Theoretical characteristics and numerical methods for a class of special piecewise quadratic optimization

Abstract

In this paper, we discuss the theoretical characteristics and numerical methods for a class of special piecewise quadratic optimization (SPQO). Theoretically, the existence, uniqueness and necessary and sufficient conditions of the SPQO's solution are studied; the coercivity and the strong convexity as well as the Lipschitz-differentiability of the objective function of the SPQO are discussed. Methodically, based on Armijo line search, an algorithm framework with sufficient descent property is proposed. The convergence, ergodic and non-ergodic convergence rates of the algorithm framework are proved. Moreover, a modified quasi-Newton direction satisfying the convergence conditions is designed, and then this together with three existing conjugate gradient methods is used to compute the search directions of the algorithm framework, respectively. Numerically, a large number of numerical experiments are carried out on the SPQO, and the results are compared with those obtained by calling the standard quadratic optimization solver in commercial software CPLEX to solve the standard quadratic optimization equivalent to the SPQO. As a result, preliminary suggestions on the selection of effective calculation methods for the SPQO are provided.