Globally convergent coderivative-based generalized Newton methods in nonsmooth optimization

This paper proposes and justifies two globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are coderivative-based and employ generalized Hessians (coderiv...

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Veröffentlicht in:Mathematical programming Jg. 205; H. 1-2; S. 373 - 429
Hauptverfasser: Khanh, Pham Duy, Mordukhovich, Boris S., Phat, Vo Thanh, Tran, Dat Ba
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2024
Springer
Schlagworte:
Algorithms
Analysis
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
Convex composite optimization
90C31
Variational analysis
Global convergence
Lasso problems
49J53
49J52
Nonsmooth optimization
Generalized Newton methods
Linear and superlinear convergence rates
ISSN:0025-5610, 1436-4646
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Zusammenfassung:This paper proposes and justifies two globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are coderivative-based and employ generalized Hessians (coderivatives of subgradient mappings) associated with objective functions, which are either of class C 1 , 1 , or are represented in the form of convex composite optimization, where one of the terms may be extended-real-valued. The proposed globally convergent algorithms are of two types. The first one extends the damped Newton method and requires positive-definiteness of the generalized Hessians for its well-posedness and efficient performance, while the other algorithm is of the regularized Newton-type being well-defined when the generalized Hessians are merely positive-semidefinite. The obtained convergence rates for both methods are at least linear, but become superlinear under the semismooth ∗ property of subgradient mappings. Problems of convex composite optimization are investigated with and without the strong convexity assumption on smooth parts of objective functions by implementing the machinery of forward–backward envelopes. Numerical experiments are conducted for Lasso problems and for box constrained quadratic programs with providing performance comparisons of the new algorithms and some other first-order and second-order methods that are highly recognized in nonsmooth optimization.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-01980-2