On implementing a primal-dual interior-point method for conic quadratic optimization

Abstract

Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case. Conic quadratic optimization problems can in theory be solved efficiently using interior-point methods. In particular it has been shown by Nesterov and Todd that primal-dual interior-point methods developed for linear optimization can be generalized to the conic quadratic case while maintaining their efficiency. Therefore, based on the work of Nesterov and Todd, we discuss an implementation of a primal-dual interior-point method for solution of large-scale sparse conic quadratic optimization problems. The main features of the implementation are it is based on a homogeneous and self-dual model, handles the rotated quadratic cone directly, employs a Mehrotra type predictor-corrector

A homogeneous interior-point algorithm for nonsymmetric convex conic optimization

“…Mehrotra's second order correction generalizes nicely to self-scaled conic problem by use of the Jordan product that can be defined on such cones, see e.g. [2]. For non-symmetric cones, this generalization seems to no longer be possible.…”

“…This approach has proven successful for self-scaled cones [2,22,26] because it implies several desirable properties, among which are the ability to detect infeasibility in the problem pair and the ease of finding a suitable starting point, eliminating the need for a phase-I method. Unlike the algorithm in [19], our algorithm uses only the primal barrier function and therefore our linear systems are no larger than those appearing in ipms for self-scaled cones.…”

“…The Mehotra second order correction [12] is known to significantly improve practical performance of ipms for linear and quadratic conic problems [2,12,23]. With the same goal in mind, we suggest a new way to compute a search direction containing second order information for the general (possibly non-self-scaled) conic problem.…”

Limit analysis of plates using the EFG method and second‐order cone programming

“…On the other hand, the objective function in the associated optimization problem is convex, but not everywhere differentiable. One of the most efficient algorithms to overcome this difficulty is the primal-dual interior-point method presented in [19,20] and implemented in commercial codes such as the Mosek software package. The limit analysis problem involving conic constraints can then be solved by this efficient algorithm [16,21,22].…”

Advances in the simulation of viscoplastic fluid flows using interior-point methods

“…We denote by α a max the maximum step length possible for this search direction. The centering parameter γ is then estimated from this maximum step length using the following heuristic [31]:…”