The rate of convergence of proximal method of multipliers for second-order cone optimization problems
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The rate of convergence of proximal method of multipliers for second-order cone optimization problems Li Chu1,2 · Bo Wang3
· Liwei Zhang4 · Hongwei Zhang4
Received: 2 November 2019 / Accepted: 30 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper we consider a proximal method of multipliers (PMM) for a nonlinear second-order cone optimization problem. With the assumptions of constraint nondegeneracy, strict complementarity and second-order sufficient condition, we estimate the local convergence rate of PMM to be linear or superlinear, which depends on the strategy of parameter selection. Keywords Rate of convergence · Proximal method · Second-order cone
1 Introduction In this paper, we consider the following nonlinear second-order cone optimization problem (NSOCP): min f (x) s.t. h(x) = 0, (1) G(x) ∈ K,
B
Bo Wang [email protected] Li Chu [email protected] Liwei Zhang [email protected] Hongwei Zhang [email protected]
1
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2
City Institute, Dalian University of Technology, Dalian 116600, China
3
Key Laboratory of Operations Research and Control of Universities in Fujian, College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China
4
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
123
L. Chu et al.
where f : Rn → R, h : Rn → Rq and G : Rn → Rm are continuously differentiable J m mappings. Cone K := Km 1 × Km 2 × · · · × Km J with m = j=1 j , which is a Cartesian product of second-order cones. The second-order cone (or ice-cream cone, or Lorentz cone) of dimension m j is defined by Km j := (x1 ; x2 ) ∈ R × Rm j −1 | x1 ≥ x2 . The NSOCP has various applications in engineering, finance, robust optimization and many other fields [1,11]. In the literature, many related methods have been proposed to solve NSOCP, e.g., Newton type methods [6,8], augmented Lagrangian method [9,10], homotopy method [15], we only mention several of them. Recently, some new theoretical and practical results have been published, e.g., existence condition for saddle points by Zhou and Chen [19], new constraint qualification for NSOCP by Zhang and Zhang [18], feasible direction algorithm by Canelas et al. [4]. In this paper, we consider the proximal method of multipliers (PMM) for NSOCP. PMM is different from but closely related to the popular augmented Lagrangian method. Augmented Lagrangian method can trace back to Hestenes [7], Powell [12] and Rockafellar [13] for non-linear programming. Based on these foundations, Rockafellar [14] proposed three proximal point methods for convex programming: the proximal point method of the primal problem, the augmented Lagrange method (proximal point method applied to the dual problem) and the proximal method of multipliers (PMM). The last one has not received much attention as the first two, until recently. Zhang et al. [16] study a PMM for equality constrained optimization problem. Linear or
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