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Impossibility of finding local realistic models for quantum correlations due to entanglement is an important fact in foundations of quantum physics, gaining now new applications in quantum information theory. We present an in-depth…
We generalize the reduction mechanism for linear programming problems and semidefinite programming problems from [arXiv:1410.8816] in two ways 1) relaxing the requirement of affineness and 2) extending to fractional optimization problems.…
Deep matrix factorizations (deep MFs) are recent unsupervised data mining techniques inspired by constrained low-rank approximations. They aim to extract complex hierarchies of features within high-dimensional datasets. Most of the loss…
Consider a problem where a set of feasible observations are provided by an expert and a cost function is defined that characterizes which of the observations dominate the others and are hence, preferred. Our goal is to find a set of linear…
We present a subgradient method for minimizing non-smooth, non-Lipschitz convex optimization problems. The only structure assumed is that a strictly feasible point is known. We extend the work of Renegar [5] by taking a different…
A fundamental theorem of linear programming states that a feasible linear program is solvable if and only if its objective function is copositive with respect to the recession cone of its feasible set. This paper demonstrates that this…
Minimization of the nuclear norm is often used as a surrogate, convex relaxation, for finding the minimum rank completion (recovery) of a partial matrix. The minimum nuclear norm problem can be solved as a trace minimization semidefinite…
In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained…
Facial reduction (FR) is an important tool in linear and semidefinite programming, providing both algorithmic and theoretical insights into these problems. The maximum length of an FR sequence for a convex set is referred to as the maximum…
The objective of this work is to study weak infeasibility in second order cone programming. For this purpose, we consider a relaxation sequence of feasibility problems that mostly preserve the feasibility status of the original problem.…
The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range…
We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be…
Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These…
We propose FlexQP, an always-feasible convex quadratic programming (QP) solver based on an $\ell_1$ elastic relaxation of the QP constraints. If the original constraints are feasible, FlexQP provably recovers the optimal solution. If the…
Interior Point Methods are widely used to solve Linear Programming problems. In this work, we present two primal affine scaling algorithms to achieve faster convergence in solving Linear Programming problems. In the first algorithm, we…
Estimation of nonlinear dynamic models from data poses many challenges, including model instability and non-convexity of long-term simulation fidelity. Recently Lagrangian relaxation has been proposed as a method to approximate simulation…
We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
We consider large linear and nonlinear fixed point problems, and solution with proximal algorithms. We show that there is a close connection between two seemingly different types of methods from distinct fields: 1) Proximal iterations for…
Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bound are not computationally…