From combinatorial optimization to real algebraic geometry and back

In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which rely on results from polynomial optimization, a sub-eld of real algebraic geometry

Community structure and the evolution of interdisciplinarity in Slovenia's scientific collaboration network

Interaction among the scientific disciplines is of vital importance in modern science. Focusing on the case of Slovenia, we study the dynamics of interdisciplinary sciences from 1960 to 2010. Our approach relies on quantifying the interdisciplinarity of research communities detected in the coauthorship network of Slovenian scientists over time. Examining the evolution of the community structure, we find that the frequency of interdisciplinary research is only proportional with the overall growth of the network. Although marginal improvements in favor of interdisciplinarity are inferable during the 70s and 80s, the overall trends during the past 20 years are constant and indicative of stalemate. We conclude that the flow of knowledge between different fields of research in Slovenia is in need of further stimulation.Comment: 11 pages, 4 figures; accepted for publication in PLoS ONE [related work available at http://arxiv.org/abs/1004.4824 and http://www.matjazperc.com/sicris/stats.html

Advancing stable set problem solutions through quantum annealers

We assess the performance of D-wave quantum solvers for solving the stable set problem in a graph, one of the most studied NP-hard problems. We perform computations on some instances from the literature with up to 125 vertices and compare the quality of the obtained solutions with known optimum solutions. It turns out that the hybrid solver gives very good results, while the Quantum Processing Unit solver shows rather modest performance overall

A Block Coordinate Descent-based Projected Gradient Algorithm for Orthogonal Non-negative Matrix Factorization

This article utilizes the projected gradient method (PG) for a non-negative matrix factorization problem (NMF), where one or both matrix factors must have orthonormal columns or rows. We penalise the orthonormality constraints and apply the PG method via a block coordinate descent approach. This means that at a certain time one matrix factor is fixed and the other is updated by moving along the steepest descent direction computed from the penalised objective function and projecting onto the space of non-negative matrices. Our method is tested on two sets of synthetic data for various values of penalty parameters. The performance is compared to the well-known multiplicative update (MU) method from Ding (2006), and with a modified global convergent variant of the MU algorithm recently proposed by Mirzal (2014). We provide extensive numerical results coupled with appropriate visualizations, which demonstrate that our method is very competitive and usually outperforms the other two methods

A new approximation hierarchy for polynomial conic optimization

In this paper we consider polynomial conic optimization problems, where the feasible set is defined by constraints in the form of given polynomial vectors belonging to given nonempty closed convex cones, and we assume that all the feasible solutions are non-negative. This family of problems captures in particular polynomial optimization problems (POPs), polynomial semi-definite polynomial optimization problems (PSDPs) and polynomial second-order cone-optimization problems (PSOCPs). We propose a new general hierarchy of linear conic optimization relaxations inspired by an extension of Pólyaʼs Positivstellensatz for homogeneous polynomials being positive over a basic semi-algebraic cone contained in the non-negative orthant, introduced in Dickinson and Povh (J Glob Optim 61(4):615-625, 2015). We prove that based on some classic assumptions, these relaxations converge monotonically to the optimal value of the original problem. Adding a redundant polynomial positive semi-definite constraint to the original problem drastically improves the bounds produced by our method. We provide an extensive list of numerical examples that clearly indicate the advantages and disadvantages of our hierarchy. In particular, in comparison to the classic approach of sum-of-squares, our new method provides reasonable bounds on the optimal value for POPs, and strong bounds for PSDPs and PSOCPs, even outperforming the sum-of-squares approach in these latter two cases.V članku obravnavamo polinomske konične optimizacijske probleme, kjer je dopustna množica definirana z omejitvami, da morajo biti dani polinomski vektorji v danih nepraznih zaprtih konveksnih stožcih. Dodatno morajo dopustne rešitve zadoščati pogoju nenegativnosti. Ta družina problemov zajema zlasti klasične probleme polinomske optimizacije (POP), probleme polinomske semidefinitne optimizacije (PSDP) in probleme polinomske optimizacije nad stožci drugega reda (PSOCP). Predlagamo novo splošno hierarhijo linearnih koničnih optimizacijskih poenostavitev, ki naravno sledijo iz razširitve Pólya-jevega izreka o pozitivnosti iz Dickinson in Povh (J Glob Optim 61 (4): 615-625, 2015). Ob nekaterih klasičnih predpostavkah te poenostavitve monotono konvergirajo k optimalni vrednosti izvirnega problema. Kot zanimivost pokažemo, da dodajanje posebne redundantne omejitve k osnovnemu problemu ne spremeni optimalne rešitve tega problema, a bistveno izboljša kvaliteto poenostavitev. V članku tudi predstavimo obsežen seznam številčnih primerov, ki jasno kažejo na prednosti in slabosti naše hierarhije

New Approach to Modelling and Its Application in Transportation in Urban Traffic

Urban and suburban transport is a transport system that combines various types of transport, transporting people and goods in the city and the nearest suburban area, as well as performing work on the improvement of the city. The urban transport system is part of a diversified urban economy and includes: vehicles (rolling stock); track devices (rail tracks, tunnels, overpasses, bridges, overpasses, stations, parking lots); marinas and boat stations; power supply devices (traction power substations, cable and contact networks, gas stations); repair shops and factories; depot, garages, service stations; car rental offices; linear communication devices, alarms, locks, traffic control. The city’s transport system also includes a bicycle, for which in civilized countries a special bicycle path on the sidewalks is allocated. The urban passenger transport is faced with the task of delivering passengers to their destination with maximum comfort at the minimum cost of time, labour and resources. The territorial development of cities at all times of their history was determined primarily by the speed characteristics of mass intracity movements. Therefore, the famous architect, creator of modern cities Le Corbusier noted that no city can grow faster than its transport. In this article, we introduce a new approach to modelling by using network theory and calculating topological properties of network, which have practical applications in transportation and urban traffic network. This work is licensed under a&nbsp;Creative Commons Attribution-NonCommercial 4.0 International License.</p

ON THE COMPLEXITY OF SEMIDEFINITE PROGRAMS ARISING IN POLYNOMIAL OPTIMIZATION

In this paper we investigate matrix inequalities which hold irrespective of the size of the matrices involved, and explain how the search for such inequalities can be implemented as a semidefinite program (SDP). We provide a comprehensive discussion of the time complexity of these SDPs

New Approach to Modelling and Its Application in Transportation in Urban Traffic

Urban and suburban transport is a transport system that combines various types of transport, transporting people and goods in the city and the nearest suburban area, as well as performing work on the improvement of the city. The urban transport system is part of a diversified urban economy and includes: vehicles (rolling stock); track devices (rail tracks, tunnels, overpasses, bridges, overpasses, stations, parking lots); marinas and boat stations; power supply devices (traction power substations, cable and contact networks, gas stations); repair shops and factories; depot, garages, service stations; car rental offices; linear communication devices, alarms, locks, traffic control. The city’s transport system also includes a bicycle, for which in civilized countries a special bicycle path on the sidewalks is allocated. The urban passenger transport is faced with the task of delivering passengers to their destination with maximum comfort at the minimum cost of time, labour and resources. The territorial development of cities at all times of their history was determined primarily by the speed characteristics of mass intracity movements. Therefore, the famous architect, creator of modern cities Le Corbusier noted that no city can grow faster than its transport. In this article, we introduce a new approach to modelling by using network theory and calculating topological properties of network, which have practical applications in transportation and urban traffic network. This work is licensed under a&nbsp;Creative Commons Attribution-NonCommercial 4.0 International License.</p

ON THE SET-SEMIDEFINITE REPRESENTATION OF NONCONVEX QUADRATIC PROGRAMS WITH CONE CONSTRAINTS

The well-known result stating that any non-convex quadratic problem over the non-negative orthant with some additional linear and binary constraints can be rewritten as linear problem over the cone of completely positive matrices (Burer, 2009) is generalizes by replacing the non-negative orthant with arbitrary closed convex and pointed cone. This set-semidefinite representation result implies new semidefinite lower bounds for quadratic problems over the Bishop-Phelps cones, based on the Euclidian norm