On the equivalence of global quadratic growth condition and second-order sufficient condition

Abstract

The main purpose of this thesis is to study various properties of quadratically constrained quadratic programming problems. We concentrate on the existence of solutions and global second-order sufficient conditions for the quadratic problems. First, we study the existence of global solutions for general quadratic programming problems. With the tool of asymptotical directions of sets, we provide an alternative proof for the nonemptiness of intersection of a sequence of nested sets defined by a finite number of convex quadratic functions, which can be applied to show the existence results of the corresponding optimization problems. We also prove the existence of global solutions of the convex quadratic program with convex quadratic constraints by an analytic approach. Next, we study the global quadratic growth condition and the global second-order sufficient condition for the quadratic programming problems. By formulating the problem in the form of minimizing a maximum function of a finite number of quadratic functions, we study the relation between the global quadratic growth condition and the global second-order sufficient condition for the maximum function. As is shown, the global second-order sufficient condition implies the global quadratic growth condition. But the reverse implication is in general not true. We show that, when the solution set is a singleton and the number of quadratic term is 2, the reverse implication holds. For the homogeneous quadratic case, these two are equivalent. We then apply the corresponding results to the constrained optimization problem in standard form. Finally, we investigate the fractional programming problems. We present the S-lemma for the fractional functions. We also obtain the attainability property of various types of fractional programming problems.