Solving a class of matrix minimization problems by linear variational inequality approaches

A class of matrix optimization problems can be formulated as a linear variational inequalities with special structures. For solving such problems, the projection and contraction method (PC method) is extended to variational inequalities with matrix variables. Then the main costly computational load in PC method is to make a projection onto the semi-definite cone. Exploiting the special structures of the relevant variational inequalities, the Levenberg–Marquardt type projection and contraction method is advantageous. Preliminary numerical tests up to 1000×1000 matrices indicate that the suggested approach is promising.