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.
Highlights
IntroductionIt is known that symmetric cone complementarity problems provide a unified framework for nonlinear complementarity problems (NCPs), semidefinite complementarity problems (SDCPs) and second-order cone complementarity problems (SOCCPs)
The symmetric cone complementarity problem ( SCCP) is to find a vector x ∈ V such that x ∈ K, F (x) ∈ K and x, F (x) = 0, (1)where V is a Euclidean Jordan algebra, K ⊂ V is a symmetric cone, ·, · denotes the usual Euclidean inner product and F is a continuous mapping from V into itself
When F reduces to a linear transformation L, i.e., F (x) = L(x) + q with q ∈ V, the above symmetric cone complementarity problem becomes x ∈ K, L(x) + q ∈ K and x, L(x) + q = 0, which is called a symmetric cone linear complementarity problem and denoted by SCLCP
Summary
It is known that symmetric cone complementarity problems provide a unified framework for nonlinear complementarity problems (NCPs), semidefinite complementarity problems (SDCPs) and second-order cone complementarity problems (SOCCPs) Along this line, there is some research work on error bounds for SCCPs. For instance, Chen [3] gives some conditions towards error bounds and bounded level sets for SOCCPs; Pan and Chen [22] consider error bound and bounded level sets of a one-parametric class of merit functions for SCCPs; Kong, Tuncel and Xiu [14] study error bounds of the implicit Lagrangian ψMS(x) for SCCPs. In general, one needs conditions such as F has the uniform Cartesian P -property and is Lipschitz continuous.
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