The Simplest Semidefinite Programs are Trivial

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 , 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.