Related papers: Penalty Decomposition Methods for $L0$-Norm Minimi…
In this paper we consider sparse approximation problems, that is, general $l_0$ minimization problems with the $l_0$-"norm" of a vector being a part of constraints or objective function. In particular, we first study the first-order…
In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first establish that a class of special rank minimization problems has closed-form solutions. Using this result,…
We reformulate the zero-norm minimization problem as an equivalent mathematical program with equilibrium constraints and establish that its penalty problem, induced by adding the complementarity constraint to the objective, is exact. Then,…
This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints…
The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models,…
In this paper, we consider the problem of minimizing a smooth function, given as finite sum of black-box functions, over a convex set. In order to advantageously exploit the structure of the problem, for instance when the terms of the…
The composite L0 function serves as a sparse regularizer in many applications. The algorithmic difficulty caused by the composite L0 regularization (the L0 norm composed with a linear mapping) is usually bypassed through approximating the…
We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range…
Many contemporary signal processing, machine learning and wireless communication applications can be formulated as nonconvex nonsmooth optimization problems. Often there is a lack of efficient algorithms for these problems, especially when…
This paper proposes an improved quasi-Newton penalty decomposition algorithm for the minimization of continuously differentiable functions, possibly nonconvex, over sparse symmetric sets. The method solves a sequence of penalty subproblems…
Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a…
In compressed sensing, the l0-norm minimization of sparse signal reconstruction is NP-hard. Recent work shows that compared with the best convex relaxation (l1-norm), nonconvex penalties can better approximate the l0-norm and can…
We consider a convex optimization problem with many linear inequality constraints. To deal with a large number of constraints, we provide a penalty reformulation of the problem, where the penalty is a variant of the one-sided Huber loss…
We consider a general decomposable convex optimization problem. By using right-hand side allocation technique, it can be transformed into a collection of small dimensional optimization problems. The master problem is a convex non-smooth…
For minimizing a strongly convex objective function subject to linear inequality constraints, we consider a penalty approach that allows one to utilize stochastic methods for problems with a large number of constraints and/or objective…
Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal, then, is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for…
The constrained $\ell_0$ regularization plays an important role in sparse reconstruction. A widely used approach for solving this problem is the penalty method, of which the least square penalty problem is a special case. However, the…
We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed $l_1$ (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates $l_0$ and $l_1$ norms through a nonnegative…
We propose a deep learning algorithm for high dimensional optimal stopping problems. Our method is inspired by the penalty method for solving free boundary PDEs. Within our approach, the penalized PDE is approximated using the Deep BSDE…
Optimization problems with norm-bounding constraints arise in a variety of applications, including portfolio optimization, machine learning, and feature selection. A common approach to these problems involves relaxing the norm constraint…