Convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search

Abstract

In this paper, we propose a smoothing algorithm for solving the monotone symmetric cone complementarity problems (SCCP for short) with a nonmonotone line search. We show that the nonmonotone algorithm is globally convergent under an assumption that the solution set of the problem concerned is nonempty. Such an assumption is weaker than those given in most existing algorithms for solving optimization problems over symmetric cones. We also prove that the solution obtained by the algorithm is a maximally complementary solution to the monotone SCCP under some assumptions.

Analysis of a non-monotone smoothing-type algorithm for the second-order cone programming

“…The numerical results are listed in Table 2 From the numerical results listed in Table 1, we may see that Algorithm 3.1 works better with the non-monotone line search than the monotone line search in the sense that the former could find the optimizer more efficiently than the latter; and the former has less number of iterations and less CPU time than the latter for most cases. Moreover, by the numerical results in Tables 1 and 2, we may find that our non-monotone smoothing algorithm seems to be more effective than Huang-Hu-Han's algorithm [11]. These demonstrate that the non-monotone smoothing-type algorithm proposed in this paper has some advantages.…”

“…For the non-monotone line search given in Algorithm 3.1, some basic results are included in the following lemma, whose proof can be found in Remark 3.4, [11].…”

“…It is well known that the non-monotone line search rule has many advantages, especially in the case of iterates trapped in a narrow curved valley of objective functions (e.g., [8], [20]). Some non-monotone line search schemes have been applied to the smoothing-type algorithms for nonlinear complementarity problems (e.g., [9], [16]) and symmetric cone complementarity problems (e.g., [11]). A crucial question in this respect is whether the smoothing-type algorithm with a non-monotone line search for the SOCP not only possesses locally fast convergence properties but also has encouraging numerical results.…”

“…Notice that by using Zhang-Hager's non-monotone scheme [20], Huang, Hu, and Han [11] proposed a non-monotone smoothing-type algorithm to solve symmetric…”

A smoothing-type algorithm for solving inequalities under the order induced by a symmetric cone

“…On one hand, smoothing-type algorithms have been developed to solve symmetric cone complementarity problems (see, for example, [10][11][12][13][14]) and symmetric cone linear programming (see, for example, [15,16]). On the other hand, smoothing-type algorithms have also been developed to solve the system of inequalities under the order induced by n + (see, for example, [17][18][19]).…”