Error bounds for symmetric cone complementarity problems

Abstract

In this paper, we investigate the issue of error bounds for symmetric cone complementarity problems (SCCPs). In particular, we show that the distance between an arbitrary point in Euclidean Jordan algebra and the solution set of the symmetric cone complementarity problem can be bounded above by some merit functions such as Fischer-Burmeister merit function, the natural residual function and the implicit Lagrangian function. The so-called R 0 -type conditions, which are new and weaker than existing ones in the literature, are assumed to guarantee that such merit functions can provide local and global error bounds for SCCPs. Moreover, when SCCPs reduce to linear cases, we demonstrate such merit functions cannot serve as global error bounds under general monotone condition, which implicitly indicates that the proposed R 0 -type conditions cannot be replaced by P -type conditions which include monotone condition as special cases.

A new model for solving stochastic second-order cone complementarity problem and its convergence analysis

“…One is a Fischer–Burimister (FB) second-order cone complementarity function, that is, , Another is a natural residual function , where denotes the metric projection of x onto . And by [14], …”

“…Based on the definition of -function in [14], we present the definition of stochastic τ - function, which will be used in the proof of boundedness of level sets.…”