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Non-convex Optimization via Strongly Convex Majorization-minimization

Part of: Integral equations, integral transforms

Published online by Cambridge University Press:  10 December 2019

Azita Mayeli
Azita Mayeli*
Affiliation:
Department of Mathematics, The City University of New York, USA Email: amayeli@gc.cuny.edu
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Abstract

In this paper, we introduce a class of nonsmooth nonconvex optimization problems, and we propose to use a local iterative minimization-majorization (MM) algorithm to find an optimal solution for the optimization problem. The cost functions in our optimization problems are an extension of convex functions with MC separable penalty, which were previously introduced by Ivan Selesnick. These functions are not convex; therefore, convex optimization methods cannot be applied here to prove the existence of optimal minimum point for these functions. For our purpose, we use convex analysis tools to first construct a class of convex majorizers, which approximate the value of non-convex cost function locally, then use the MM algorithm to prove the existence of local minimum. The convergence of the algorithm is guaranteed when the iterative points $x^{(k)}$ are obtained in a ball centred at $x^{(k-1)}$ with small radius. We prove that the algorithm converges to a stationary point (local minimum) of cost function when the surregators are strongly convex.

Keywords

Cost functionlocal majorizer and minimizersurregatorMoreau envelopeinfimal convolutionconvex functionstationary point

MSC classification

Primary: 65R32: Inverse problems
Secondary: 90C26: Nonconvex programming, global optimization
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020 , pp. 726 - 737
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Copyright
© Canadian Mathematical Society 2019

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