A New QP-free Algorithm Without a Penalty Function or a Filter for Nonlinear Semidefinite Programming
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Acta Mathemacae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2020
A New QP-free Algorithm Without a Penalty Function or a Filter for Nonlinear Semidefinite Programming Jian-ling LI1 , Zhen-ping YANG2 , Jia-qi WU1 , Jin-bao JIAN3,† 1 College 2 School
of Mathematics and Information Science, Guangxi University, Nanning 530004, China of Mathematics, Jiaying University, Meizhou 514015, China
3 College
of Science, Guangxi University for Nationalities, Nanning 530006, China († E-mail: [email protected])
Abstract In this paper, we present a QP-free algorithm without a penalty function or a filter for nonlinear semidefinite programming. At each iteration, two systems of linear equations with the same coefficient matrix are solved to determine search direction; the nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced. There is no feasibility restoration phase in our algorithm, which is necessary for traditional filter methods. The proposed algorithm is globally convergent under some mild conditions. Preliminary numerical results indicate that the proposed algorithm is comparable. Keywords vergence
nonlinear semidefinite programming; QP-free; penalty-free; nonmonotone line search; global con-
2000 MR Subject Classification
1
90C30; 65K05
Introduction
Consider the following nonlinear semidefinite programming (NLSDP for short): min f (x) s.t. G(x) ≼ 0; hj (x) = 0, j ∈ E = {1, 2, · · · , l};
(1.1)
where f : Rn → R, hj : Rn → R and G : Rn → Sm are twice continuously differentiable functions, not necessarily convex. Sm is the set of m × m real symmetric matrices, G(x) ≼ 0 means that G(x) is a negative semidefinite matrix. Nonlinear semidefinite programming has many applications, e.g., optimal control problems, optimal structural design, truss design problems (see [2, 10, 25]). In recent years, nonlinear semidefinite programming has been attracting a great deal of research attention (cf. [1, 2, 5, 10, 11]). As we know, NLSDP is an extension of nonlinear programming and some efficient numerical methods for the latter are generalized to solve NLSDP. For example, Fare et al. in [5] applied the sequential linear SDP method to robust control problems. Kanzow et al. in [11] presented a successive linearization method with a trust region-type globalization strategy. Freund et al. in [6] also developed a sequential SDP method. Correa and Ramirez in [4] proposed a sequential SDP algorithm. Gomez and Ramirez in [7], Zhu and Zhu in [30] extended the filter SQP method for nonlinear programming to NLSDP, respectively. Zhao and Chen Manuscript received March 20, 2019. Accepted on November 25, 2019. The research is supported by the National Natural Science Foundation (No.11561005), the National Science Foundation of Guangxi (No.2016GXNSFAA380248). † Corresponding author.
A New QP-free Algorithm for Nonlinear Semidefinite Programming
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in [29] proposed a penalty-free method for NLSDP. They introduced second-order corre
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