Extremal problems on the set of nonnegative definite matrices
The problem of minimizing a function of symmetric matrix elements, subject to the matrix being positive semi-definite, is formulated as a semi-infinite program. Optimality conditions are then stated under the usual assumptions (convexity, Slater's condition). For the linear program with the objective function being the inner product \((A,X)=tr AX\) and the constraints \(S+X\) being positive semi-definite matrices (S is fixed symmetric), conditions are given for uniqueness of the solution and differentiability of the optimal value function. The corresponding duality results recover some of \textit{I. Olkin} and \textit{F. Pukelsheim}'s results [ibid. 48, 257-263 (1982; Zbl 0527.60015)] as special cases.
differentiability of the optimal value function
scientific article; zbMATH DE number 3902405