Cutting Plane Methods and Subgradient Methods

Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposition methods for many classes of optimization problems. We discuss the use of interior point methods in a column generation scheme. Semidefinite programming relaxations of combinatorial optimization problems are often tighter than linear programming relaxations. We describe some research in using SDP relaxations to find exact solutions to combinatorial optimization problems. Semidefinite programs are expensive to solve directly, so we also consider cutting surface approaches to solving them. Finally, we look at recent smoothing techniques for solving nonsmooth optimization problems using a subgradient approach; these methods have some links to cutting surface approaches.