In these lectures, I will describe a simple iterative method that supplies new proofs of integrality of linear characterizations of various basic problems in combinatorial optimization, and also allows adaptations to design approximation algorithms for NP-hard variants of these problems involving extra “degree-like” budget constraints. It is inspired by Jain’s iterative rounding method for desi...

In the last decade our capability of solving integer programming problems has increased dramatically due to the effectiveness of cutting plane methods based on polyhedral investigations. Polyhedral cutting planes have become central features in optimization software packages for integer programming. Here we present some of the important polyhedral methods used in discrete optimization. We discu...

We study a class of overrelaxed bound optimization algorithms, and their relationship to standard bound optimizers, such as ExpectationMaximization, Iterative Scaling, CCCP and Non-Negative Matrix Factorization. We provide a theoretical analysis of the convergence properties of these optimizers and identify analytic conditions under which they are expected to outperform the standard versions. B...

Kernel methods are learning systems which employ embedding of the input data in highdimensional feature spaces. The embedding is implemented via kernel functions which act as a dot product in Reproducing Kernel Hilbert Spaces (RKHS). Learning algorithms which require only the canonical dot product of data represented in finite dimensional linear spaces can be generalized to perform in RKHS. Thi...