A Globally Convergent QP-Free Algorithm for Inequality Constrained Minimax Optimization
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
A GLOBALLY CONVERGENT QP-FREE ALGORITHM FOR INEQUALITY CONSTRAINED MINIMAX OPTIMIZATION∗
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Jinbao JIAN (
College of Mathematics and Physics, Guangxi University for Nationalities, Nanning 530006, China E-mail : [email protected]
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Guodong MA (
†
School of Mathematics and Statistics, Yulin Normal University, Yulin 537000, China E-mail : [email protected] Abstract Although QP-free algorithms have good theoretical convergence and are effective in practice, their applications to minimax optimization have not yet been investigated. In this article, on the basis of the stationary conditions, without the exponential smooth function or constrained smooth transformation, we propose a QP-free algorithm for the nonlinear minimax optimization with inequality constraints. By means of a new and much tighter working set, we develop a new technique for constructing the sub-matrix in the lower right corner of the coefficient matrix. At each iteration, to obtain the search direction, two reduced systems of linear equations with the same coefficient are solved. Under mild conditions, the proposed algorithm is globally convergent. Finally, some preliminary numerical experiments are reported, and these show that the algorithm is promising. Key words
minimax optimization; inequality constraints; QP-free algorithm; global convergence
2010 MR Subject Classification
1
49K35; 90C30; 65K05
Introduction
In this work, we discuss the nonlinear minimax optimization with inequality constraints of the form min F (x) (1.1) s.t. fi (x) ≤ 0, i ∈ I = {l + 1, l + 2, · · · , l + m}, where F (x) = max{fj (x), j ∈ J} with J = {1, 2, · · · , l}. The discussed type of problem may occur in engineering design [1], control system design [2], subproblems in semi-infinite minimax ∗ Received
August 2, 2019; revised July 15, 2020. This work was supported by the Natural Science Foundation of Guangxi Province (2018GXNSFAA281099), the National Natural Science Foundation of China (11771383) and the Yulin Normal University Research Grant (2019YJKY16). † Corresponding author: Guodong MA.
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
algorithms [3] and portfolio optimization [4]. The non-differentiability of the objective function possesses the main challenge for solving minimax problems, as the standard constrained optimization algorithms cannot be applied directly. On the basis of smooth transformation problems, many algorithms have been proposed, and these can be grouped into two classes of methods. The first method is the regularization technique in which the objective function is approximated by the exponential smooth functions [5–7]; a fundamental challenge of these smoothing algorithms is that the smoothed problem becomes increasingly ill-conditioned as the approximation becomes more accurate. In [6], the authors proposed an adaptive precision-parameter adjustment scheme to ensure that the smoothing precision parameter is kept small (thus control
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