Optimality criteria without constraint qualications for linear semidenite problems

We consider two closely related optimization problems: a problem of convex Semi- Infinite Programming with multidimensional index set and a linear problem of Semidefinite Programming. In study of these problems we apply the approach suggested in our recent paper [14] and based on the notions of immobile indices and their immobility orders. For the linear semidefinite problem, we define the subspace of immobile indices and formulate the first order optimality conditions in terms of a basic matrix of this subspace. These conditions are explicit, do not use constraint qualifications, and have the form of criterion. An algorithm determining a basis of the subspace of immobile indices in a finite number of steps is suggested. The optimality conditions obtained are compared with other known optimality conditions