Polyaks convexity theorem, Yuans lemma and S-lemma: extensions and applications

We extend Polyaks theorem on the convexity of joint numerical range from three to any number of quadratic forms on condition that they can be generated by three quadratic forms with a positive definite linear combination. Our new result covers the fundamental Diness theorem. As applications, we further extend Yuans lemma and S-lemma, respectively. Our extended Yuans lemma is used to build a more generalized assumption than that of Haeser (J. Optim. Theory Appl. 174(3): 641-649, 2017), under which the standard second-order necessary optimality condition holds at local minimizer. The extended S-lemma reveals strong duality of homogeneous quadratic optimization problem with two bilateral quadratic constraints.