Let S: = {M ∈ Sn | M 0}. Then

Many authors [1-7] have studied semidefinite scalar optimization problems since the probelems have many engineering applications and many kinds of optmization problems can be reduced to the problems. In this talk, we discuss optimality conditions for a convex semidefinite vector optimization problem which consists of more than two convex objective convex functions over a linear matrix inequality and a closed convex subset. Our optimality conditions, which can be applied without any constraint qualification, are expressed with sequences. So our optimality conditions can be called sequential optimality conditions. FORMULATION Now we consider the following convex semidefinite vector optimization problem(SDVP): (SDVP) Minimize f(x): = (f1(x), · · · , fp(x)) subject to S = {x ∈ C | F0 +∑mi=1 xiFi 0}. where f: Rm → R is a convex function, and for i = 0, 1, · · ·,m, Fi ∈ Sn, the space of (n × n) real symmetric matrices, and C is a closed convex subset of Rm. The space Sn is partially ordered by the Löwner order; that is, for M,N ∈ Sn, M N if and only if M−N is positive semidefinite. The inner product in Sn is defined by (M,N) = Tr[MN], where Tr[·] is the trace operation