On the analyticity properties of infeasible-interior-point paths for monotone linear complementarity problems

Abstract

Infeasible-interior-point paths \(\{(x(r,\eta),y(r,\eta))\}_{r>0}\), \(0 0, y>0, x_iy_i &=& r\eta_i, & 1\le i\le n. \end{array} \leqno(LCP)_r \) If the path \(z(r,\eta)=(x(r,\eta),y(r,\eta))\) converges for \(r\downarrow 0\), then it converges to a solution of \((LCP)_0\). This paper deals with the analyticity properties of \(z(r,\eta)\) and its derivatives with respect to r near r = 0 for solvable monotone complementarity problems \((LCP)_0\). It is shown for \((LCP)_0\) with a strictly complementary solution that the path \(z(r,\eta)\), \(r\downarrow0\), has an extension to \(r=0\) which is analytic also at \(r=0\). If \((LCP)_0\) has no strictly complementary solution, then \(\hat z(\rho,\eta):=z(\rho^2,\eta)\), \(\rho=\sqrt r\), has an extension to \(\rho=0\) that is analytic at \(\rho=0\).