B - 290 Linear Algebra for Semide nite Programming

Let M n (IK) denote the set of all n n matrices with elements in IK, where IK represents the eld IR of real numbers, the eld 0 C of complex numbers or the (noncommutative) eld IH of quaternion numbers. We call a subset T of M n (IK) a *-subalgebra of M n (IK) over the eld IR (or simply a *-subalgebra) if (i) T forms a subring of M n (IK) with the usual addition A + B and multiplication AB of matrices A; B 2 M n (IK); speci cally the zero matrix O and the identity matrix I belong to T . (ii) T is an IR-module, i.e., a vector space over the eld IR; A+ B 2 T for every ; 2 IR and A; B 2 T , (iii) A 2 T if A 2 T , where A denotes the conjugate transpose of A 2 M n (IK). The introduction of *-subalgebras T provides us with a uni ed and compact way of handling LPs (linear programs) in IR n , SDPs (semide nite programs) in M n (IR), M n ( 0 C) and M n (IH), and monotone SDLCPs (semide nite linear complementarity problems) in those spaces. We can extend the duality theory to SDPs in T , and the existence of the central trajectory to monotone SDLCPs in T . This paper presents adaptation of interior-point methods which were recently proposed by Kojima, Shindoh and Hara for a monotone SDLCP in M n (IR) to a monotone SDLCP in a *-subalgebra T of