Conic optimization is a significant and thriving research area within the optimization community. Conic optimization is the general class of problems concerned with optimizing a linear function over the intersection of an affine space and a closed convex cone. One special case of great interest is the choice of the cone of positive semidefinite matrices for which the resulting optimization problem is called a semidefinite optimization problem.
Semidefinite optimization, or semidefinite programming (SDP), has been studied (under different names) since at least the 1940s. Its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve SDP problems (and more generally, to solve convex optimization problems with efficiently computable self-concordant barrier functions). Some of the earliest applications of SDP that followed this development were the solution of linear matrix inequalities in control theory, and the design of polynomial-time approximation schemes for hard combinatorial problems such as the maximum-cut problem.
The objective of this Handbook on Semidefinite, Conic and Polynomial Optimization is to provide the reader with a snapshot of the state of the art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. Our intention is to provide a compendium of the research activity that has taken place since the publication of the seminal Handbook mentioned above. It is our hope that this will motivate more researchers, especially doctoral students and young graduates, to become involved in these thrilling areas