A homotopy method based on penalty function for nonlinear semidefinite programming
A homotopy method based on penalty function for nonlinear semidefinite programming
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DOI：
10.1007/s10898-015-0276-5
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发表时间：
2015-02
影响因子：
1.8
通讯作者：
Li Yang-;Bo Yu;Yanxi Li
中科院分区：
文献类型：
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作者：
Li Yang-;Bo Yu;Yanxi Li
提出了一种求解非线性半定规划问题的基于罚函数的同伦方法。罚函数是指数罚函数、特征值函数和非线性算子映射的复合函数。给出了它的一阶和二阶导数的表示式。利用罚函数构造了一个新的同伦。在较温和的条件下，证明了由同伦决定的光滑曲线的全局收敛。在数值跟踪曲线的过程中，该方法只需要一个线性维度系统的解，而由Yang和Yu提出的同伦方法(Comput OpTim Appl 56(1)：81-96,2013)需要求解一个维度系统，其中变量数是约束矩阵的阶数。因此，该方法有望提高杨和余等人提出的方法的效率。初步的数值实验表明，所提出的算法对某些非线性半定规划问题是有效的。
This paper proposes a homotopy method based on a penalty function for solving nonlinear semidefinite programming problems. The penalty function is the composite function of an exponential penalty function, the eigenvalue function and a nonlinear operator mapping. Representations of its first and second order derivatives are given. Using the penalty function, a new homotopy is constructed. Global convergence of a smooth curve determined by the homotopy is proven under mild conditions. In the process of numerically tracing the curve, the method requires just the solution of a linear system of dimension, whereas a homotopy method proposed by Yang and Yu (Comput Optim Appl 56(1):81–96, 2013) requires a system of dimensionto be solved, whereis the number of variables whileis the order of constraint matrix. So, it is expected that the proposed method can improve the efficiency of the method proposed by Yang and Yu. Preliminary numerical experiments are presented and show that the considered algorithm is efficient for some nonlinear semidefinite programming problems.