Classical stochastic programming has already been used with large scale LP models for long-term analysis of energy-environment systems. We propose a Minimax Regret formulation suitable for large scale linear programming models. It has been experimentally verified that the minimax regret strategy depends only on the extremal scenarios and not on the intermediate ones, thus making the approach co...

We present a topological minimax theorem (Theorem 2.2). The topological assumptions on the spaces involved are somewhat weaker than those usually found in the literature. Even when reinterpreted in the convex setting of topological vector spaces, our theorem yields nonnegligible improvements, for example, of the Passy–Prisman theorem and consequently of the Sion theorem, contrary to most result...

We consider the estimation of regression coefficients in a high-dimensional linear model. A lower bound of the minimax `q risk is provided for regression coefficients in `r balls, along with a minimax lower bound for the tail of the `q loss. Under certain conditions on the design matrix and penalty level, we prove that these minimax convergence rates are attained by both the Lasso and Dantzig e...

In this paper, we propose a trust region filter method for minimax problems. Based on the filter technique, the minimax problem is transformed to a constrained optimization problem and solved by the traditional filter idea. In the presented algorithm, the acceptable criterion of the trial points is relaxed, so compared to the existed SQP and Newton-type methods for minimax method, this method i...

It is well known that a near minimax polynomial approximation p is obtained by truncating the Chebyshev series of a function fafter n + 1 terms. It is shown that if /'E C' " + " [-1, I], then 1I.f-pII may be expressed in terms off' " ' I) in the same manner as the error of minimax approximation. The result is extended to other types of near minimax approximation.

We consider the random design regression with square loss. We propose a method that aggregates empirical minimizers (ERM) over appropriately chosen random subsets and reduces to ERM in the extreme case, and we establish exact oracle inequalities for its risk. We show that, under the −p growth of the empirical -entropy, the excess risk of the proposed method attains the rate n− 2 2+p for p ∈ (0,...

IMP-minimax is the analog to minimax for games with imperfect information, like card games such as bridge or poker. It computes an optimal strategy for the game if the game has a single player and a certain natural property called perfect recall. IMP-minimax is described fully in a companion paper in this proceedings. Here we introduce an algorithm IMP-alpha-beta that is to IMP-minimax as alpha...

Cai et al. (2010) [4] have studied the minimax optimal estimation of a collection of large bandable covariance matrices whose off-diagonal entries decay to zero at a polynomial rate. They have shown that the minimax optimal procedures are fundamentally different under Frobenius and spectral norms, regardless of the rate of polynomial decay. To gain more insight into this interesting problem, we...

Redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length. In the minimax scenario one designs the best code for the worst source within the class. Such minimax redundancy comes in two flavors: average minimax or worst case minimax. We study the worst case minimax redundancy of universal block codes for Markovian sources o...