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
We develop first-order smoothing techniques for saddle-point problems that arise in the Nash equilibria computation of sequential games. The crux of our work is a construction of suitable prox-functions for a certain class of polytopes that encode the sequential nature of the games. An implementation based on our smoothing techniques computes approximate Nash equilibria for … Read more
We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive … Read more
In this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems. We present a unified treatment of both problems using convex semi–infinite programming. We establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics. The duality results … Read more
Given ${\cal A} := \{a^1,\ldots,a^m\} \subset \R^n$ with corresponding positive weights ${\cal W} := \{\omega_1,\ldots,\omega_m\}$, the weighted Euclidean one-center problem, which is a generalization of the minimum enclosing ball problem, involves the computation of a point $c_{\cal A} \in \R^n$ that minimizes the maximum weighted Euclidean distance from $c_{\cal A}$ to each point in ${\cal … Read more