In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dualactiveset type for a class of nonconvex sparsitypromotin...
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In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dualactiveset type for a class of nonconvex sparsitypromoting penalties, including l(0), bridge, smoothly clipped absolute deviation, capped l(1) and minimax concavity penalty. First, we establish the existence of a global minimizer for the related optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. The solutions to the optimality system are coordinatewise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the activeset can be determined using the primal and dual variables together. Further, this relation lends itself to an iterative algorithm of activeset type which at each step involves first updating the primal variable only on the activeset and then updating the dual variable explicitly. When combined with a continuation strategy on the regularization parameter, the primaldualactiveset method is shown to converge globally to the underlying regression target under certain regularity conditions. Extensive numerical experiments with both simulated and real data demonstrate its superior performance in terms of computational efficiency and recovery accuracy compared with the existing sparse recovery methods.
We design multigrid methods for an elliptic distributed optimal control problem with pointwise state constraints. They are based on the P1 finite element method from Brenner et al. (2021), where the optimal control pr...
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We design multigrid methods for an elliptic distributed optimal control problem with pointwise state constraints. They are based on the P1 finite element method from Brenner et al. (2021), where the optimal control problem was reformulated as a variational inequality for the state variable. The performance of the multigrid methods is illustrated by numerical experiments. (c) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).
Motivated by an obstacle problem for a membrane subject to cohesion forces, constrained minimization problems involving a nonconvex and nondifferentiable objective functional representing the total potential energy ar...
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Motivated by an obstacle problem for a membrane subject to cohesion forces, constrained minimization problems involving a nonconvex and nondifferentiable objective functional representing the total potential energy are considered. The associated first-order optimality system leads to a hemivariational inequality, which can also be interpreted as a special complementarity problem in function space. Besides an analytical investigation of first-order optimality, a primal-dualactiveset solver is introduced. It is associated to a limit case of a semismooth Newton method for a regularized version of the underlying problem class. For the numerical algorithms studied in this paper, global as well as local convergence properties are derived and verified numerically.
The mathematical model of a crack with non-penetration conditions is considered in the framework of 3D elasticity. The spatial crack problem is investigated with respect to its numerical realization in the context of ...
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The mathematical model of a crack with non-penetration conditions is considered in the framework of 3D elasticity. The spatial crack problem is investigated with respect to its numerical realization in the context of constrained optimization. Specifically, for homogeneous isotropic solids with planar cracks, a Papkovich-Neuber-based representation is adopted. It allows to employ a primal-dualactiveset strategy with an unconditional global and monotone convergence property. The iterates turn out to be primally feasible. Illustrative numerical examples are presented.
This paper deals with convergence results of Howard's algorithm for the resolution of min(a is an element of A)(B(a)x-c(a)) = 0, where B-a is a matrix, c(a) is a vector, and A is a compact set. We show a global su...
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This paper deals with convergence results of Howard's algorithm for the resolution of min(a is an element of A)(B(a)x-c(a)) = 0, where B-a is a matrix, c(a) is a vector, and A is a compact set. We show a global superlinear convergence result, under a monotonicity assumption on the matrices Ba. Extensions of Howard's algorithm for a max-min problem of the form max(b is an element of B) min(a is an element of A)(B(a,b)x-c(a,b)) = 0 are also proposed. In the particular case of an obstacle problem of the form min(Ax-b, x-g) = 0, where A is an N x N matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than N iterations, instead of the usual 2(N) bound. Still in the case of an obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm [M. Hintermuller, K. Ito, and K. Kunisch, SIAM J. Optim., 13 (2002), pp. 865 888]. The algorithms are illustrated on the discretization of nonlinear PDEs arising in the context of mathematical finance (American option and Merton's portfolio problem), of front propagation problems, and for the double-obstacle problem.
In this paper a simplified friction problem and iterative second-order algorithms for its solution are analyzed in infinite dimensional function spaces. Motivated from the dual formulation, a primal-dualactiveset st...
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In this paper a simplified friction problem and iterative second-order algorithms for its solution are analyzed in infinite dimensional function spaces. Motivated from the dual formulation, a primal-dualactiveset strategy and a semismooth Newton method for a regularized problem as well as an augmented Lagrangian method for the original problem are presented and their close relation is analyzed. Local as well as global convergence results are given. By means of numerical tests, we discuss among others convergence properties, the dependence on the mesh, and the role of the regularization and illustrate the efficiency of the proposed methodologies.
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