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On monotone and primal-dual active set schemes for l^p-type problems, p(0,1]

COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2019年第1期72卷 45-85页

作者： Ghilli, Daria Kunisch, Karl Karl Franzens Univ Graz Inst Math & Sci Comp Heinrichstr 36 A-8010 Graz Austria Austrian Acad Sci Johann Radon Inst Computat & Appl Math RICAM Altenbergerstr 69 A-4040 Linz Austria

Nonsmooth nonconvex optimization problems involving the p quasi-norm, p(0,1], of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developed and necessary optimality conditions for the original problem in the form of a complementary system amenable for computation are given. Then an algorithm for solving the above mentioned necessary optimality conditions is proposed. It is based on a combination of the monotone scheme and a primal-dual active set strategy. The performance of the two algorithms is studied by means of a series of numerical tests in different cases, including optimal control problems, fracture mechanics and microscopy image reconstruction.

关键词： Nonsmooth nonconvex optimization Active-set method monotone algorithm Optimal control problems Image reconstruction Fracture mechanics

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On a monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第2期183卷 609-641页

作者： Ghilli, Daria Kunisch, Karl Karl Franzens Univ Graz Inst Math & Sci Comp Graz Austria Johann Radon Inst Linz Austria

A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is also compared to two alternative algorithms for nonsmooth nonconvex optimization arising in optimal control and mathematical imaging.

关键词： Nonsmooth nonconvex optimization monotone algorithm Fracture mechanics Sparse recovery

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A monotone scheme for sparsity optimization in lp with p Є (0,1]

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IFAC-PapersOnLine 2017年第1期50卷 494-499页

作者： Ghilli D. Kunisch K. Institute of Mathematics and Scientific Computing Karl-Franzens-Universitaät Heinrichstrasse 36 Graz A8010 Austria Johann Radon Institute for Computational and Applied Mathematics(RICAM) Austrian Academy of Sciences Altenbergerstrasse 69 Linz A4040 Austria

Nonsmooth nonconvex optimization problems are considered in infinite dimensional sequence spaces lp with p Є (0,1]. Our starting points are necessary optimality conditions in the form of a complementary system and a monotonically convergent algorithm for a regularized version of the original problem. We propose an algorithm for solving the necessary optimality condition based on a combination of the monotone scheme and an active-set strategy. Numerical results for different test cases are provided, e.g. for optimal control problems and microscopy image reconstruction. © 2017

关键词： active-set method image reconstruction monotone algorithm nonsmooth nonconvex optimization optimal control problems sparsity optimization

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