A Theorem on Strict Separability of Convex Polyhedra and Its Applications in Optimization

We propose a new approach to the strict separation of convex polyhedra. This approach is based on the construction of the set of normal vectors for the hyperplanes, such that each one strict separates the polyhedra A and B. We prove the necessary and sufficient conditions of strict separability for convex polyhedra in the Euclidean space and present its applications in optimization. © 2010 Springer Science+Business Media, LLC

On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems

In this paper we propose a distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we prove that under the assumption of strong convexity and Lipshitz continuity of the gradient of the primal objective function we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme, i.e. a gradient scheme based on a weighted step size, for which we derive global linear rate of convergence for both dual and primal suboptimality and for primal feasibility violation. Many real applications, e.g. distributed model predictive control, network utility maximization or optimal power flow, can be posed as linearly constrained separable convex problems for which dual gradient type methods from literature have sublinear convergence rate. In the present paper we prove for the first time that in fact we can achieve linear convergence rate for such algorithms when they are used for solving these applications. Numerical simulations are also provided to confirm our theory.Comment: 14 pages, 4 figures, submitted to Automatica Journal, February 2014. arXiv admin note: substantial text overlap with arXiv:1401.4398. We revised the paper, adding more simulations and checking for typo

DECOMPOSITION METHODS FOR SOLVING ONE TYPE OF VARIATIONAL INEQUALITIES

В настоящей работе мы изучаем декомпозиционные методы решения вариационных неравенств, тесно связанных с задачей линейного отделения множеств. Эти методы позволяют решать независимые подзадачи, на которые может быть разложено исходное вариационное неравенство, как последовательно, так и параллельно.In present paper, we treat the decomposition methods for solving the variational inequalities which are closely connected with the linear separation problem of sets. These methods allow one to solve the independent subproblems, into which the original variational inequality problem can be decoupled, successively as well as in parallel.178-17

Solving of a projection problem for convex polyhedra given by a system of linear constraints

© 2017 IEEE. We propose a novel approach to solving the problem which is referred to as the polyhedral projection problem (PPP) and serves to find a projection of a point onto a polyhedron given by the linear inequality constraints. The basic idea of this approach is to utilize a reduction of the PPP to the problem of projecting the origin of Euclidean space onto the Minkowski difference of the considered polyhedron and point. We make use our previous results related to the concept of the Minkowski difference for the above-mentioned objects. The proposed approach is new (relative to the traditional ones) thanks to further reducing the PPP to the problem of projecting the origin onto the convex hull of some vectors corresponding to the gradients of the constraint functions. In the paper, this reduction is justified for the case when all of constraints of the PPP are violated at the point being projected onto the originally given polyhedron. In this case, the presented reduction makes broader a spectrum of the powerful tools of mathematical programming which may be operated for solving the PPP

An algorithm for data analysis via polyhedral optimization

© 2017 IEEE. We propose the novel data analysis algorithm which allows to identify exactly the position of a given point as exterior, interior, or boundary relatively to an intersection of the finite number of pattern sets. Due to the special structure of the problem under study, this algorithm can be realized not only by sequential, but also by parallel computing on the basis of appropriate model decomposition