Second Order Cone problems (SOCPs)

Second order cone problems (SOCP) are a special class of quadratically constrained problems, where the quadratic matrix Q is not required to be semi–definite.

The FICO Xpress Optimizer supports (mixed integer) second order cone problems that satisfy the following requirements.

Each quadratic constraint satisfies one of the following two forms:

1. Second order (or Lorentz) cone: x₁²+x₁²+...+x_k²-t²≤0 where t≥0
2. Rotated second order (or Lorentz) cone: x₁²+x₁²+...+x_k²-2t₁t₂≤0 where t₁,t₂≥0

All of the cone coefficients must be exactly one, except for the coefficient of 2 for the t₁t₂ product. Constants or linear terms are not allowed.

Cones cannot be overlapping. That is, a variable x can appear in at most one second–order cone constraint.

Second order cone problems are loaded using the same API functions as for quadratic constraints, and the conic constraints are auto–detected by the optimizer at run time.