Related papers: Sparse Copositive Polynomial Optimization
We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert--Artin, Reznick, Putinar, and Putinar--Vasilescu Positivstellens\"atze. First, we…
Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse…
In view of the KS-tensor complementarity problem, the sparse solution of this problem is studied. Due to the nonconvexity and noncontinuity of the l_0-norm, it is a NP hard problem to find the sparse solution of the KS-tensor…
This paper studies moment and tensor recovery problems whose decomposing vectors are contained in some given semialgebraic sets. We propose Moment-SOS relaxations with generic objectives for recovering moments and tensors, whose…
In this work, we consider optimal control problems constrained by elliptic partial differential equations (PDEs) with lognormal random coefficients, which are represented by a countably infinite-dimensional random parameter with i.i.d.…
Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…
Recent research has shown that performance in signal processing tasks can often be significantly improved by using signal models based on sparse representations, where a signal is approximated using a small number of elements from a fixed…
In this paper, we propose new methods to efficiently solve convex optimization problems encountered in sparse estimation, which include a new quasi-Newton method that avoids computing the Hessian matrix and improves efficiency, and we prove…
Our work is focused on the joint sparsity recovery problem where the common sparsity pattern is corrupted by Poisson noise. We formulate the confidence-constrained optimization problem in both least squares (LS) and maximum likelihood (ML)…
Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function…
This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and…
In this paper we present an efficient active-set method for the solution of convex quadratic programming problems with general piecewise-linear terms in the objective, with applications to sparse approximations and risk-minimization. The…
In an effort to develop an alternative approach to traditional sparse reformulations, we will provide a new type of convex reformulation of a large class of stochastic quadratically constrained quadratic optimization problems that is…
Sparse principal component analysis addresses the problem of finding a linear combination of the variables in a given data set with a sparse coefficients vector that maximizes the variability of the data. This model enhances the ability to…
Detection of a signal under noise is a classical signal processing problem. When monitoring spatial phenomena under a fixed budget, i.e., either physical, economical or computational constraints, the selection of a subset of available…
We use convex relaxation techniques to produce lower bounds on the optimal value of subset selection problems and generate good approximate solutions. We then explicitly bound the quality of these relaxations by studying the approximation…
We present the framework of slowly varying regression under sparsity, allowing sparse regression models to exhibit slow and sparse variations. The problem of parameter estimation is formulated as a mixed-integer optimization problem. We…
In theory, hierarchies of semidefinite programming (SDP) relaxations based on sum of squares (SOS) polynomials have been shown to provide arbitrarily close approximations for a general polynomial optimization problem (POP). However, due to…
In this paper, we study a class of fractional semi-infinite polynomial programming problems involving s.o.s-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules…
Sparse optimization is a central problem in machine learning and computer vision. However, this problem is inherently NP-hard and thus difficult to solve in general. Combinatorial search methods find the global optimal solution but are…