Advances in Convex Optimization

In this talk I will give an overview of general convex optimization, which can be thought of as an extension of linear programming, and some recently developed subfamilies such as second-order cone, semidefinite, and geometric programming. Like linear programming, we have a fairly complete duality theory, and very effective numerical methods for these problem classes; in addition, recently developed software tools considerably reduce the effort of specifying and solving convex optimization problems. There is a steadily expanding list of new applications of convex optimization, in areas such as circuit design, signal processing, statistics, machine learning, communications, control, finance, and other fields. Convex optimization is also emerging as an important tool for hard, non-convex problems, where it can be used to generate lower bounds on the optimal value, and as a heuristic method for generating suboptimal points.