Linear Conic Programming

A little story in the development of semidefinite programming. One day in 1990, I visited the Computer Science Department of the University of Min-nesota and met a young graduate student, Farid Alizadeh. He, working then on combinatorial optimization, introduced me " semidefinite optimization " or linear programming over the positive definite matrix cone. We had a very extensive discussion that afternoon and concluded that interior-point linear programming algorithms could be applicable to solving SDP. I urged Farid to look at the LP potential functions and to develop an SDP primal potential reduction algorithm. He worked hard for several months, and one afternoon showed up in my office in Iowa City, about 300 miles from Minneapolis. He had everything worked out, including potential, algorithm, complexity bound, and even a " dictionary " from LP to SDP, but was stuck on one problem which was how to keep the symmetry of the matrix. We went to a bar nearby on Clinton Street in Iowa City (I paid for him since I was a third-year professor then and eager to demonstrate that I could take care of students). After chatting for a while, I suggested that he use X −1/2 ∆X −1/2 to keep symmetry instead of X −1 ∆ which he was using, where X is the current symmetric positive definite matrix and ∆ is the symmetric directional matrix. He returned to Minneapolis and moved to Berkeley shortly after, and few weeks later sent me an e-mail message telling me that everything had worked out beautifully. At the same time or even earlier, Nesterov and Nemirovskii developed a more general and powerful theory in extending interior-point algorithms for solving conic programs, where SDP was a special case. Boyd and his group later presented a wide range of SDP applications and formulations, many of which were incredibly novel and elegant. Then came the primal-dual algorithm, the Max-Cut, ... and SDP established its full popularity.