Higher-Order Cone Programming

TL;DR

This paper introduces a hierarchy of convex cones and cone programs that interpolate between linear, second-order cone, and semidefinite programming, unifying various optimization frameworks through a conic embedding condition.

Contribution

It proposes a new conic embedding condition that generates a hierarchy of cones, connecting LP, SOCP, and SDP, and extends to polynomial optimization.

Findings

Hierarchy of cones interpolates LP, SOCP, SDP

Cones realized as symmetric matrices or polynomial cones

Framework unifies various convex optimization methods

Abstract

We introduce a conic embedding condition that gives a hierarchy of cones and cone programs. This condition is satisfied by a large number of convex cones including the cone of copositive matrices, the cone of completely positive matrices, and all symmetric cones. We discuss properties of the intermediate cones and conic programs in the hierarchy. In particular, we demonstrate how this embedding condition gives rise to a family of cone programs that interpolates between LP, SOCP, and SDP. This family of $k$th order cones may be realized either as cones of $n$-by-$n$ symmetric matrices or as cones of $n$-variate even degree polynomials. The cases $k = 1, 2, n$ then correspond to LP, SOCP, SDP; or, in the language of polynomial optimization, to DSOS, SDSOS, SOS.