A semi-smooth Newton method for conic projection equations. - Armijo, Nicolas Esteban Fuentealba (2024)
Semi-smooth Newtons method has been a powerful tool for solving various types of problems. Its convergence theory for solving non-smooth, non-linear systems of equations renders it particularly valuable, especially in recent years. The main objective of this thesis is to solve non-linear conic programming problems using a system of conic projection equations and adapt a semi-smooth Newtons method to solve them. The study begins by investigating the projection equation for the nonnegative orthant in , closely related to quadratic conic programming, reviewing its properties, and proposing two alternatives in addition to the semi-smooth Newtons method. The fast convergence of the semi-smooth Newtons method motivates the generalization of the projection equations to the quadratic linearly constrained programming problem and the adaptation of the method to this new problem, applying it to the Nearest Correlation Matrix problem. Finally, a general form of the conic projection equations is presented to solve the non-linear conic programming problem, applying it to linear conic programming and comparing it to state-of-the-art solvers for second-order programming and semidefinite programming.