A Line Search Penalty-Free Method for Nonlinear Second-Order Cone Programming
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A Line Search Penalty-Free Method for Nonlinear Second-Order Cone Programming Qi Zhao1 · Zhongwen Chen2
Received: 22 June 2019 / Accepted: 7 May 2020 © Springer Nature B.V. 2020
Abstract In this paper, we propose a line search penalty-free method for solving nonlinear second-order cone programming (NSOCP) problem. Compared with the traditional SQPtype method for NSOCP, our method does not need the assumption that the subproblem is feasible. Besides that, it does not use penalty-function or filter technique. We first use a robust linear second-order cone programming subproblem to get a detective step and then compute an optimal step from a quadratic optimization subproblem. The search direction is a convex combination of the detective step and optimal step. This two-phase strategy and the penalty-free technique are employed to promote global convergence which is analyzed under mild assumptions. We report some numerical experiments whose results show that the proposed algorithm is applicable and efficient. Keywords Nonlinear second-order cone programming · Two-phase strategy · Penalty-free technique · Global convergence Mathematics Subject Classification 90C22 · 90C55
1 Introduction We consider the following nonlinear second-order cone programming (NSOCP) problem The work was supported by Chinese NSF grant 11871362 and Overseas Study Fund and Start-up Fund for doctoral research by JiangSu University of Science and Technology.
B Z. Chen
[email protected] Q. Zhao [email protected]
1
Jiangsu University of Science and Technology, ZhenJiang, 212001, P.R. China
2
School of Mathematics Science, Soochow University, Suzhou, 215006, P.R. China
Q. Zhao, Z. Chen
min
f (x)
s.t.
hi (x) = 0, i ∈ E = {1, 2, . . . , p},
x∈Rn
(1.1)
gi (x) ∈ Ki , Ki ⊆ Rmi , i ∈ I = {1, 2, . . . , l}, where f : Rn → R and h : Rn → Rp , h(x) = (h1 (x), h2 (x), . . . , hp (x))T , gi : Rn → Rmi (i ∈ I ) are smooth functions. Denote a vector z as (z(1) ; z¯ ), where z(1) is the first entry of z and z¯ is the subvector that consists of the remaining entries. The mi -dimensional second-order cone Ki (i ∈ I ) is defined by {z ∈ R | z ≥ 0}, mi = 1, Ki = {(z(1) ; z¯ ) ∈ R × Rmi −1 | z(1) ≥ ||¯z||}, mi ≥ 2, where || · || denotes the Euclidean norm. The NSOCP problem is a particular case of nonlinear semidefinite programming [4] and has many applications [18]. Many efficient software packages (SDPT3, SeDuMi, MOSEK, PENNON) have been developed for solving linear second-order cone programming (LSOCP) [19, 21, 22], which is a special case of NSOCP since all f (x), h(x), gi (x) are affine functions. Theoretical properties of NSOCP have been studied in [4, 5] and different algorithms have been proposed for solving NSOCP. For example, Kanzow, Ferenczi and Fukushima [14] reformulated NSOCP as a nonsmooth system of equations using a projection mapping. Their analysis was done for a particular case where the constraints are linear. Yamashita and Yabe [25] proposed a primal-dual interior point algorithm by combining the barrier penalty function and the p
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