Characterizations of Asymptotic Cone of the Solution Set of a Composite Convex Optimization Problem

We characterize the asymptotic cone of the solution set of a convex composite optimization problem. We then apply the obtained results to study the necessary and sufficient conditions for the nonemptiness and compactness of the solution set of the problem. Our results generalize and improve some known results in literature

Strong Metric (Sub)regularity of KKT Mappings for Piecewise Linear-Quadratic Convex-Composite Optimization

This work concerns the local convergence theory of Newton and quasi-Newton methods for convex-composite optimization: minimize f(x):=h(c(x)), where h is an infinite-valued proper convex function and c is C^2-smooth. We focus on the case where h is infinite-valued piecewise linear-quadratic and convex. Such problems include nonlinear programming, mini-max optimization, estimation of nonlinear dynamics with non-Gaussian noise as well as many modern approaches to large-scale data analysis and machine learning. Our approach embeds the optimality conditions for convex-composite optimization problems into a generalized equation. We establish conditions for strong metric subregularity and strong metric regularity of the corresponding set-valued mappings. This allows us to extend classical convergence of Newton and quasi-Newton methods to the broader class of non-finite valued piecewise linear-quadratic convex-composite optimization problems. In particular we establish local quadratic convergence of the Newton method under conditions that parallel those in nonlinear programming when h is non-finite valued piecewise linear

Optimization of Photonic Band Structures

In this work we study mathematical optimization problems that arise in the design of photonic crystals, whose band structures should exhibit specific properties. To this end we develop a mathematical model for time-harmonic wave propagation in three-dimensional, periodic media. We investigate the dependency of band structures on the medium structure and develop two types of optimization algorithms. The performance of these algorithms is demonstrated through several of numerical experiments