This paper develops Jellyfish, an algorithm for solving data-processing problems with matrix-valued decision variables regularized to have low rank. Particular examples of problems solvable by Jellyfish include matrix completion problems and least-squares problems regularized by the nuclear norm or the max-norm. Jellyfish implements a projected incremental gradient method with a biased, random ordering of the … Read more
We first show that a continuous function “f” is nonnegative on a closed set K if and only if (countably many) moment matrices of some signed measure dnu = fdmu are all positive semidefinite (if K is compact mu is an arbitrary finite Borel measure with support exactly K). In particular, we obtain a convergent … Read more
This paper focuses on the study of the second-order directional derivative of a symmetric matrix-valued function of the form $F(X)=P\mbox{diag}[f(\lambda_1(X)),\cdots,f(\lambda_n(X))]P^T$. For this purpose, we first adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a matrix in Torki \cite{Tor01}; Second, we establish a formula for the (parabolic) … Read more
We introduce a flexible, open source implementation that provides the optimal sensitivity of solutions of nonlinear programming (NLP) problems, and is adapted to a fast solver based on a barrier NLP method. The program, called sIPOPT evaluates the sensitivity of the KKT system with respect to model parameters. It is paired with the open-source IPOPT … Read more
We propose new stochastic gradient algorithms for solving convex composite optimization problems. In each iteration, our algorithms utilize a stochastic oracle of the gradient of the smooth component in the objective function. Our algorithms are based on a stochastic version of the estimate sequence technique introduced by Nesterov (Introductory Lectures on Convex Optimization: A Basic … Read more
This paper proposes efficient algorithms for group sparse optimization with mixed L21-regularization, which arises from the reconstruction of group sparse signals in compressive sensing, and the group Lasso problem in statistics and machine learning. It is known that encoding the group information in addition to sparsity will lead to better signal recovery/feature selection. The L21-regularization … Read more
Given a family of real or complex monic polynomials of fixed degree with one affine constraint on their coefficients, consider the problem of minimizing the root radius (largest modulus of the roots) or root abscissa (largest real part of the roots). We give constructive methods for efficiently computing the globally optimal value as well as … Read more
A cutting-plane procedure for integer programming (IP) problems usually involves invoking a black-box procedure (such as the Gomory-Chvatal (GC) procedure) to compute a cutting-plane. In this paper, we describe an alternative paradigm of using the same cutting-plane black-box. This involves two steps. In the first step, we design an inequality cx = d + 1\} … Read more
This is a progress report on an implementation of the active-set method for nonlinear programming proposed in [6] that employs piecewise linear models in the active-set prediction phase. The motivation for this work is to develop an algorithm that is capable of solving large-scale problems, including those with a large reduced space. Unlike SQP methods, … Read more
We propose new versions of accelerated first order methods for convex composite optimization, where the prox parameter is allowed to increase from one iteration to the next. In particular we show that a full backtracking strategy can be used within the FISTA \cite{Beck-Teboulle-2009} and FALM algorithms \cite{Goldfarb-Ma-Scheinberg-2010} while preserving their worst-case iteration complexities of $O(\sqrt{L(f)/\epsilon})$. … Read more