SIAM Journal on Optimization, Volume 30, Issue 3, Page 2628-2658, January 2020.
We introduce a new feasible corrector-predictor (CP) interior-point algorithm (IPA), which is suitable for solving linear complementarity problem (LCP) with $P_{*} (\kappa)$-matrices. We use the method of algebraically equivalent transformation (AET) of the nonlinear equation of the system which defines the central path. The AET is based on the function $\varphi(t) = t - \sqrt{t}$ and plays a crucial role in the calculation of the new search direction. We prove that the algorithm has $O((1+2 \kappa) \sqrt{n} \log \frac{9n \mu^0}{8\epsilon} )$ iteration complexity, where $\kappa$ is an upper bound of the handicap of the input matrix. To the best of our knowledge, this is the first CP IPA for $P_*(\kappa)$-LCPs which is based on this search direction. We implement the proposed CP IPA in the C++ programming language with specific parameters and demonstrate its performance on three families of LCPs. The first family consists of LCPs with $P_{*} (\kappa)$-matrices. The second family of LCPs has the $P$-matrix defined by Csizmadia. Eisenberg-Nagy and de Klerk [Math. Program., 129 (2011), pp. 383--402] showed that the handicap of this matrix should be at least $2^{2 n - 8} - \frac14$. Namely, from the known complexity results for $P_{*} (\kappa)$-LCPs it might follow that the computational performance of IPAs on LCPs with the matrix defined by Csizmadia could be very poor. Our preliminary computational study shows that an implemented variant of the theoretical version of the CP IPA (Algorithm 4.1) presented in this paper, finds a $\epsilon$-approximate solution for LCPs with the Csizmadia matrix in a very small number of iterations. The third family of problems consists of the LCPs related to the copositivity test of 88 matrices from [C. Brás, G. Eichfelder, and J. Júdice, Comput. Optim. Appl., 63 (2016), pp. 461--493]. For each of these matrices we create a special LCP and try to solve it using our IPA. If the LCP does not have a solution, then the related matrix is strictly copositive, otherwise it is on the boundary or outside the copositive cone. For these LCPs we do not know whether the underlying matrix is $P_{*} (\kappa)$ or not, but we could reveal the real copositivity status of the input matrices in 83 out of 88 cases (accuracy $\ge 94\%$). The numerical test shows that our CP IPA performs well on the sets of test problems used in the paper.

中文翻译：

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基于新搜索方向的$ P _ {*}（\ kappa）$线性互补问题的可行校正器-预测器内点算法

SIAM优化杂志，第30卷，第3期，第2628-2658页，2020年1月。
我们引入了一种新的可行的校正预测器（CP）内点算法（IPA），该算法适用于解决带有$ P _ {**（\ kappa）$矩阵的线性互补问题（LCP）。我们使用定义中心路径的系统非线性方程的代数等效变换（AET）方法。AET基于函数$ \ varphi（t）= t-\ sqrt {t} $，并且在计算新的搜索方向时起着至关重要的作用。我们证明该算法具有$ O（（1 + 2 \ kappa）\ sqrt {n} \ log \ frac {9n \ mu ^ 0} {8 \ epsilon}）$迭代复杂度，其中$ \ kappa $是上限值输入矩阵的让分的界限。据我们所知，这是基于此搜索方向的$ P _ *（\ kappa）$-LCP的第一个CP IPA。我们使用具有特定参数的C ++编程语言来实现建议的CP IPA，并在三个LCP系列上演示其性能。第一个家庭由具有$ P _ {**（\ kappa）$个矩阵的LCP组成。LCP的第二族具有Csizmadia定义的$ P $矩阵。Eisenberg-Nagy和de Klerk [数学。Program。129（2011），第383--402页]显示，此矩阵的让分应至少为$ 2 ^ {2 n-8}-\ frac14 $。即，从$ P _ {*}（\ kappa）$-LCP的已知复杂性结果来看，可能会得出这样的结论：在Csizmadia定义的矩阵的LCP上，IPA的计算性能可能非常差。我们的初步计算研究表明，本文介绍的CP IPA理论版本（算法4.1）的已实现变体，使用极少的迭代次数，就可以找到具有Csizmadia矩阵的LCP的\ epsilon $近似解。第三类问题由LCP组成，这些LCP与[C. Co. Brás，G.Eichfelder和J.Júdice，Comput。最佳 Appl。，63（2016），461--493页。对于这些矩阵中的每一个，我们创建一个特殊的LCP并尝试使用IPA对其进行求解。如果LCP没有解，则相关矩阵严格为正正，否则它在正正锥的边界上或外部。对于这些LCP，我们不知道底层矩阵是否为$ P _ {*}（\ kappa）$，但是我们可以在88个案例中的83个案例中揭示输入矩阵的真实共存状态（精度$ \ ge 94 \ ％$）。数值测试表明，我们的CP IPA在本文中使用的一系列测试问题上表现良好。