Conic optimization software

We give an overview of cone optimization software, with special attention to the differences between the existing packages. We assume the reader is familiar with the theory and algorithms of cone optimization, thus technical details are kept at a minimum. We also outline current research trends and area of potential improvement. 1 Problem description Conic optimization solvers target problems of the form min cx max b y Ax = b A y + s = c (1.1) x ∈ K s ∈ K∗, where b, y ∈ Rm, c, x, s ∈ RN , A ∈ Rm×N , K,K∗ ⊂ RN , and K∗ is the dual of K. In general solvers exist if K is one of, or the product of a few copies of the following cones: nonnegative orthant: the set of nonnegative vectors, R+; Lorentz cone: the set Ln+1 = {(u0, u) ∈ R+ × Rn : u0 ≥ ‖u‖}, also called the quadratic or ice-cream cone; rotated Lorentz cone: the set Ln+1 = { (u0, u1, u) ∈ R+ × Rn : u0u1 ≥ ‖u‖ , u0 ≥ 0 } ; positive semidefinite cone: the cone PSn×n of n×n real symmetric positive semidefinite matrices; complex Hermitian cone: the cone n × n complex Hermitian positive semidefinite matrices. ∗Lehigh University, [email protected]