A Unified Bregman Alternating Minimization Algorithm for Generalized DC Programs with Application to Imaging

In this paper, we consider a class of nonconvex (not necessarily differentiable) optimization problems called generalized Difference-of-Convex functions (DC) programs, which is minimizing the sum of two separable DC parts and one two-block-variable coupling function. To circumvent the nonconvexity a...

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Veröffentlicht in:	Journal of scientific computing Jg. 101; H. 3; S. 76
Hauptverfasser:	He, Hongjin, Zhang, Zhiyuan
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.12.2024 Springer Nature B.V
Schlagworte:	Algorithms Approximation Artificial intelligence Closed form solutions Computational Mathematics and Numerical Analysis Convexity Coupling Critical point First order algorithms Kernel functions Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Optimization Regularization Theoretical Variables Nonconvex optimization Kurdyka-Łojasiewicz property 90C30 Image processing 65K10 DC programs 90C26 49M37 Bregman distance First-order methods Alternating minimization algorithm
ISSN:	0885-7474, 1573-7691
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Abstract	In this paper, we consider a class of nonconvex (not necessarily differentiable) optimization problems called generalized Difference-of-Convex functions (DC) programs, which is minimizing the sum of two separable DC parts and one two-block-variable coupling function. To circumvent the nonconvexity and nonseparability of the problem under consideration, we accordingly introduce a unified Bregman alternating minimization algorithm by maximally exploiting the favorable DC structure of the objective. Specifically, we first follow the spirit of alternating minimization to update each block variable in a sequential order, which can efficiently tackle the nonseparablitity caused by the coupling function. Then, we employ the Fenchel–Young inequality to approximate the second DC components (i.e., concave parts) so that each subproblem reduces to a convex optimization problem, thereby alleviating the computational burden of the nonconvex DC parts. Moreover, each subproblem absorbs a Bregman proximal regularization term, which is usually beneficial for inducing closed-form solutions of subproblems for many cases via choosing appropriate Bregman kernel functions. It is remarkable that our algorithm not only provides an algorithmic framework to understand the iterative schemes of some recently proposed algorithms, but also enjoys implementable schemes with easier subproblems than some state-of-the-art first-order algorithms developed for generic nonconvex and nonsmooth optimization problems. Theoretically, we prove that the sequence generated by our algorithm globally converges to a critical point under the Kurdyka–Łojasiewicz (KŁ) condition. Besides, we estimate the local convergence rates of our algorithm when we further know the prior information of the KŁ exponent. A series of numerical experiments on imaging data demonstrate the reliability and efficiency of the proposed algorithmic framework.
AbstractList	In this paper, we consider a class of nonconvex (not necessarily differentiable) optimization problems called generalized Difference-of-Convex functions (DC) programs, which is minimizing the sum of two separable DC parts and one two-block-variable coupling function. To circumvent the nonconvexity and nonseparability of the problem under consideration, we accordingly introduce a unified Bregman alternating minimization algorithm by maximally exploiting the favorable DC structure of the objective. Specifically, we first follow the spirit of alternating minimization to update each block variable in a sequential order, which can efficiently tackle the nonseparablitity caused by the coupling function. Then, we employ the Fenchel–Young inequality to approximate the second DC components (i.e., concave parts) so that each subproblem reduces to a convex optimization problem, thereby alleviating the computational burden of the nonconvex DC parts. Moreover, each subproblem absorbs a Bregman proximal regularization term, which is usually beneficial for inducing closed-form solutions of subproblems for many cases via choosing appropriate Bregman kernel functions. It is remarkable that our algorithm not only provides an algorithmic framework to understand the iterative schemes of some recently proposed algorithms, but also enjoys implementable schemes with easier subproblems than some state-of-the-art first-order algorithms developed for generic nonconvex and nonsmooth optimization problems. Theoretically, we prove that the sequence generated by our algorithm globally converges to a critical point under the Kurdyka–Łojasiewicz (KŁ) condition. Besides, we estimate the local convergence rates of our algorithm when we further know the prior information of the KŁ exponent. A series of numerical experiments on imaging data demonstrate the reliability and efficiency of the proposed algorithmic framework.
ArticleNumber	76
Author	Zhang, Zhiyuan He, Hongjin
Author_xml	– sequence: 1 givenname: Hongjin orcidid: 0000-0001-7672-4943 surname: He fullname: He, Hongjin email: hehongjin@nbu.edu.cn organization: School of Mathematics and Statistics, Ningbo University – sequence: 2 givenname: Zhiyuan surname: Zhang fullname: Zhang, Zhiyuan organization: Department of Operations Research and Engineering Management, Southern Methodist University
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Keywords	Nonconvex optimization Kurdyka-Łojasiewicz property 90C30 Image processing 65K10 DC programs 90C26 49M37 Bregman distance First-order methods Alternating minimization algorithm
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Snippet	In this paper, we consider a class of nonconvex (not necessarily differentiable) optimization problems called generalized Difference-of-Convex functions (DC)...
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SourceType	Aggregation Database Enrichment Source Index Database Publisher
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SubjectTerms	Algorithms Approximation Artificial intelligence Closed form solutions Computational Mathematics and Numerical Analysis Convexity Coupling Critical point First order algorithms Kernel functions Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Optimization Regularization Theoretical Variables
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Title	A Unified Bregman Alternating Minimization Algorithm for Generalized DC Programs with Application to Imaging
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Volume	101
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