We study the asymptotic behavior of almost-orbits of abstract evolution systems in Banach spaces with or without a Lipschitz assumption. In particular, we establish convergence, convergence in average and almost-convergence of almost-orbits both for the weak and the strong topologies based on the behavior of the orbits. We also analyze the set of almost-stationary points. … Read more
The cone of completely positive matrices $C^*$ is the convex hull of all symmetric rank-1-matrices $xx^T$ with nonnegative entries. Determining whether a given matrix $B$ is completely positive is an $\cal NP$-complete problem. We examine a simple algorithm which — for a given input $B$ — either determines a certificate proving that $B\in C^*$ or … Read more
We focus on a permutation betting market under parimutuel call auction model where traders bet on the final ranking of n candidates. We present a Proportional Betting mechanism for this market. Our mechanism allows the traders to bet on any subset of the n x n ‘candidate-rank’ pairs, and rewards them proportionally to the number … Read more
In applications such as signal processing and statistics, many problems involve finding sparse solutions to under-determined linear systems of equations. These problems can be formulated as a structured nonsmooth optimization problems, i.e., the problem of minimizing L_1-regularized linear least squares problems. In this paper, we propose a block coordinate gradient descent method (abbreviated as CGD) … Read more
We study general algorithmic frameworks for online learning tasks. These include binary classification, regression, multiclass problems and cost-sensitive multiclass classification. The theorems that we present give loss bounds on the behavior of our algorithms that depend on general conditions on the iterative step sizes. CitationInternational Journal of Pure and Applied Mathematics, Vol. 46 (2008), pp. … Read more
We present a new algorithm, Iterative Estimation Maximization (IEM), for stochastic linear programs with Conditional Value-at-Risk constraints. IEM iteratively constructs a sequence of compact-sized linear optimization problems, and solves them sequentially to find the optimal solution. The problem size IEM solves in each iteration is unaffected by the size of random samples, which makes it … Read more
We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self-adapting manner. This strategy leaves enough user-flexibility but gives a mathematical guarantee for the algorithm’s behavior in … Read more
This paper describes a parallel space decomposition PSD technique for the mesh adaptive direct search MADS algorithm. MADS extends a generalized pattern search for constrained nonsmooth optimization problems. The objective of the present work is to obtain good solutions to larger problems than the ones typically solved by MADS. The new method PSD-MADS is an … Read more
he purpose of this paper is to introduce a new way of choosing directions for the mesh adaptive direct search (Mads) class of algorithms. The advantages of this new OrthoMads instantiation of Mads are that the polling directions are chosen deterministically, ensuring that the results of a given run are repeatable, and that they are … Read more
We propose an iterated version of Nesterov’s first-order smoothing method for the two-person zero-sum game equilibrium problem $$\min_{x\in Q_1} \max_{y\in Q_2} \ip{x}{Ay} = \max_{y\in Q_2} \min_{x\in Q_1} \ip{x}{Ay}.$$ This formulation applies to matrix games as well as sequential games. Our new algorithmic scheme computes an $\epsilon$-equilibrium to this min-max problem in $\Oh(\kappa(A) \ln(1/\epsilon))$ first-order iterations, … Read more