Grant: Discovery Grants Program - Individual

We are interested in good barrier and penalty functions for new classes of optimization problems over matrix domains. Such functions are crucial for the speed of the optimization algorithms using them. The recently developed nonsmooth Newton's method shows enormous potential for finding quickly optimal solutions of large-scale optimization problems. Naturally, the speed of convergence of the method depends on the properties of the function being optimized. We are interested in finding new applications of the Nonsmooth Newton's method as well as new classes of functions ensuring fast quadratic convergence. Hyperbolic polynomials are a rich class of multivariate polynomials having only real roots when restricted to any one-dimensional line parallel to particular directions. The roots of these polynomials naturally generalize the eigenvalues of symmetric matrices and every polynomial has an associated hyperbolicity cone, a notion that generalizes the positive semi-definite cone. Understanding the structure of the set of these polynomials and their properties may lead to methods for solving optimization problems over hyperbolicity cones.