Elliptic cone optimization and primal–dual path-following algorithms. (2nd December 2017)

• Record Type: Journal Article
• Authors: Alzalg, Baha; Pirhaji, Mohammad
• Is Part Of: Optimization. Volume 66:Number 12(2017)
• Page Start: 2245
• Page End: 2274
• Publication Date: 2017-12-02
• Subjects: Conic optimization -- elliptic cone optimization -- duality -- primal–dual path-following methods -- optimization models; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2017.1360888
• Languages: English
• ISSNs: 0233-1934

Abstract

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.