Gaussian Mixture Penalty for Trajectory Optimization Problems

Manifold Optimization for Gaussian Mixture Models

We take a new look at parameter estimation for Gaussian Mixture Models (GMMs). In particular, we propose using Riemannian manifold optimization as a powerful counterpart to Expectation Maximization (EM). An out-of-the-box invocation of manifold optimization, however, fails spectacularly: it converges to the same solution but vastly slower. Driven by intuition from manifold convexity, we then pr...

Penalty Methods for a Class of Non-Lipschitz Optimization Problems

We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range of applications in data science, where the objective is used for inducing sparsity in the solutions while the constraint set models the noise tolerance and ...

Hessian Matrices of Penalty Functions for Solving Constrained-optimization Problems

This paper deals with the Hessian matrices of penalty functions, evaluated at their minimizing point. It is concerned with the condition number of these matrices for small values of a controlling parameter. At the end of the paper a comparison is made between different types of penalty functions on the grounds of the results obtained. 1. Classification of penalty-function techniques Throughout ...

Adaptive Penalty Methods for Genetic Optimization of Constrained Combinatorial Problems

Smoothed Lower Order Penalty Function for Constrained Optimization Problems

The paper introduces a smoothing method to the lower order penalty function for constrained optimization problems. It is shown that, under some mild conditions, an optimal solution of the smoothed penalty problem is an approximate optimal solution of the original problem. Based on the smoothed penalty function, an algorithm is presented and its convergence is proved under some mild assumptions....