We consider a recently proposed optimization formulation of multi-task learning based on trace norm regularized least squares. While this problem may be formulated as a semidefinite program (SDP), its size is beyond general SDP solvers. Previous solution approaches apply proximal gradient methods to solve the primal problem. We derive new primal and dual reformulations of this problem, includin...

In this article we study support vector machine (SVM) classifiers in the face of uncertain knowledge sets and show how data uncertainty in knowledge sets can be treated in SVM classification by employing robust optimization. We present knowledge-based SVM classifiers with uncertain knowledge sets using convex quadratic optimization duality. We show that the knowledge-based SVM, where prior know...

In this paper we present a new class of theorems of the alternative for SOS-convex inequality systems without any qualifications. This class of theorems provides an alternative equations in terms of sums of squares to the solvability of the given inequality system. A strong separation theorem for convex sets, described by convex polynomial inequalities, plays a key role in establishing the clas...

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation QCQP is tight. We begin by outlining general framework for proving such sufficient conditions. Then using framework, show that SDP tight whenever eigenvalue mult...

We investigate an alterative solution method to the joint signal-beamformer optimization problem considered by Setlur and Rangaswamy [1]. First, we directly demonstrate that the problem, which minimizes the recieved noise, interference, and clutter power under a minimum variance distortionless response (MVDR) constraint, is generally non-convex and provide concrete insight into the nature of th...

The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex ...

We develop a natural generalization to the notion of the central path – a notion that lies at the heart of interior-point methods for convex optimization. The generalization is accomplished via the “derivative cones” of a “hyperbolicity cone,” the derivatives being direct and mathematicallyappealing relaxations of the underlying (hyperbolic) conic constraint, be it the non-negative orthant, the...

We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if U i...

For a symmetric positive semidefinite linear system of equations Qx = b, where x = (x1, . . . , xs) is partitioned into s blocks, with s ≥ 2, we show that each cycle of the classical block symmetric Gauss-Seidel (block sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term of the form 12‖x−x ‖T , where T is a symmetric positive semidef...

We develop a scalable, computationally efficient method for the task of energy disaggregation for home appliance monitoring. In this problem the goal is to estimate the energy consumption of each appliance over time based on the total energy-consumption signal of a household. The current state of the art is to model the problem as inference in factorial HMMs, and use quadratic programming to fi...