On simultaneous diagonalization via congruence of real symmetric matrices

Simultaneous diagonalization via congruence (SDC) for more than two symmetric matrices has been a long standing problem. So far, the best attempt either relies on the existence of a semidefinite matrix pencil or casts on the complex field. The problem now is resolved without any assumption. We first propose necessary and sufficient conditions for SDC in case that at least one of the matrices is nonsingular. Otherwise, we show that the singular matrices can be decomposed into diagonal blocks such that the SDC of given matrices becomes equivalently the SDC of the sub-matrices. Most importantly, the sub-matrices now contain at least one nonsingular matrix. Applications to simplify some difficult optimization problems with the presence of SDC are mentioned

On the complexity of quadratic programming with two quadratic constraints

The complexity of quadratic programming problems with two quadratic constraints is an open problem. In this paper we show that when one constraint is a ball constraint and the Hessian of the quadratic function defining the other constraint is positive definite, then, under quite general conditions, the problem can be solved in polynomial time in the real-number model of computation through an approach based on the analysis of the dual space of the Lagrange multipliers. However, the degree of the polynomial is rather large, thus making the result mostly of theoretical interest