Minimax Theory Research Articles

Fixed points, coincidence points and maximal elements with applications to generalized equilibrium problems and minimax theory

In this paper, using a generalization of the Fan–Browder fixed point theorem, we obtain a new fixed point theorem for multivalued maps in generalized convex spaces from which we derive several coincidence theorems and existence theorems for maximal elements. Applications of these results to generalized equilibrium problems and minimax theory will be given in the last sections of the paper.

Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions

This paper is concerned with deconvolution from error or blurring densities whose Fourier transforms have isolated zeros and show oscillatory behaviour; unlike conventional approaches where the Fourier transform decays about monotonously. We introduce specific estimation procedures based on local polynomial approximation in the Fourier domain. Under combined moment and smoothness conditions, we are able to improve the convergence rates compared to existing methods in density deconvolution. The corresponding minimax theory is derived. In compactly supported models as in signal deconvolution and Berkson regression, nearly optimal rates are achieved under conditions which are significantly weaker than those assumed in earlier papers.

An intersection theorem with applications in minimax theory and equilibrium problem

In this paper we obtain a very general intersection theorem for the values of a map. From this we derive an analytic alternative, a new type of minimax inequality and an alternative theorem concerning three abstract equilibrium problems.

On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory

In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems.

Subharmonic oscillations of a class of Hamiltonian systems

In this work, we prove the existence of subharmonics of the Hamiltonian system x ̇ = J H ′ ( t , x ) when the Hamiltonian H is coercive or resonant in a part of the variables uniformly in the other part. We derive the existence of such solutions by using minimax theory.

On the variational approach to the periodic n-body problem

This expository paper gathers some of the results obtained by the author in recent works in collaboration with Davide Ferrario and Vivina Barutello, focusing on the periodic n-body problem from the perspective of the calculus of variations and minimax theory. These researches were aimed at developing a systematic variational approach to the equivariant periodic n-body problem in the two and three-dimensional space. The purpose of this paper is to expose the main problems and achievements of this approach. The material here was exposed in the talk that given at the Meeting CELMEC IV promoted by SIMCA (Societa italiana di Meccanica Celeste).

Savage vs. Wald: Was Bayesian Decision Theory the Only Available Alternative for Postwar Economics?

The paper compares the two main approaches to decision-making under uncertainty in the early 1950s, Wald's minimax theory (Wald 1950) and Savage's subjective expected utility theory (Savage 1954), from the viewpoint of postwar neoclassical economics. It is argued that, while obvious reasons led to the success of Savage's approach - the main one being of course its dependence upon the familiar notion of utility maximization - the latter also entailed a significant change in the traditional neoclassical representation of rational behavior, in particular as far as the role of subjectivism and of consistency restrictions were concerned. Despite its severe limitations, Wald's approach was instead immune from these shortcomings and pointed at an objectivization of decision-making.

Optimization Techniques for State-Constrained Control and Obstacle Problems

The design of control laws for systems subject to complex state constraints still presents a significant challenge. This paper explores a dynamic programming approach to a specific class of such problems, that of reachability under state constraints. The problems are formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given in terms of Hamilton-Jacobi-Bellman (HJB) equations or variational inequalities. The solution of these relations is complicated in general; however, for linear systems, the value functions may be described also in terms of duality relations of convex analysis and minmax theory. Consequently, solution techniques specific to systems with a linear structure may be designed independently of HJB theory. These techniques are illustrated through two examples.

Variational Eigenvalues of Degenerate Eigenvalue Problems for the Weighted p-Laplacian

Abstract We prove the existence of nondecreasing sequences of positive eigenvalues of the homogeneous degenerate quasilinear eigenvalue problem − div(a(x)|▽u|p-2▽u) = λb(x)|u|p-2u, λ > 0 subject to Dirichlet boundary conditions on a bounded domain . The diffusion coefficient a(x) is a function in L1 loc(Ω) and b(x) is a nontrivial function in Lr(Ω) (r depending on a, p and N) and may change sign. We use Ljusternik-Schnirelman theory, minimax theory and the theory of weighted Sobolev spaces to establish our results.

The level set method of Joó and its use in minimax theory

In this paper we discuss the level set method of Joó and how to use it to give an elementary proof of the well-known minimax theorem of Sion. Although this proof technique was initiated by Joó and based on the intersection of upper level sets and a clever use of the topological notion of connectedness, it is not very well known and accessible for researchers in optimization. At the same time we simplify the original proof of Joó and give a more elementary proof of the celebrated minimax theorem of Sion.

On the existence of nontrivial solutions for a fourth‐order semilinear elliptic problem

By means of Minimax theory, we study the existence of one nontrivial solution and multiple nontrivial solutions for a fourth‐order semilinear elliptic problem with Navier boundary conditions.

An adaptation theory for nonparametric confidence intervals

A nonparametric adaptation theory is developed for the construction of confidence intervals for linear functionals. A between class modulus of continuity captures the expected length of adaptive confidence intervals. Sharp lower bounds are given for the expected length and an ordered modulus of continuity is used to construct adaptive confidence procedures which are within a constant factor of the lower bounds. In addition, minimax theory over nonconvex parameter spaces is developed.

On equivalent results in minimax theory

In this paper we review known minimax theorems with applications in game theory and show that these theorems can be proved using the first minimax theorem for a two-person zero-sum game with finite strategy sets published by von Neumann in 1928. Among these results are the well known minimax theorems of Wald, Ville and Kneser and their generalizations due to Kakutani, Ky Fan, König, Neumann and Gwinner–Oettli. Actually, it is shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. To show the implications the authors only use simple properties of compact sets and the well-known Weierstrass–Lebesgue lemma.

Minimax estimation of linear functionals over nonconvex parameter spaces

The minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces. Upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity. However in contrast to the theory for convex parameter spaces rate optimal procedures are often required to be nonlinear. A construction of such nonlinear procedures is given. The results developed in this paper have important applications to the theory of adaptation.

Transversal sets

This paper presents a new concept of sets which we call it transversal (upper, lower or medial) sets. We introduce this concept as a natural extension of ordinary, fuzzy and transversal fuzzy sets. Transversal sets are a new way in the nonlinear analysis. Applications in minimax theory, algebra, topology, analysis, games theory, algebraic equations theory, ideal theory of BCC-algebras, theory of measures and integration, and convex analysis are considered. First time in history of the theory sets, we give a technology of an arbitrary set, in the sense that every set has "three sides" which are invisible but they de facto existing.

Kinematic analysis and optimal design of 3-PPR planar parallel manipulator

This paper proposes a 3-PPR planar parallel manipulator, which consists of three active prismatic joints, three passive prismatic joints, and three passive rotational joints. The analysis of the kinematics and the optimal design of the manipulator are also discussed. The proposed manipulator has the advantages of the closed type of direct kinematics and a void-free workspace with a convex type of borderline. For the kinematic analysis of the proposed manipulator, the direct kinematics, the inverse kinematics, and the inverse Jacobian of the manipulator are derived. After the rotational limits and the workspaces of the manipulator are investigated, the workspace of the manipulator is simulated. In addition, for the optimal design of the manipulator, the performance indices of the manipulator are investigated, and then an optimal design procedure is carried out using Min-Max theory. Finally, one example using the optimal design is presented.

Visualization and Genetic Algorithms in Minimax Theory for Nonlinear Functionals

In this paper, evolution and visualization of the existence of saddle points of nonlinear functionals or multi-variable functions in finite dimensional spaces are presented. New algorithms are developed based on the mountain pass lemma and link thery in nonlinear analysis. Further more, a simple comparison of the steepest descent algorithm and the genetic algorithm is given. The process of the saddle point finding is visualised in an inteactive graphical interface.

Ultimate periodicity of orbits for min-max systems

The ultimate periodicity theorem is an important result in min-max systems theory. It was first proved by Olsder and Perennes in their unpublished work. In this paper, we present a new proof. This proof is also based on two important theorems: the existence of cycle time for any min-max function and the Nussbaum-Sine theorem. However, two different techniques, pure min-max function and conditional redundancy, are used to obtain two important intermediate results. The purpose of this paper is to provide a simple alternate proof to the ultimate periodicity theorem.

Random rates in anisotropic regression (with a discussion and a rejoinder by the authors)

In the context of minimax theory, we propose a new kind of risk, normalized by a random variable, measurable with respect to the data. We present a notion of optimality and a method to construct optimal procedures accordingly. We apply this general setup to the problem of selecting significant variables in Gaussian white noise. In particular, we show that our method essentially improves the accuracy of estimation, in the sense of giving explicit improved confidence sets in L2-norm. Links to adaptive estimation are discussed. 1. Introduction. Searching for significant variables is certainly one of the oldest and most popular problems in statistics. One of the simplest models where the issue of selecting significant variables was first stated mathematically is linear regression. A vast literature has been devoted to this topic since and different approaches have been proposed over the last forty years, both for estimation and for hypothesis testing. Among many authors, we refer to Akaike [1], Breiman and Freedman [3], Chernoff [5], Csiszar and Korner [6], Dychakov [10], Patel [42], Renyi [46], Freidlina [13], Meshalkin [35], Malyutov and Tsitovich [34], Schwarz [47] and Stone [48]. In classical parametric regression, if we consider a linear model, we first have to measure the possible gain of “searching for a limited number of significant variables.” If the model comes from a specific field of application, then only an adequate description together with its solution is relevant. However, from a mathematical point of view, a theory of selecting significant variables does not lead—at least asymptotically—to a substantial improvement of the accuracy of estimation: in a regular parametric model, the classical √ n rate of convergence is not affected by the number of significant variables. (However, even in this setup, let us emphasize that “asymptotically” has to be understood as “up to a constant” and that the correct choice of significant variables may possibly improve this constant.) If instead of a linear model we consider a nonparametric regression model, the search for significant variables becomes crucial for estimating the regression function: the rate of convergence explicitly depends on the set of significant variables. Let us develop this statement with the following example of multivariate regression: suppose we observe Z (n) = (Xi ,Y i ,i = 1 ,... , n)in the model

Sign-Changing Solutions and Multiple Solutions Theorems for Semilinear Elliptic Boundary Value Problems with a Reaction Term Nonzero at Zero

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, in particular, the local mountain pass lemma to study asymptotically linear and superlinear elliptic boundary value problems with a reaction term nonzero at zero, and get new multiple solutions and sign-changing solutions theorems. Moreover, the sign-changing solutions change sign exactly once under some conditions.