Convex approximations in stochastic programming by semidefinite programming

The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in $L_1$, $L_\infty$ and $L_2$ norm. Extensive numerical experiments … Read more

The Globally Uniquely Solvable Property of Second-Order Cone Linear Complementarity Problems

The globally uniquely solvable (GUS) property of the linear transformation of the linear complementarity problems over symmetric cones has been studied recently by Gowda et al. via the approach of Euclidean Jordan algebra. In this paper, we contribute a new approach to characterizing the GUS property of the linear transformation of the second-order cone linear … Read more

The tracial moment problem and trace-optimization of polynomials

The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f, what is the smallest trace f(A) can attain for a tuple of matrices A? A relaxation using semidefinite programming (SDP) based on sums of hermitian squares and commutators is proposed. While this relaxation is not … Read more

Dual Averaging Methods for Regularized Stochastic Learning and Online Optimization

We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as $\ell_1$-norm for promoting sparsity. We develop extensions of Nesterov’s dual averaging method, that can exploit the … Read more

Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a hBcmatrix

The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) $0\leq x\perp(Mx+q)\geq0$ can be viewed as a nonsmooth Newton algorithm without globalization technique to solve the system of piecewise linear equations $\min(x,Mx+q)=0$, which is equivalent to the LCP. When $M$ is an $M$-matrix of order~$n$, the algorithm is known to converge in … Read more

Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube

We consider the problem of minimizing a polynomial on the hypercube [0,1]^n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmuedgen (1991). The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates … Read more

Approximating the minimum directed tree cover

Given a directed graph $G$ with non negative cost on the arcs, a directed tree cover of $G$ is a directed tree such that either head or tail (or both of them) of every arc in $G$ is touched by $T$. The minimum directed tree cover problem (DTCP) is to find a directed tree cover … Read more

A biased random-key genetic algorithm for routing and wavelength assignment

The problem of routing and wavelength assignment (RWA) in wavelength division multiplexing (WDM) optical networks consists in routing a set of lightpaths and assigning a wavelength to each of them, such that lightpaths whose routes share a common fiber are assigned different wavelengths. This problem was shown to be NP-hard when the objective is to … Read more

Prediction Range Estimation from Noisy Raman Spectra

Inferences need to be drawn in biological systems using experimental multivariate data. The number of samples collected in many such experiments is small, and the data is noisy. We present and study the performance of a robust optimization (RO) model for such situations. We adapt this model to generate a minimum and a maximum estimation … Read more

Semi-algebraic functions have small subdifferentials

We prove that the subdifferential of any semi-algebraic extended-real-valued function on $\R^n$ has $n$-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems. Citation Cornell University, School of Operations Research and Information Engineering, 206 Rhodes Hall Cornell University Ithaca, NY 14853. April 2010. Article Download View Semi-algebraic functions have small subdifferentials