Solving a Class of LP Problems with a Primal-Dual Logarithmic Barrier Method

Applying a higher order primal-dual logarithmic barrier method for solving large real-life linear programming problems is addressed in this paper. The efficiency of interior point algorithm on these problems is compared with the one of the state-of-the-art simplex code MINOS version 5.3. Based on such experience, a wide class of LP problems is identified for which logarithmic barrier approach seems advantageous over the simplex one. Additionally, some practical rules for model builders are derived that should allow them to create problems that can easily be solved with logarithmic barrier algorithms

HOPDM Modular Solver for LP Problems User's Guide to version 2.12

The paper provides a description of HOPDM, a library of routines for solving large scale linear programming problems and its implementation at IIASA. HOPDM stands for Higher Order Primal Dual Method. The algorithm implemented in HOPDM is a new variant of a primal-dual logarithmic barrier method that uses multiple correctors of centrality. The newest version of the library -- HOPDM 2.12 -- is a robust and efficient LP code that compares favorably with the up to date commercial solvers. The paper contains an outline of the algorithm implemented in HOPDM and information about results of tests done with large LP problems developed at IIASA. Finally, the paper provides with details of the implementation of HOPDM and its use at IIASA, as well as with information about availability of the portable version of the HOPDM library

A note on bound entanglement and local realism

We show using a numerical approach that gives necessary and sufficient conditions for the existence of local realism, that the bound entangled state presented in Bennett et. al. Phys. Rev. Lett. 82, 5385 (1999) admits a local and realistic description. We also find the lowest possible amount of some appropriate entangled state that must be ad-mixed to the bound entangled state so that the resulting density operator has no local and realistic description and as such can be useful in quantum communication and quantum computation.Comment: 5 page

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra's d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities

A Relaxed Interior Point Method for Low-Rank Semidefinite Programming Problems with Applications to Matrix Completion

A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated Newton directions can be computed. In particular, it admits the application of both the first- and the second-order methods in this context. The convergence of the method is established. A prototype implementation is discussed and encouraging preliminary computational results are reported for solving the SDP-reformulation of matrix-completion problems

Violations of local realism with quNits up to N=16

Predictions for systems in entangled states cannot be described in local realistic terms. However, after admixing some noise such a description is possible. We show that for two quNits (quantum systems described by N dimensional Hilbert spaces) in a maximally entangled state the minimal admixture of noise increases monotonically with N. The results are a direct extension of those of Kaszlikowski et. al., Phys. Rev. Lett. {\bf 85}, 4418 (2000), where results for $N\leq 9$ were presented. The extension up to N=16 is possible when one defines for each N a specially chosen set of observables. We also present results concerning the critical detectors efficiency beyond which a valid test of local realism for entangled quNits is possible.Comment: 5 pages, 3 ps picture

On the Solution of Linear Programming Problems in the Age of Big Data

The Big Data phenomenon has spawned large-scale linear programming problems. In many cases, these problems are non-stationary. In this paper, we describe a new scalable algorithm called NSLP for solving high-dimensional, non-stationary linear programming problems on modern cluster computing systems. The algorithm consists of two phases: Quest and Targeting. The Quest phase calculates a solution of the system of inequalities defining the constraint system of the linear programming problem under the condition of dynamic changes in input data. To this end, the apparatus of Fejer mappings is used. The Targeting phase forms a special system of points having the shape of an n-dimensional axisymmetric cross. The cross moves in the n-dimensional space in such a way that the solution of the linear programming problem is located all the time in an "-vicinity of the central point of the cross.Comment: Parallel Computational Technologies - 11th International Conference, PCT 2017, Kazan, Russia, April 3-7, 2017, Proceedings (to be published in Communications in Computer and Information Science, vol. 753

Violations of local realism by two entangled quNits are stronger than for two qubits

Tests of local realism vs quantum mechanics based on Bell's inequality employ two entangled qubits. We investigate the general case of two entangled quNits, i.e. quantum systems defined in an N-dimensional Hilbert space. Via a numerical linear optimization method we show that violations of local realism are stronger for two maximally entangled quNits (N=3,4,...,9), than for two qubits and that they increase with N. The two quNit measurements can be experimentally realized using entangled photons and unbiased multiport beamsplitters.Comment: 5 pages, 2 pictures, LaTex, two columns; No changes in the result