Minimax Theorem Research Articles

A flexible minimax theorem

A flexible minimax theorem

Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions

We establish the following theorems: (i) an existence theorem for weak type generalized saddle points; (ii) an existence theorem for strong type generalized saddle points; (iii) a generalized minimax theorem for a vector-valued function. These theorems are generalizations and extensions of the author's recent results. For such extensions, we propose new concepts of convexity and continuity of vector-valued functions, which are weaker than ordinary ones. Some of the proofs are based on a few key observations and also on the Browder coincidence theorem or the Tychonoff fixed-point theorem. Also, the minimax theorem follows from the existence theorem for weak type generalized saddle points. The main spaces with mathematical structures considered are real locally convex spaces and real ordered topological vector spaces.

Nonuniform spatial mesh adaptation using a posteriori error estimates: applications to forward and inverse problems

In this paper we present an adaptive method for the solution of large scale forward and inverse problems occurring in biomedicine. Specifically, we describe models based on magnetic resonance images of the human thorax that are used to solve electric and potential field imaging problems. We introduce two- and three-dimensional nonuniform spatial mesh adaptation schemes based upon a posteriori error estimates from the finite element approximation, an algorithm that utilizes a minimax theorem based on estimates of the electric and potential fields and a three-dimensional, Delaunay triangulation program. Initial results are presented for two- and three-dimensional models.

Relativistic quantum chemistry and rigorous variational analysis

A brief review of relativistic quantum chemistry is given here. Relativistic effects and their importance in chemistry are discussed. An outline of different theoretical aspects is presented. Aspects of variation techniques relevant to relativistic calculations are discussed in detail. These involve the derivation of min-max theorems for Dirac, Dirac-Hartree-Fock and Dirac-Coulomb calculations. The consequence of relativistic Hamiltonians being unbounded are also discussed for other lines of investigation. The upper bounds derived are physically interpreted. Sample Dirac-Hartree-Fock results for the Be atom, calculated using both STO and GTO bases for the nonrelativistic orbitals and the upper components of the relativistic orbitals, are given. The inadequacy of the so-called kinetically balanced basis set is discussed and illustrated with these results. The importance of the variational or dynamical balance and hence the merit of the LCAS-MS scheme is pointed out. The possibility of calculating quantum electrodynamical pair energy from relativistic configuration interaction calculations on a two-electron atom is discussed and exemplified. The present status of relativistic molecular calculations is briefly reviewed. Conclusions on the aspects of variational analysis and molecular calculations are enclosed.

A Topological KKM Theorem and Minimax Theorems

A more general version of the KKM theorem and some minimax theorems are obtained using only topological conditions.

The fundamental theorem of game theory revisited

This paper offers an alternative proof of the so-called fundamental theorem of the theory of games or the minimax theorem. The proof uses a particularization of Farkas' theorem involving the expression of one vector as a convex combination of a set of vectors. It is demonstrated that the minimax theorem holds as a consequence of this specialization of Farkas' theorem of the alternative.

On weighted multiway cuts in trees

A min—max theorem is developed for the multiway cut problem of edge-weighted trees. We present a polynomial time algorithm to construct an optimal dual solution, if edge weights come in unary representation. Applications to biology also require some more complex edge weights. We describe a dynamic programming type algorithm for this more general problem from biology and show that our min—max theorem does not apply to it.

Section Theorems on H-Spaces with Applications

Some further generalizations of Fan′s and Shih-Tan′s geometric theorems for convex sets are obtained. As applications, in Sections 3-5, we utilize these results to study the minimax inequalities, and the fixed point theorems for multi-valued mappings. The results presented in this paper generalize the von Neumann and the Fan-Kneser minimax theorems and some others recent results.

Intersection Theorems and Minimax Theorems Based on Connectedness

Intersection Theorems and Minimax Theorems Based on Connectedness

Optimal control of a class of DEDS: Flow-shops with state-dependent processing times

We consider a permutation flow-shop withn jobs andm machines, where the job processing times are given by a monotone nondecreasing function of the time elapsed since the release of the jobs. In this class of discrete event dynamic systems (DEDS), the dynamics is bothevent dependent, andtime dependent. Unless processing times are constant, it cannot be linearized using the (max, +) algebra. In order to schedule the jobs, we first need to know how to compute the (optimal) value of each schedule, i.e., when the jobs should be released. This optimal control problem is solved here by proving apredictive feedback control law, which holds for any regular performance criterion. Then we derive aBellman principle for this type of DEDS, and prove closed form control formulas for the minimum of: the makespan (Cmax), the maximum lateness (Lmax), and the maximum tardiness (Tmax). We also prove some minimax theorems for a class of nonconvex maps derived from the dynamics of the system. The optimal control problem is solved in polynomial time, provided the inverse of the processing time functions can be computed in polynomial time.

Graph Searching and a Min-Max Theorem for Tree-Width

The tree-width of a graph G is the minimum k such that G may be decomposed into a "tree-structure" of pieces each with at most k + l vertices. We prove that this equals the maximum k such that there is a collection of connected subgraphs, pairwise intersecting or adjacent, such that no set of ≤ k vertices meets all of them. A corollary is an analogue of LaPaugh′s "monotone search" theorem for cops trapping a robber they can see (LaPaugh′s robber was invisible).

Topological intersection theorems

Let {Cx: x e X} be a family of subsets of some set Y. A purely topological condition is given that is both necessary and sufficient for nf{Cx x e X} to be nonvoid. Applications to minimax theorems are sketched.

Some topological mini-max theorems via an alternative principle for multifunctions

does hold. The relevant literature is by now really impressive. For a first approach to it, we refer the reader to the references quoted in [4], [5], [7], [9]. However, as well stressed by the lucid Introduction of [9], it is rather easy to recognize a few research currents within which practically each existing contribution can be located. In particular, one of these currents is that concerning the so called topological mini-max theorems, the ancestors of which are the results of [13] and, more properly, of [16] (see also [2]). For a sharp discussion on such kind 0f theorems, we refer to a very recent paper by H. K6nig [8] which also contains the best results in this area, up to now. Since the aim of the present paper is to establish some new topological mini-max theorems as applications of an alternative principle for multifunctions (several further consequences of which will be presented in successive papers), we consider just [8] as a starting point to introduce the main novelties of our results. So, we state now a particular case of the main theorem of [8].

Correlated Equilibria in Nonzero-Sum Differential Games

This paper is our second study of correlated equilibria in nonzero-sum differential games of prescribed duration. Our first work, based on relaxed controls of the players, dealt with linear differential games. In this paper, we use mixed correlated strategies with finite supports and consider nonzero-sum differential games with the open-loop information pattern, for which no equilibrium solution has been known to exist so far. Assuming very mild conditions about the primitive data (dynamical systems, payoff functionals), we prove that all games under consideration possess correlated ϵ-equilibria for any ϵ > 0. For linear differential games one can use such ϵ-equilibria to construct chattering ϵ-equilibrium solutions in the class of relaxed correlated strategies of the players. Our main result also implies Wilson′s minimax theorem for zero-sum differential games with no information.

A minimax principle for the optimal error of Monte Carlo methods

Some results are proved showing that the average case error with least favorable distribution and the optimal error of appropriate Monte Carlo methods coincide in several cases. The results are based on a special Minimax Theorem, derived from Ky Fan's Lemma. Finally, an application is given to the case of Banach spaces having “few” extreme points.

On minimax inequalities on spaces having certain contractible subsets

The concept of a convex space is extended to an H-space; that is, a space having certain family of contractible subsets. For such spaces the KKM type theorems, the Fan-Browder fixed point theorem, the Ky Fan type matching theorem, and minimax inequalities are given. Moreover, applications to a von Neumann-Sion type minimax theorem, a saddle point theorem, a quasi-variational inequality, and a Kakutani type fixed point theorem are obtained.

A generalized duality and applications

The aim of this paper is to present a nonconvex duality with a zero gap and its connection with convex duality. Since a convex program can be regarded as a particular case of convex maximization over a convex set, a nonconvex duality can be regarded as a generalization of convex duality. The generalized duality can be obtained on the basis of convex duality and minimax theorems. The duality with a zero gap can be extended to a more general nonconvex problems such as a quasiconvex maximization over a general nonconvex set or a general minimization over the complement of a convex set. Several applications are given.

Coincidence theorems and minimax theorems (II)

Coincidence theorems and minimax theorems (II)

Convexlike and concavelike conditions in alternative, minimax, and minimization theorems

Convexlike and concavelike conditions are of interest for extensions of the Von Neumann minimax theorem. Since the beginning of the 80's, these conditions also play a certain role in deriving generalized alternative theorems of the Gordan, Motzkin, and Farkas type and Lagrange multiplier results for constrained minimization problems.

Asymptotic linearity of minimax estimators

The celebrated local asymptotic minimax (LAM) theorem due to HÁjek (1972) also includes the statement that a LAM estimator Is necessarily asymptotically linear. A similar result. is true for semi‐parametric models, but Hájek's result doesn't apply to this case as the efficient influence function is often not contained in the (proper) tangent space. This note gives a simple, elementary proof of both the LAM theorem and the necessity of asymptotic linearity of a LAM estimator sequence.