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We present a logarithmic barrier interior-point method for solving a second-order cone programming problem. Newton’s method is used to compute the descent direction. The main contribution of this paper is that it uniquely uses the so-called majorant functions as an efficient alternative to line search methods to determine the displacement step along the direction while solving second-order cone programs. The efficiency of our method is shown by presenting numerical experiments.

Alzalg (J Optim Theory Appl 163(1):148–164, 2014) derived a homogeneous self-dual algorithm for stochastic second-order cone programs with finite event space. In this paper, we derive an infeasible interior-point algorithm for the same stochastic optimization problem by utilizing the work of Rangarajan (SIAM J Optim 16(4), 1211–1229, 2006) for deterministic symmetric cone programs. We show that the infeasible interior-point algorithm developed in this paper has complexity less than that of the homogeneous self-dual algorithm mentioned above. We implement the proposed algorithm to show that they are efficient.

In this paper, we develop a primal-dual central trajectory interior-point algorithm for symmetric programming problems and establish its complexity analysis. The main contribution of the paper is that it uniquely equips the central trajectory algorithm with various selections of the displacement step while solving symmetric programming. To show the efficiency of the proposed algorithm, these selections of calculating the displacement step are compared in numerical examples for second-order cone programming, which is a special case of symmetric programming.

We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization pro...

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.

We study and analyze the algebraic structure of the arbitrary-order cones. We show that, unlike popularly perceived, the arbitrary-order cone is self-dual for any order greater than or equal to 1. We establish a spectral decomposition, consider the Jordan algebra associated with this cone, and prove that this algebra forms a Euclidean Jordan algebra with a certain inner product. We generalize some important notions and properties in the Euclidean Jordan algebra of the second-order cone to the Euclidean Jordan algebra of the arbitrary-order cone.

Homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space have been proposed by Jin et al. in [17]. Alzalg [8] has adopted their work to derive homogeneous selfdual algorithms for stochastic second-order programs with finite event space. In this paper, we generalize these to derive homogeneous selfdual algorithms for stochastic programs with finite event space over the much wider class of all symmetric cones. They include among others, stochastic semidefinite programs and stochastic second-order cone programs.