Quadratic Optimization for Nonsmooth Optimization Algorithms: Theory and Numerical Experiments

Nonsmooth optimization arises in many scientific and engineering applications, such as optimal control, neural network training, and others. Gradient sampling and bundle methods are two ef- ficient types of algorithms for solving nonsmooth optimization problems. Quadratic optimization (commonly referred to as QP) problems arise as subproblems in both types of algorithms. This thesis introduces an algorithm for solving the types of QP problems that arise in such methods. The proposed algorithm is inspired by one proposed in a paper written by Krzysztof C. Kiwiel in the 1980s. Improvements are proposed so that the algorithm may solve problems with addi- tional bound constraints, which are often required in practice. The solver also allows for general quadratic terms in the objective. Our QP solver has been implemented in C++. This thesis not only covers the theoretical background related to the QP solver; it also contains the results of numerical experiments on a wide range of randomly generated test problems.