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

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 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

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

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

A Computational Approach to the Detection and Prediction of (Ir)Regularity in Children's Folk Songs

We examine (ir)regularity in the musical structure of 736 monophonic children's folk songs from 22 European countries, by simulating and detecting (ir)regularity with the computational model, IDyOM, and our own algorithm, Ir_Reg, which classifies melodies according to regularity of their musical structure. IDyOM offers a range of viewpoints which allow observation and prediction of various musical features. We used five viewpoints to measure the information content and entropy of musical events in songs. Analysis across the data shows absence of irregular musical structure in children's folk songs from Croatia, Serbia, Turkey, Portugal, Hungary, and Romania. Conversely, absence of regular structure in children's folk songs was found in Great Britain, Norway and Switzerland. Further analysis of (ir)regularity, by individual country, revealed the importance of patterns repeated at pitch in regular songs, and a higher occurrence of transposed repeated patterns in irregular songs. Principal component analysis (PCA) shows the salience of pitch and pitch intervals in the perception of (ir)regular structure. Neither rhythm nor contour affects the perception of regularity. Recurring pulse/meter and arch-like melodic structure were found in the majority of children's folk songs. The study shows that irregularity exists in children's folk songs, and that this genre can be complex