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Coordinate descent algorithms solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. They have been used in applications for many years, and their popularity…
In this paper, we describe a low-rank matrix completion method based on matrix decomposition. An incomplete matrix is decomposed into submatrices which are filled with a proposed trimming step and then are recombined to form a low-rank…
We study the minimum weight basis problem on matroid when elements' weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are…
We consider the problem of approximating a semialgebraic set with a sublevel-set of a polynomial function. In this setting, it is standard to seek a minimum volume outer approximation and/or maximum volume inner approximation. As there is…
We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$.…
We propose a conditional gradient framework for a composite convex minimization template with broad applications. Our approach combines smoothing and homotopy techniques under the CGM framework, and provably achieves the optimal…
We show that the max-min-angle polygon in a planar point set can be found in time $O(n\log n)$ and a max-min-solid-angle convex polyhedron in a three-dimensional point set can be found in time $O(n^2)$. We also study the maxmin-angle…
Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables.…
Low-rank modeling plays a pivotal role in signal processing and machine learning, with applications ranging from collaborative filtering, video surveillance, medical imaging, to dimensionality reduction and adaptive filtering. Many modern…
The problem of convex optimization is studied. Usually in convex optimization the minimization is over a d-dimensional domain. Very often the convergence rate of an optimization algorithm depends on the dimension d. The algorithms studied…
This paper studies the convergence properties of a family of Relaxed $\ell$-Minimal Gradient Descent methods for quadratic optimization; the family includes the omnipresent Steepest Descent method, as well as the Minimal Gradient method.…
In this paper, we will present a generalization for a minimization problem from I. Daubechies, M. Defrise, and C. Demol [3]. This generalization is useful for solving many practical problems in which more than one constraint are involved.…
In this paper, we consider the problem of partitioning a polygon into a set of connected disjoint sub-polygons, each of which covers an area of a specific size. The work is motivated by terrain covering applications in robotics, where the…
This paper presents an algorithmic framework for the minimization of strictly convex quadratic functions. The framework is flexible and generic. At every iteration the search direction is a linear combination of the negative gradient, as…
Let $P$ be a set of $n$ points in $\mathbb{R}^2$. For a given positive integer $w<n$, our objective is to find a set $C \subset P$ of points, such that $CH(P\setminus C)$ has the smallest number of vertices and $C$ has at most $n-w$ points.…
This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…
The paper studies decentralized optimization over networks, where agents minimize a composite objective consisting of the sum of smooth convex functions--the agents' losses--and an additional nonsmooth convex extended value function. We…
Making the gradients small is a fundamental optimization problem that has eluded unifying and simple convergence arguments in first-order optimization, so far primarily reserved for other convergence criteria, such as reducing the…
Convex optimization over the spectrahedron, i.e., the set of all real $n\times n$ positive semidefinite matrices with unit trace, has important applications in machine learning, signal processing and statistics, mainly as a convex…
Exotic behaviour of mechanical metamaterials often relies on an internal transformation of the underlying microstructure triggered by its local instabilities, rearrangements, and rotations. Depending on the presence and magnitude of such a…