We consider optimization problems of the following type: min(tr(CX): A(X)=B, X positive semidefinite. Here, tr(.) denotes the trace operator, C and X are symmetric n×n matrices, B is a symmetric m×m matrix and A(.) denotes a linear operator. Such problems are called semidefinite programs and have recently become the object of considerable interest due to important connections with max–min eigenvalue problems and with new bounds for integer programming. In the context of symmetric matrices, the simplest linear operators have the following form: A(X)=MXM(T), where M is an arbitrary m×n matrix. In this paper, we show that for such linear operators the optimization problem is trivial in the sense that an explicit solution can be given.
