Related papers: Approximation Bounds for Sparse Programs
Results on two different settings of asymptotic behavior of approximation characteristics of individual functions are presented. First, we discuss the following classical question for sparse approximation. Is it true that for any individual…
We consider the problem of minimizing a sum of non-convex functions over a compact domain, subject to linear inequality and equality constraints. Approximate solutions can be found by solving a convexified version of the problem, in which…
We present a novel analysis of semidefinite programs (SDPs) with positive duality gaps, i.e. different optimal values in the primal and dual problems. These SDPs are extremely pathological, often unsolvable, and also serve as models of more…
We investigate time-dependent optimization problems in fractional Sobolev spaces with the sparsity promoting $L^p$-pseudo norm for $0<p<1$ in the objective functional. In order to avoid computing the fractional Laplacian on the time-space…
We present a general method for obtaining strong bounds for discrete optimization problems that is based on a concept of branching duality. It can be applied when no useful integer programming model is available, and we illustrate this with…
In recent years, information relaxation and duality in dynamic programs have been studied extensively, and the resulted primal-dual approach has become a powerful procedure in solving dynamic programs by providing lower-upper bounds on the…
Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel…
A memory-efficient framework is described for the cardinality-constrained structured data-fitting problem. Dual-based atom-identification rules are proposed that reveal the structure of the optimal primal solution from near-optimal dual…
We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares…
Decision makers routinely use constrained optimization technology to plan and operate complex systems like global supply chains or power grids. In this context, practitioners must assess how close a computed solution is to optimality in…
This paper studies distributed convex optimization with both affine equality and nonlinear inequality couplings through the duality analysis. We first formulate the dual of the coupling-constraint problem and reformulate it as a consensus…
Motivated by applications of large embedding models, we study differentially private (DP) optimization problems under sparsity of individual gradients. We start with new near-optimal bounds for the classic mean estimation problem but with…
In this paper we consider a distributed optimization scenario in which a set of agents has to solve a convex optimization problem with separable cost function, local constraint sets and a coupling inequality constraint. We propose a novel…
We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the…
This paper studies dynamic stochastic optimization problems parametrized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of…
The support recovery problem consists of determining a sparse subset of a set of variables that is relevant in generating a set of observations, and arises in a diverse range of settings such as compressive sensing, and subset selection in…
We present a new duality theory for non-convex variational problems, under possibly mixed Dirichlet and Neumann boundary conditions. The dual problem reads nicely as a linear programming problem, and our main result states that there is no…
Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing. Computations over sparse structures are less complex compared to their dense counterparts, and their storage consumes less space. This…
The constrained minimization (respectively maximization) of directed distances and of related generalized entropies is a fundamental task in information theory as well as in the adjacent fields of statistics, machine learning, artificial…
Signal processing is rich in inherently continuous and often nonlinear applications, such as spectral estimation, optical imaging, and super-resolution microscopy, in which sparsity plays a key role in obtaining state-of-the-art results.…