NEW COMPLEXITY ANALYSIS OF IPM FOR $P_*({\kappa})$ LCP BASED ON KERNEL FUNCTIONS

In this paper we extend primal-dual interior point algorithm for linear optimization (LO) problems to $P_*({\kappa})$ linear complementarity problems(LCPs) ([1]). We define proximity functions and search directions based on kernel functions, ${\psi}(t)=\frac{t^{p+1}-1}{p+1}-{\log}\;t$, $p{\in}$[0, 1], which is a generalized form of the one in [16]. It is the first to use this class of kernel functions in the complexity analysis of interior point method(IPM) for $P_*({\kappa})$ LCPs. We show that if a strictly feasible starting point is available, then new large-update primal-dual interior point algorithms for $P_*({\kappa})$ LCPs have $O((1+2{\kappa})nlog{\frac{n}{\varepsilon}})$ complexity which is similar to the one in [16]. For small-update methods, we have $O((1+2{\kappa})\sqrt{n}{\log}{\frac{n}{\varepsilon}})$ which is the best known complexity so far.