Elliptic cone optimization and primal–dual path-following algorithms

Abstract

In elliptic cone optimization problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of the so-called elliptic cones. We present some general classes of optimization problems that can be cast as elliptic cone programmes such as second-order cone programmes and circular cone programmes. We also describe some real-world applications of this class of optimization problems. We study and analyse the Jordan algebraic structure of the elliptic cones. Then, we present a glimpse of the duality theory associated with elliptic cone optimization. A primal–dual path-following interior-point algorithm is derived for elliptic cone optimization problems. We prove the polynomial convergence of the proposed algorithms by showing that the logarithmic barrier is a strongly self-concordant barrier. The numerical examples show the path-following algorithms are efficient.

Keywords:

• Conic optimization
• elliptic cone optimization
• duality
• primal–dual path-following methods
• optimization models

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the first author was supported in part by the Deanship of Scientific Research at the University of Jordan.