Conic optimization: a unified framework for structured convex optimization

Among optimization problems, convex problems form a special subset with two important and useful properties: (1) the existence of a strongly related dual problem that provides certified bounds and (2) the possibility to find an optimal solution using polynomial-time algorithms. In the first part of this talk, we will outline how the framework of conic optimization, which formulates structured convex problems using convex cones, facilitates the exploitation of those two properties.

In the second part of this talk, we will introduce a specific cone (called the power cone) that allows the formulation of a large class of convex problems (including linear, quadratic, entropy, sum-of-norm and geometric optimization). For this class of problems, we present a primal-dual interior-point algorithm, which focuses on preserving the perfect symmetry between the primal and dual sides of the problem (arising from the self-duality of the power cone).