An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization

Ruyu Liu (), Shaohua Pan (), Yuqia Wu () and Xiaoqi Yang ()
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Ruyu Liu: South China University of Technology
Shaohua Pan: South China University of Technology
Yuqia Wu: The Hong Kong Polytechnic University
Xiaoqi Yang: The Hong Kong Polytechnic University

Computational Optimization and Applications, 2024, vol. 88, issue 2, No 7, 603-641

Abstract: Abstract This paper focuses on the minimization of a sum of a twice continuously differentiable function f and a nonsmooth convex function. An inexact regularized proximal Newton method is proposed by an approximation to the Hessian of f involving the $$\varrho $$ ϱ th power of the KKT residual. For $$\varrho =0$$ ϱ = 0 , we justify the global convergence of the iterate sequence for the KL objective function and its R-linear convergence rate for the KL objective function of exponent 1/2. For $$\varrho \in (0,1)$$ ϱ ∈ ( 0 , 1 ) , by assuming that cluster points satisfy a locally Hölderian error bound of order q on a second-order stationary point set and a local error bound of order $$q>1\!+\!\varrho $$ q > 1 + ϱ on the common stationary point set, respectively, we establish the global convergence of the iterate sequence and its superlinear convergence rate with order depending on q and $$\varrho $$ ϱ . A dual semismooth Newton augmented Lagrangian method is also developed for seeking an inexact minimizer of subproblems. Numerical comparisons with two state-of-the-art methods on $$\ell _1$$ ℓ 1 -regularized Student’s t-regressions, group penalized Student’s t-regressions, and nonconvex image restoration confirm the efficiency of the proposed method.

Keywords: Nonconvex and nonsmooth optimization; Regularized proximal Newton method; Global convergence; Convergence rate; KL function; Metric q-subregularity; 90C26; 49M15; 90C55 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-024-00560-0

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