A Double Projection Algorithm for Solving Non-Monotone Variational Inequalities
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摘要:
投影算法是求解变分不等式问题的主要方法之一。目前，有关投影算法的研究通常需要假设映射是单调且Lipschitz连续的，然而在实际问题中，往往不满足这些假设条件。该文利用线搜索方法，提出了一种新的求解非单调变分不等式问题的二次投影算法。在一致连续假设下，证明了算法产生的迭代序列强收敛到变分不等式问题的解。数值实验结果表明了该文所提算法的有效性和优越性。
Abstract:The projection algorithm is one of the main methods to solve variational inequality problems. At present, the research on projection algorithms usually requires the assumptions that the mapping is monotone and Lipschitz continuous, but in practical problems, these assumptions are often unsatisfied. A new double projection algorithm for solving non-monotone variational inequality problems was proposed with the line search method. Under the assumption that the mapping is uniformly continuous, the sequence generated by the algorithm was proved to strongly converge to the solution of the variational inequality. The numerical experiments illustrate the effectiveness and superiority of the proposed algorithm.
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表 1
$ \varepsilon _{\rm{err}} = 10^{-4}$ 时不同算法关于维数的比较Table 1. For
$ \varepsilon _{\rm{err}} = 10^{-4},$ comparison of different algorithms about dimensions
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表 2
$\varepsilon_ {\rm{err}} = 10^{-4}$ 时不同算法关于初始点的比较Table 2. For
$ \varepsilon_{\rm{err}} = 10^{-4},$ comparison of different algorithms about the initial point$ m = 100$ $ x_{1} = {\rm{rand}}(100,1)$ $x_{1} = 2\times{\rm{rand} }(100,1)$ $x_{1} = 5\times{\rm{rand} }(100,1)$ iter Ni CPU time t/s iter Ni CPU time t/s iter Ni CPU time t/s alg. 1 93 0.6047 119 0.7330 195 1.0379 alg. 3.3 in ref. [14] $ 10^{5}$ 758.4269 $ 10^{5}$ 757.2559 $ 10^{5}$ 748.3109 alg. 4 in ref. [15] 489 5.4683 699 7.6306 924 9.3821
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表 3
$ m = 100$ 时不同算法关于允许误差的比较Table 3. For
$ m = 100,$ comparison of different algorithms about allowable errors$ x_{1} = (1,1,\cdots ,1)$ $\varepsilon_{ {\rm{err} } }=10^{-3}$ $ \varepsilon_{ {\rm{err} } }=10^{-5}$ $ \varepsilon_{ {\rm{err} } }=10^{-8}$ iter Ni CPU time t/s iter Ni CPU time t/s iter Ni CPU time t/s alg. 1 59 0.0142 145 0.0215 166 1.1689 alg. 3.3 in ref. [14] 54460 407.0808 $ 10^{5}$ 764.3791 $ 10^{5}$ 751.3783 alg. 4 in ref. [15] 622 7.1918 1133 13.1318 1418 16.5836
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