Existence of Solutions and Variational Principles for Generalized Vector Systems

By means of generalized KKM theory, we prove a result on the existence of solutions and we establish general variational principles, that is, vector optimization formulations of set-valued maps for vector generalized systems. A perturbation function is involved in general variational principles. We extend the theory of gap functions for vector variational inequalities to vector generalized systems and we prove that the solution sets of the related vector optimization problems of set-valued maps contain the solution sets of vector generalized systems. A further vector optimization problem is defined in such a way that its solution set coincides with the solution set of a weak vector generalized system. Research carried on within the agreement between National Sun Yat-Sen University of Kaohsiung, Taiwan and Pisa University, Pisa, Italy, 2007. L.C. Ceng research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J.C. Yao research was partially supported by the National Science Center for Theoretical Sciences at Tainan.     

Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

The noncooperative multi-leader-follower game can be formulated as a generalized Nash equilibrium problem where each player solves a nonconvex mathematical program with equilibrium constraints. Two major deficiencies exist with such a formulation: One is that the resulting Nash equilibrium may not exist, due to the nonconvexity in each players problem; the other is that such a nonconvex Nash game is computationally intractable. In order to obtain a viable formulation that is amenable to practical solution, we introduce a class of remedial models for the multi-leader-follower game that can be formulated as generalized Nash games with convexified strategy sets. In turn, a game of the latter kind can be formulated as a quasi-variational inequality for whose solution we develop an iterative penalty method. We establish the convergence of the method, which involves solving a sequence of penalized variational inequalities, under a set of modest assumptions. We also discuss some oligopolistic competition models in electric power markets that lead to multi-leader-follower games.Jong-Shi Pang: The work of this authors research was partially supported by the National Science Foundation under grant CCR-0098013 and ECS-0080577 and by the Office of Naval Research under grant N00014-02-1-0286.Masao Fukushima: The work of this authors research was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Culture and Sports of Japan.     

A likelihood-MPEC approach to target classification

In this paper we develop a method for classifying an unknown data vector as belonging to one of several classes. This method is based on the statistical methods of maximum likehood and borrowed strength estimation. We develop an MPEC procedure (for Mathematical Program with Equilibrium Constraints) for the classification of a multi-dimensional observation, using a finite set of observed training data as the inputs to a bilevel optimization problem. We present a penalty interior point method for solving the resulting MPEC and report numerical results for a multispectral minefield classification application. Related approaches based on conventional maximum likehood estimation and a bivariate normal mixture model, as well as alternative surrogate classification objective functions, are described. Received: October 26, 1998 / Accepted: June 11, 2001?Published online March 24, 2003 RID="***" ID="***"The authors of this work were all partially supported by the Wright Patterson Air Force Base via Veda Contract F33615-94-D-1400. The first and third author were also supported by the National Science Foundation under grant DMS-9705220. RID="*" ID="*"The work of this author was based on research supported by the U.S. National Science Foundation under grant CCR-9624018. RID="**" ID="**"The work of this author was supported by the Office of Naval Research under grant N00014-95-1-0777.     

Topological spaces related to the Banach-Mazur game and the generic well-posedness of optimization problems

The existence of special kind of winning strategies in the Banach-Mazur game in a completely regular topological spaceX is shown to be equivalent to generic stability properties of optimization problems generated by the continuous bounded real-valued functions inX.Research partially supported by the National Foundation for Scientific Research at the Bulgarian Ministry of Education and Science under Grant Number MM-408/94.     

Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems

In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set. The extension of Robust Optimization methodology developed in this paper opens up new possibilities to solve efficiently multi-stage finite-horizon uncertain optimization problems, in particular, to analyze and to synthesize linear controllers for discrete time dynamical systems. Research was supported by the Binational Science Foundation grant #2002038     

Optimality conditions for non-finite valued convex composite functions

Burke (1987) has recently developed second-order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued. In this note we present a technique for reducing the infinite valued case to the finite valued one. We then use this technique to extend the results in Burke (1987) to the case in which the convex function may take infinite values. We conclude by comparing these results with those established by Rockafellar (1989) for the piecewise linear-quadratic case.Dedicated to the memory of Robin W. ChaneyResearch supported in part by the National Science Foundation under grants DMS-8602399 and DMS-8803206, and by the Air Force Office of Scientific Research under grant ISSA-860080.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983.     

Robust game theory

We present a distribution-free model of incomplete-information games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our ``robust game' model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative distribution-free equilibrium concept, which we call ``robust-optimization equilibrium,' to that of the ex post equilibrium. We prove that the robust-optimization equilibria of an incomplete-information game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robust-optimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results. The research of the author was partially supported by a National Science Foundation Graduate Research Fellowship and by the Singapore-MIT Alliance. The research of the author was partially supported by the Singapore-MIT Alliance.     

On a class of nonzero-sum,linear-quadratic differential games

A class of two-player, nonzero-sum, linear-quadratic differential games is investigated for Nash equilibrium solutions when both players use closed-loop control and when one or both of the players are required to use open-loop control. For three formulations of the game, necessary and sufficient conditions are obtained for a particular strategy set to be a Nash equilibrium strategy set. For a fourth formulation of the game, where both players use open-loop control, necessary and sufficient conditions for the existence of a Nash equilibrium strategy set are developed. Several examples are presented in order to illustrate the differences between this class of differential games and its zero-sum analog.This research was supported by the National Science Foundation under Grant No. GK-3341.     

Hidden convexity in some nonconvex quadratically constrained quadratic programming

We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent convex programs. This author's work was partially supported by GIF, the German-Israeli Foundation for Scientific Research and Development and by the Binational Science Foundation. This author's work was partially supported by National Science Foundation Grants DMS-9201297 and DMS-9401871.     

A hybrid extragradient method for general variational inequalities

In this paper, we introduce and study a hybrid extragradient method for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. An iterative algorithm is proposed by virtue of the hybrid extragradient method. Under two sets of quite mild conditions, we prove the strong convergence of this iterative algorithm to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively. L. C. Zeng’s research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J. C. Yao’s research was partially supported by a grant from the National Science Council of Taiwan.     

An algorithm for the fast solution of symmetric linear complementarity problems

This paper studies algorithms for the solution of mixed symmetric linear complementarity problems. The goal is to compute fast and approximate solutions of medium to large sized problems, such as those arising in computer game simulations and American options pricing. The paper proposes an improvement of a method described by Kocvara and Zowe (Numer Math 68:95–106, 1994) that combines projected Gauss–Seidel iterations with subspace minimization steps. The proposed algorithm employs a recursive subspace minimization designed to handle severely ill-conditioned problems. Numerical tests indicate that the approach is more efficient than interior-point and gradient projection methods on some physical simulation problems that arise in computer game scenarios. The research of J. L. Morales was supported by Asociación Mexicana de Cultura AC and CONACyT-NSF grant J110.388/2006. The research of J. Nocedal was supported by National Science Foundation grant CCF-0514772, Department of Energy grant DE-FG02-87ER25047-A004 and a grant from the Intel Corporation.     

Deciding uniqueness in norm maximization

NP-hardness is established for the problem whose instance is a system of linear inequalities defining a polytopeP, and whose question is whether, onP, the global maximum of the Euclidean norm is attained at more than one vertex ofP. The NP-hardness persists even for the restricted problem in whichP is a full-dimensional parallelotope with one vertex at the origin. This makes it possible to establish NP-hardness for other uniqueness problems, including some from pseudoboolean programming and computational convexity.Research of the first author was supported in part by the Deutsche Forschungsgemeinschaft. Research of the second author was supported in part by the National Science Foundation.     

Mathematical Programs with Vector Optimization Constraints

In this paper, we introduce mathematical programs with vector optimization constraints. For these problems, we establish two models in both the weak Pareto solution and Pareto solution setting. Some new existence results are obtained under rather weak conditions. We establish also equivalences between mathematical programs with vector optimization constraints and mathematical programs with vector variational inequality constraints.This research was partially supported by a grant from the National Science Council of the ROC. The authors thank the referees for helpful suggestions and comments.     

A solution for noncooperative games

In this paper, we study solutions of strict noncooperative games that are played just once. The players are not allowed to communicate with each other. The main ingredient of our theory is the concept of rationalizing a set of strategies for each player of a game. We state an axiom based on this concept that every solution of a noncooperative game is required to satisfy. Strong Nash solvability is shown to be a sufficient condition for the rationalizing set to exist, but it is not necessary. Also, Nash solvability is neither necessary nor sufficient for the existence of the rationalizing set of a game. For a game with no solution (in our sense), a player is assumed to recourse to a standard of behavior. Some standards of behavior are examined and discussed.This work was sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS-75-17385-A01. The author is grateful to J. C. Harsanyi for his comments and to S. M. Robinson for suggesting the problem.     

Non-Zenoness of a class of differential quasi-variational inequalities

The Zeno phenomenon of a switched dynamical system refers to the infinite number of mode switches in finite time. The absence of this phenomenon is crucial to the numerical simulation of such a system by time-stepping methods and to the understanding of the behavior of the system trajectory. Extending a previous result for a strongly regular differential variational inequality, this paper establishes that a certain class of non-strongly regular differential variational inequalities is devoid of the Zeno phenomenon. The proof involves many supplemental results that are of independent interest. Specialized to a frictional contact problem with local compliance and polygonal friction laws, this non-Zenoness result is of fundamental significance and the first of its kind. This work was based on research partially supported by the National Science Foundation under grants DMS-0508986 and IIS-0413227 awarded to Rensselaer Polytechnic Institute, where the original version of the paper was first written. The revision was based on research partially supported by the National Science Foundation under grant DMS awarded to the University of Illinois at Urbana-Champaign.     

Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with coulomb friction

In this paper, we study the problem of predicting the acceleration of a set of rigid, 3-dimensional bodies in contact with Coulomb friction. The nonlinearity of Coulomb's law leads to a nonlinear complementarity formulation of the system model. This model is used in conjunction with the theory of quasi-variational inequalities to prove for the first time that multi-rigid-body systems with all contacts rolling always has a solution under a feasibility-type condition. The analysis of the more general problem with sliding and rolling contacts presents difficulties that motivate our consideration of a relaxed friction law. The corresponding complementarity formulations of the multi-rigid-body contact problem are derived and existence of solutions of these models is established. The research of this author was based on work supported by the National, Science Foundation under grants DDM-9104078 and CCR-9213739. The research of this author was partially supported by the National Science Foundation under grant IRI-9304734, by the Texas Advanced Research Program grant 999903-078, and by the Texas Advanced Technology Program under grant 999903-095.     

Multicriterion Aerodynamic Shape Design Optimization and Inverse Problems Using Control Theory and Nash Games

Multicriterion design is gaining importance in aeronautics in order to cope with new needs of society. In the literature, contributions to single discipline and/or single-point design optimization abound. The goal of this paper is to introduce a new approach combining the adjoint method with a formulation derived from game theory for multipoint aerodynamic design problems. Transonic flows around lifting airfoils are analyzed via Euler computations. Airfoil shapes are optimized according to various aerodynamic criteria. The notion of player is introduced. In a competitive Nash game, each player attempts to optimize its own criterion through a symmetric exchange of information with others. A Nash equilibrium is reached when each player, constrained by the strategy of the others, cannot improve further its own criterion. Specific real and virtual symmetric Nash games are implemented to set up an optimization strategy for design under conflict. This work has benefited partially from the support of the National Science Foundation of China Grant NSFC-10372040 and Scientific Research Foundation (SRF) for Returned Overseas Chinese Scholars (ROCS) Grant 2003-091. The first author acknowledges the support of INRIA (Institut National de Recherche en Information et en Automatique), France.     

Proto-derivative formulas for basic subgradient mappings in mathematical programming

Subgradient mappings associated with various convex and nonconvex functions are a vehicle for stating optimality conditions, and their proto-differentiability plays a role therefore in the sensitivity analysis of solutions to problems of optimization. Examples of special interest are the subgradients of the max of finitely manyC ² functions, and the subgradients of the indicator of a set defined by finitely manyC ² constraints satisfying a basic constraint qualification. In both cases the function has a property called full amenability, so the general theory of existence and calculus of proto-derivatives of subgradient mappings associated with fully amenable functions is applicable. This paper works out the details for such examples. A formula of Auslender and Cominetti in the case of a max function is improved in particular.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983 for the first author and by the National Science Foundation under grant DMS-9200303 for the second author.     

Iterative Algorithm for Generalized Set-Valued Strongly Nonlinear Mixed Variational-Like Inequalities

The auxiliary principle technique is extended to study the generalized strongly nonlinear mixed variational-like inequality problem for set-valued mappings without compact values. We establish first the existence of a solution of the related auxiliary problem. Then, the iterative algorithm for solving that problem is given by using this existence result. Moreover, the existence of a solution of the original problem and the convergence of iterative sequences generated by the algorithm are both derived.Research partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai, China. Research partially supported by a grant from the National Science Council of Taiwan     

n-Person games with only 1, n− 1, and n-person coalitions

A symmetric solution is presented for any von Neumann-Morgenstern n-person game when the only coalitions that are not completely defeated contain n– 1 or n players.Portions of this research were supported by a National Science Foundation grant at the University of Michigan and by a Fulbright grant at the Middle East Technical University, Ankara.Mathematics Research Center, The University of Wisconsin, Madison.