Calabi-Polyak convexity theorem, Yuan’s lemma and S-lemma: extensions and applications

We extend the Calabi-Polyak theorem on the convexity of joint numerical range from three to any number of matrices on condition that each of them is a linear combination of three matrices having a positive definite linear combination. Our new result covers the fundamental Dines’s theorem. As applications, the further extended Yuan’s lemma and S-lemma are presented. The former is used to establish a more generalized assumption under which the standard second-order necessary optimality condition holds at the local minimizer in nonlinear programming, and the latter reveals hidden convexity of the homogeneous quadratic optimization problem with two bilateral quadratic constraints and its fractional extension.