This is a convex optimization question.

We consider the convex) vector optimization problem, with respect to the positive semi-definite cone, minimize (w.rt. Sm) X subject to X A; i = 1,...,m where A, E S", i = 1,...,m, are given. The constraints mean the X is an upper bound on the the given matrices A;..., Ami a Pareto optimal solution of this is a minimal upper bound on the matrices. The equation above means a) b) c) Find matrix X with the smallest possible positive elements Find a positive semidefinite matrix X with the smallest possible positive elements Find a positive semidefinite matrix X such that , for every positive semidefinite matrix Y, X-Y0 S Find a positive semidefinite matrix X such that, for every positive semidefinite matrix Y, Y-X0 S d) We consider the convex) vector optimization problem, with respect to the positive semi-definite cone, minimize (w.rt. Sm) X subject to X A; i = 1,...,m where A, E S", i = 1,...,m, are given. The constraints mean the X is an upper bound on the the given matrices A;..., Ami a Pareto optimal solution of this is a minimal upper bound on the matrices. The equation above means a) b) c) Find matrix X with the smallest possible positive elements Find a positive semidefinite matrix X with the smallest possible positive elements Find a positive semidefinite matrix X such that , for every positive semidefinite matrix Y, X-Y0 S Find a positive semidefinite matrix X such that, for every positive semidefinite matrix Y, Y-X0 S d)