Applications of second-order cone programming

Abstract

In a second-order cone program (SOCP) a linear function is minimized over the intersection of an affine set and the product of second-order (quadratic) cones. SOCPs are nonlinear convex problems that include linear and (convex) quadratic programs as special cases, but are less general than semidefinite programs (SDPs). Several efficient primal-dual interior-point methods for SOCP have been developed in the last few years. After reviewing the basic theory of SOCPs, we describe general families of problems that can be recast as SOCPs. These include robust linear programming and robust least-squares problems, problems involving sums or maxima of norms, or with convex hyperbolic constraints. We discuss a variety of engineering applications, such as filter design, antenna array weight design, truss design, and grasping force optimization in robotics. We describe an efficient primal-dual interior-point method for solving SOCPs, which shares many of the features of primal-dual interior-point methods for linear programming (LP): Worst-case theoretical analysis shows that the number of iterations required to solve a problem grows at most as the square root of the problem size, while numerical experiments indicate that the typical number of iterations ranges between 5 and 50, almost independent of the problem size.