability of the dynamical system developed by Dupuis and Nagurney has been studied completely (Ref. 3). This paper analyzes and proves the global asymptotic stability of the dynamical system proposed by Friesz et al. under monotone and symmetric mapping conditions. Furthermore, the dynamical system is shown to be globally exponentially stable under stronger conditions. Finally, we show that the dynamical system proposed by Friesz et al. can be applied easily to neural networks for solving a class of optimization problems.     

Massive Data Classification via Unconstrained Support Vector Machines

A highly accurate algorithm, based on support vector machines formulated as linear programs (Refs. 1–2), is proposed here as a completely unconstrained minimization problem (Ref. 3). Combined with a chunking procedure (Ref. 4), this approach, which requires nothing more complex than a linear equation solver, leads to a simple and accurate method for classifying million-point datasets. Because a 1-norm support vector machine underlies the proposed approach, the method suppresses input space features as well. A state-of-the-art linear programming package (CPLEX, Ref. 5) fails to solve problems handled by the proposed algorithm.This research was supported by National Science Foundation Grants CCR-0138308 and IIS-0511905.     

The Classical Average-Cost Inventory Models of Iglehart and Veinott–Wagner Revisited

This paper revisits the classical papers of Iglehart (Ref. 1) and Veinott and Wagner (Ref. 2) devoted to stochastic inventory problems with the criterion of long-run average cost minimization. We indicate some of the assumptions that are used implicitly without verification in their stationary distribution approach to the problems and provide the missing (nontrivial) verification. In addition to completing their analysis, we examine the relationship between the stationary distribution approach and the dynamic programming approach to the average-cost stochastic inventory problems.     

二级线性价格控制问题的满意解的求解思路及实例

本文就文献 [1]对线性二级价格控制问题 (BLP) 2 研究的结果及文献 [2 ]提出的问题进行了进一步的讨论。用反例指出文 [1]求出的极点最优解是错的 ,以及一般的 (BL P) 2 问题求出的最优解 ,可能是下层决策者根本无法接受的。因此 ,本文提出 (BLP) 2 的求下层目标最优解的边界搜索法及在此基础上用多目标的观点来求 (BLP) 2 的满意解的思路及实例     

Pareto Analysis vis-à-vis Balance Space Approach in Multiobjective Global Optimization

There is much controversy about the balance space approach, introduced first in Ref. 1, pp. 138–140, with the consideration of the balance number and balance vectors, and then further developed in Ref. 2, with the consideration of balance points and balance sets. There were attempts to identify the balance space approach with some other methods of multiobjective optimization, notably the method proposed in Ref. 3 and most recently Pareto analysis, as presented in Ref. 4. In this paper, we compare Pareto analysis with the balance space approach on several examples to demonstrate the interrelation and the differences of the two methods. As a byproduct, it is shown that, in some cases, the entire Pareto sets, proper and adjoint, can be determined very simply, without any special investigation of the (nonscalarized, nonconvex) multiobjective global optimization problem. The method of parameter introduction is presented in application to determining the Pareto sets and balance set. The use of computer graphics software complemented with the Gauss–Jordan matrix reduction algorithm is proposed for a class of otherwise intractable problems with nonconvex constraint sets.     

On the Tikhonov Well-Posedness of Concave Games and Cournot Oligopoly Games

The purpose of this paper is to investigate whether theorems known to guarantee the existence and uniqueness of Nash equilibria, provide also sufficient conditions for the Tikhonov well-posedness (T-wp). We consider several hypotheses that ensure the existence and uniqueness of a Nash equilibrium (NE), such as strong positivity of the Jacobian of the utility function derivatives (Ref. 1), pseudoconcavity, and strict diagonal dominance of the Jacobian of the best reply functions in implicit form (Ref. 2). The aforesaid assumptions imply the existence and uniqueness of NE. We show that the hypotheses in Ref. 2 guarantee also the T-wp property of the Nash equilibrium.As far as the hypotheses in Ref. 1 are concerned, the result is true for quadratic games and zero-sum games. A standard way to prove the T-wp property is to show that the sets of -equilibria are compact. This last approach is used to demonstrate directly the T-wp property for the Cournot oligopoly model given in Ref. 3. The compactness of -equilibria is related also to the condition that the best reply surfaces do not approach each other near infinity.     

Generalized Surrogate Problem Methodology for Online Stochastic Discrete Optimization

We consider stochastic discrete optimization problems where the decision variables are nonnegative integers and propose a generalized surrogate problem methodology that modifies and extends previous work in Ref. 1. Our approach is based on an online control scheme which transforms the problem into a surrogate continuous optimization problem and proceeds to solve the latter using standard gradient-based approaches while simultaneously updating both the actual and surrogate system states. In contrast to Ref. 1, the proposed methodology applies to arbitrary constraint sets. It is shown that, under certain conditions, the solution of the original problem is recovered from the optimal surrogate state. Applications of this approach include solutions to multicommodity resource allocation problems; in these problems, exploiting the convergence speed of the method, one can overcome the obstacle posed by the presence of local optima.     

Generalized KKM theorems,minimax inequalities,and their applications

This paper extends the well-known KKM theorem and variational inequalities by relaxing the closedness of values of a correspondence and lower semicontinuity of a function. The approach adopted is based on Michael's continuous selection theorem. As applications, we provide theorems for the existence of maximum elements of a binary relation, a price equilibrium, and the complementarity problem. Thus our theorems, which do not require the openness of lower sections of the preference correspondences and the lower semicontinuity of the excess demand functions, generalize many of the existence theorems such as those in Sonnenschein (Ref. 1), Yannelis and Prabhakar (Ref. 2), and Border (Ref. 3).The author is grateful to Professor Franco Giannessi for helpful comments and suggestions.     

Global stability result for the generalized quasivariational inequality problem

This paper studies some stability properties for the generalized quasivariational inequality problem. The study of this topic is motivated by the work of Harker and Pang (Ref. 1). A global stability result is obtained for problems satisfying certain conditions.The author would like to express his gratitude to the referees for helpful suggestions for the revision of the paper.     

Preservation of persistence and stability under intersections and operations,part 2: Stability

The stability of the image of a moving convex set by a multifunction is considered. This study completes the persistence results obtained in Part 1 of this paper (Ref. 1). The results can be applied to epigraphs, hence to the convergence of convex functions.     

On Parametric Generalized Quasi-Variational Inequalities

In this paper, by using the projection method of Dafermos (Ref. 1), we study the behavior and sensitivity analysis of the solution set for a class of parametric generalized quasi-variational inequalities. Our approach and results are new and have a strong geometric flavor.     

On solving discontinuous extremal problems

An approach to solving discontinuous problems of optimization and control is described. The approach is based on the concept of approximate gradient introduced in Ref. 1. Generalizations of the theorems of Kuhn-Tucker and Dubovitsky-Milyutin and the maximum principle of Pontryagin are proved. The mathematical constructions described allow one to solve a wide variety of applied problems of optimization and control within the class of nonsmooth (including discontinuous) functions. The paper continues the investigations of Refs. 1–2.     

An exponential penalty method for nondifferentiable minimax problems with general constraints

A well-known approach to constrained minimization is via a sequence of unconstrained optimization computations applied to a penalty function. This paper shows how it is possible to generalize Murphy's penalty method for differentiable problems of mathematical programming (Ref. 1) to solve nondifferentiable problems of finding saddle points with constraints. As in mathematical programming, it is shown that the method has the advantages of both Fiacco and McCormick exterior and interior penalty methods (Ref. 2). Under mild assumptions, the method has the desirable property that all trial solutions become feasible after a finite number of iterations. The rate of convergence is also presented. It should be noted that the results presented here have been obtained without making any use of differentiability assumptions.     

An Observation on Two Methods of Obtaining Solutions to Variational Problems

In this paper, we compare two methods for obtaining solutions for free problems in the calculus of variations. The first is due to Carathéodory (Ref. 1) and the second due to Leitmann (Ref. 2). Both methods introduce the notion of equivalent variational problems. Using either approach, an auxiliary problem is obtained for which the solution is more easily obtained. We compare both approaches by using each to solve the same class of examples. We conclude our discussion by unifying the two approaches into one and illustrating the potential of this new method through the use of an elementary example.     

On the Marginal Cost Approach in Maintenance

In this paper we investigate the conditions under which the marginal cost approach of Refs. 1–3 holds. As observed in Ref. 4, the validity of the marginal cost approach gives rise to a useful framework of single-component maintenance optimization models, which covers almost all models used in practice. For the class of unimodal finite-valued marginal cost functions, we show that these optimization models are easy to solve.     

An applied cell mapping method for optimal control problems

From the application point of view, a series of modifications are proposed for the cell mapping method discussed in Ref. 1 for the optimal control analysis of dynamical systems. The cell order around the target set is rearranged. A set of common discriminate principles is used for the selection of the optimal one among competing control strategies of the same cost. Inequality constraints of the system are taken into account. The number of elements in the set of allowable time intervals is not prescribed, but left open. These modifications seem to make the cell mapping method more efficient for analyzing feedback systems and for obtaining their global optimal control information. The algorithms presented in this paper could broaden the application of the cell mapping approach of Ref. 1 to a wider class of engineering problems.     

Basis Property of a Rayleigh Beam with Boundary Stabilization

A Rayleigh beam equation with boundary stabilization control is considered. Using an abstract result on the Riesz basis generation of discrete operators in Hilbert spaces, we show that the closed-loop system is a Riesz spectral system; that is, there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis in the state Hilbert space. The spectrum-determined growth condition, distribution of eigenvalues, as well as stability of the system are developed. This paper generalizes the results in Ref. 1.     

Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization,Part 2: Application

In this paper, we describe an application of the planar conjugate gradient method introduced in Part 1 (Ref. 1) and aimed at solving indefinite nonsingular sets of linear equations. We prove that it can be used fruitfully within optimization frameworks; in particular, we present a globally convergent truncated Newton scheme, which uses the above planar method for solving the Newton equation. Finally, our approach is tested over several problems from the CUTE collection (Ref. 2).This work was supported by MIUR, FIRB Research Program on Large-Scale Nonlinear Optimization, Rome, Italy.The author acknowledges Luigi Grippo and Stefano Lucidi, who contributed considerably to the elaboration of this paper. The exchange of experiences with Massimo Roma was a constant help in the investigation. The author expresses his gratitude to the Associate Editor and the referees for suggestions and corrections.     

An Extension of the Coordinate Transformation Method for Open-Loop Nash Equilibria

In Leitmann (Ref. 1), a coordinate transformation method was introduced to obtain global solutions for free problems in the calculus of variations. This direct method was extended and broadened in Carlson (Ref. 2) and later in Leitmann (Ref. 3). The applicability of the original work of Leitmann (Ref. 1) was further developed in Dockner and Leitmann (Ref. 4) to include the class of open-loop dynamic games. In the present work, we improve the results of Ref. 4 in two directions. First, we enlarge the class of open-loop dynamic games to permit coupling among the dynamic equations via the states of the players; second, we incorporate the modifications given in Refs. 2 and 3. Our results greatly increase the applicability of this method. An example arising from the harvesting of a renewable resource is presented to illustrate the utility of our results.     

Some investigations about a unified approach to quadratically convergent algorithms for function minimization

In 1969, Huang (Ref. 1) proposed a unified approach to quadratically convergent algorithms for function minimization without constraints; if we slightly modify a few points of his development, some other algorithms can be generated, among which several were published after 1969. It is also possible to generate a class of algorithms which provide quadratic convergence without linear search at each step.