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Linear and quadratic optimization are crucial in numerous real-world applications, ranging from training machine learning models to solving integer linear programs. Recently, learning-to-optimize methods (L2O) for linear (LPs) or quadratic…
We study the problem of estimating precision matrices in Gaussian distributions that are multivariate totally positive of order two ($\mathrm{MTP}_2$). The precision matrix in such a distribution is an M-matrix. This problem can be…
Interior point methods for solving linearly constrained convex programming involve a variable projection matrix at each iteration to deal with the linear constraints. This matrix often becomes ill-conditioned near the boundary of the…
The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: i) Many well-known operator splitting methods, such as…
We consider minimizing an objective function subject to constraints defined by the intersection of lower-level sets of convex functions. We study two cases: (i) strongly convex and Lipschitz-smooth objective function and (ii) convex but…
Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables.…
Classical existence theorems and solution methods for quadratic programming traditionally rely on the analytical properties of real numbers, specifically compactness and completeness. These tools are unavailable in general linearly ordered…
We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly…
We introduce randomized algorithms to Clifford's Geometric Algebra, generalizing randomized linear algebra to hypercomplex vector spaces. This novel approach has many implications in machine learning, including training neural networks to…
In this paper, we consider the nonconvex quadratically constrained quadratic programming (QCQP) with one quadratic constraint. By employing the conjugate gradient method, an efficient algorithm is proposed to solve QCQP that exploits the…
Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the…
This paper introduces a new way to calculate distance-based statistics, particularly when the data are multivariate. The main idea is to pre-calculate the optimal projection directions given the variable dimension, and to project…
In this paper, we propose a novel distributed data-driven optimization scheme. In detail, we focus on the so-called aggregative framework, a scenario in which a set of agents aim to cooperatively minimize the sum of local costs, each…
A polyhedral convex set optimization problem is given by a set-valued objective mapping from the $n$-dimensional to the $q$-dimensional Euclidean space whose graph is a convex polyhedron. This problem can be seen as the most elementary…
Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…
We study the problem of policy synthesis for uncertain partially observable Markov decision processes (uPOMDPs). The transition probability function of uPOMDPs is only known to belong to a so-called uncertainty set, for instance in the form…
In this paper, we investigate a class of constrained saddle point (SP) problems where the objective function is nonconvex-concave and smooth. This class of problems has wide applicability in machine learning, including robust multi-class…
We consider the problem of solving a large-scale system of linear equations in a distributed or federated manner by a taskmaster and a set of machines, each possessing a subset of the equations. We provide a comprehensive comparison of two…
This paper explores a method for solving constrained optimization problems when the derivatives of the objective function are unavailable, while the derivatives of the constraints are known. We allow the objective and constraint function to…
While the optimization landscape of policy gradient methods has been recently investigated for partially observed linear systems in terms of both static output feedback and dynamical controllers, they only provide convergence guarantees to…