In the present paper, a new generalization of Browder fixed point theorem is obtained. As its applications, we obtain some generalized versions of Browder’s theorems for quasi-variational inequality and Ky Fan’s minimax inequality and minimax principle.

In the present paper, we are concerned with a minimax fractional programming problem containing arbitrary norms and establish duality theorems for a dual model related to minimax fractional programming problem under the assumptions of second order (Φ, ρ)-invexity.

Let X be a Banach space and Φ : X → R of class C1. We are interested in finding critical points for the restriction of Φ to the manifold M = {u ∈ X : G(u) = 1}, where G : X → R is a C1 function having 1 as a regular value. A point u ∈M is a critical point of the restriction of Φ to M if and only if dΦ(u)|TuM = 0 (see the definition in Section 2). Our purpose is to prove two general minimax prin...

We consider the problem of estimating a random vector x, with covariance uncertainties, that is observed through a known linear transformation H and corrupted by additive noise. We first develop the linear estimator that minimizes the worst-case meansquared error (MSE) across all possible covariance matrices. Although the minimax approach has enjoyed widespread use in the design of robust metho...

We consider the problem of discrete distribution estimation under l1 loss. We provide tight upper and lower bounds on the maximum risk of the empirical distribution (the maximum likelihood estimator), and the minimax risk in regimes where the support size S may grow with the number of observations n. We show that among distributions with bounded entropy H , the asymptotic maximum risk for the e...

Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained `1 minimization is proposed and its rate o...

We study the problem of aggregation of estimators when the estimators are not independent of the data used for aggregation and no sample splitting is allowed. If the estimators are deterministic vectors, it is well known that the minimax rate of aggregation is of order log(M), where M is the number of estimators to aggregate. It is proved that for affine estimators, the minimax rate of aggregat...

This article presents the results of an empirical experiment designed to gain insight into what is the effect of the minimax algorithm on the evaluation function. The experiment’s simulations were performed upon the KRK chess endgame. Our results show that dependencies between evaluations of sibling nodes in a game tree and an abundance of possibilities to commit blunders present in the KRK end...

In this paper we propose a new derivative-free algorithm for linearly constrained finite minimax problems. Due to the nonsmoothness of this class of problems, standard derivative-free algorithms can only locate points which satisfy weak necessary optimality conditions. In this work we define a new derivative-free algorithm which is globally convergent toward standard stationary points of the fi...

We consider the problem of testing hypotheses on the copula density from n bidimensional observations. We wish to test the null hypothesis characterized by a parametric class against a composite nonparametric alternative. Each density under the alternative is separated in the L2-norm from any density lying in the null hypothesis. The copula densities under consideration are supposed to belong t...