Applications of Second-order Cone Programming

In a second-order cone program (SOCP) a linear function is minimized over the intersection of an a ne 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 semide nite programs (SDPs). Several e cient 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 SOCP. These include robust linear programming, robust least-squares, problems involving sums or maxima of norms, and problems with convex hyperbolic constraints. We discuss a variety of engineering applications, such as lter design, antenna array weight design, truss design, and grasping force optimization in robotics. We describe an e cient 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. (Final revised version) to appear in Linear Algebra and Applications, special issue on linear algebra in control, signals and image processing. Research supported in part by the Portuguese Government (under Praxis XXI), AFOSR (under F49620-95-1-0318), NSF (under ECS-9222391 and EEC-9420565), and MURI (under F49620-95-10525). Associated software is available at URL http://www-isl.stanford.edu/people/boyd and from anonymous FTP to isl.stanford.edu in pub/boyd/socp.