Interior-Point Methods for Large-Scale Cone Programming - Martin Andersen; Joachim Dahl; Zhang Liu; Lieven Vandenberghe (Optimization for Machine Learning, 2011)
The cone programming formulation has been popular in the recent literature on convex optimization. In this chapter we define acone linear program(cone LP or conic LP) as an optimization problem of the form minimize${c^T}x$ subject to$Gx{\underline \prec _C}h$(3.1) $Ax = b$ with optimization variablex. The inequality$Gx{\underline \prec _C}h$is ageneralized inequality, which means that$h - Gx \in C$, whereCis a closed, pointed, convex cone with nonempty interior. We will also encountercone quadratic programs(cone QPs), minimize$(1/2){x^T}Px + {c^T}x$(3.2) subject to$Gx{\underline \prec _C}h$ $Ax = b$, withPpositive semidefinite. If$C = R_ + ^p$(the nonnegative orthant in${R^p}$), the generalized inequality is a componentwise