Strong duality in minimizing a quadratic form subject to two homogeneous quadratic inequalities over the unit sphere

In this paper, we study the strong duality for an optimization problem to minimize a homogeneous quadratic function subject to two homogeneous quadratic constraints over the unit sphere, called Problem (P) in this paper. When a feasible (P) fails to have a Slater point, we show that (P) always adopts the strong duality. When (P) has a Slater point, we propose a set of conditions, called "Property J", on an SDP relaxation of (P) and its conical dual. We show that (P) has the strong duality if and only if there exists at least one optimal solution to the SDP relaxation of (P) which fails Property J. Our techniques are based on various extensions of S-lemma as well as the matrix rank-one decomposition procedure introduced by Ai and Zhang. Many nontrivial examples are constructed to help understand the mechanism.