We investigate a mixed 0 − 1 conic quadratic optimization problem with indicator variables arising in mean-risk optimization. The indicator variables are often used to model non-convexities such as fixed charges or cardinality constraints. Observing that the problem reduces to a submodular function minimization for its binary restriction, we derive three classes of strong convex valid inequalit...

Copositive programming is a relative young field which has evolved into a highly active research area in mathematical optimization. An important line of research is to use semidefinite programming to approximate conic programming over the copositive cone. Two major drawbacks of this approach are the rapid growth in size of the resulting semidefinite programs, and the lack of information about t...

Gauge functions significantly generalize the notion of a norm, and gauge optimization, as defined by [R. M. Freund, Math. Programming, 38 (1987), pp. 47–67], seeks the element of a convex set that is minimal with respect to a gauge function. This conceptually simple problem can be used to model a remarkable array of useful problems, including a special case of conic optimization, and related pr...

1.1. Lagrange Duality . . . . . . . . . . . . . . . . . . p. 2 1.1.1. Separable Problems – Decomposition . . . . . . . p. 7 1.1.2. Partitioning . . . . . . . . . . . . . . . . . . p. 9 1.2. Fenchel Duality and Conic Programming . . . . . . . . p. 10 1.2.1. Linear Conic Problems . . . . . . . . . . . . . p. 15 1.2.2. Second Order Cone Programming . . . . . . . . . p. 17 1.2.3. Semidefinite Progr...

We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms whose iteration complexity we analyse are so-called short-step algorithms. Our iteration complexity bounds match the current best iteration complexity bounds for primal-dual symmetric interior-point algorithm of Nesterov and Todd, for symmetric cone programming...

We propose a class of quadratic optimization problems whose exact optimal objective values can be computed by their completely positive cone programming relaxations. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one inhomogeneous equality whe...

A spherical object has been introduced into central catadioptric camera calibration through utilizing the properties of an image conic, which is the projection of the occluding contour of a sphere in the catadioptric image. However, previous calibration method using sphere images employs nonlinear optimization method and requires a good initial estimation to start the minimization. In this pape...

In this paper we develop a technique for constructing self-concordant barriers for convex cones. We start from a simple proof for a variant of standard result [1] on transformation of a ν-self-concordant barrier for a set into a self-concordant barrier for its conic hull with parameter (3.08 √ ν + 3.57)2. Further, we develop a convenient composition theorem for constructing barriers directly fo...

The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of t...

Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressi...