Euclidean Jordan-algebra is a commonly used tool in designing interior point algorithms for symmetric cone programs. T-algebra, on the other hand, has rarely been used in symmetric cone programming. In this paper, we use both algebraic characterizations of symmetric cones to extend the target-following framework of linear programming to symmetric cone programming. Within this framework, … Read more
In this paper, we analyze the limiting behavior of the weighted least squares problem $\min_{x\in\Re^n}\sum_{i=1}^p\|D_i(A_ix-b_i)\|^2$, where each $D_i$ is a positive definite diagonal matrix. We consider the situation where the magnitude of the weights are drastically different block-wisely so that $\max(D_1)\geq\min(D_1) \gg \max(D_2) \geq \min(D_2) \gg \max(D_3) \geq \ldots \gg \max(D_{p-1}) \geq \min(D_{p-1}) \gg \max(D_p)$. … Read more
The convex cone of $n \times n$ completely positive (CPP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CPP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for $n \le 4$ only, every DNN matrix is CPP. … Read more
Given an approximation $\{f_n\}$ of a given objective function $f$, we provide simple and fairly general conditions under which a diagonal proximal point algorithm approximates the value $\inf f$ at a reasonable rate. We also perform some numerical tests and present a short survey on finite convergence. CitationTo appear in Journal of Convex Analysis, 16 … 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
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
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