scholarly journals APPLICATION OF SEMIDEFINITE PROGRAMMING TO TRUSS DESIGN OPTIMIZATION / SANTVAROS OPTIMIZAVIMO UŽDAVINIŲ SPRENDIMAS TAIKANT PUSIAU APIBRĖŽTĄ PROGRAMAVIMĄ

2015 ◽  
Vol 7 (3) ◽  
pp. 280-284
Author(s):  
Rasa Giniūnaitė
Keyword(s):  
Semidefinite Programming ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Algebraic Manipulation ◽  
Truss Structure ◽  
Convex Constraints ◽  
Positive Semidefinite Matrices ◽  
Combinatorial Optimization Problems ◽  
Linear Problems ◽  
The One

Semidefinite Programming (SDP) is a fairly recent way of solving optimization problems which are becoming more and more important in our fast moving world. It is a minimization of linear function over the intersection of the cone of positive semidefinite matrices with an affine space, i.e. non-linear but convex constraints. All linear problems and many engineering and combinatorial optimization problems can be expressed as SDP, so it is highly applicable. There are many packages that use different algorithms to solve SDP problems. They can be downloaded from internet and easily learnt how to use, two of these are SeDuMi and SDPT-3. In this paper truss structure optimization problem with the goal of minimizing the mass of the truss structure was solved. After doing some algebraic manipulation the problem was formulated suitably for Semidefinite Programming. SeDuMi and SDPT-3 packages were used to solve it. The choice of the initial solution had a great impact on the result using SeDuMi. The mass obtained using SDPT-3 was on average smaller than the one obtained using SeDuMi. Moreover, SDPT-3 worked more efficiently. However, the comparison of my approach and two versions of particle swarm optimization algorithm implied that semidefinite programming is in general more appropriate for solving such problems. Pusiau apibrėžtas programavimas yra iškiliojo optimizavimo posritis, kuriame tikslo funkcija tiesinė, o leistinoji sritis – pusiau teigiamai apibrėžtų matricų kūgio ir afininės erdvės sankirta. Tai gana naujas optimizavimo problemų sprendimo būdas, tačiau jau plačiai taikomas sprendžiant inžinerinius bei kombinatorinius optimizavimo uždavinius. Tokiems uždaviniams spręsti yra daug skirtingų paketų, taikančių įvairius algoritmus. Šiame darbe buvo naudojami SeDuMi ir SDPT-3 paketai, kuriuos, kaip ir daugumą kitų, galima parsisiųsti iš interneto. Tikslas buvo rasti minimalią santvaros masę atsižvelgiant į numatytus apribojimus. Naudojant SDPT-3 gauta optimali masė buvo vidutiniškai mažesnė nei naudojant SeDuMi. SDPT-3 veikė efektyviau ir pradinių sąlygų pasirinkimas neturėjo tokios didelės įtakos sprendiniui kaip naudojant SeDuMi paketą. Palyginus rezultatus su sprendiniais, gautais taikant dalelių spiečiaus optimizavimo algoritmą, nustatyta, kad tokio tipo uždaviniams pusiau apibrėžtas programavimas yra tinkamesnis.

scholarly journals On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings

2017 ◽  
Vol 32 ◽  
pp. 15-40 ◽  
Author(s):  
Sabine Burgdorf ◽  
Monique Laurent ◽  
Teresa Piovesan
Keyword(s):  
Optimization Problems ◽  
Positive Semidefinite ◽  
Quantum Correlations ◽  
Positive Semidefinite Matrices ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Positive Semidefinite Cone ◽  
The One ◽  
Graph Parameters ◽  
Semidefinite Cone

We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+$, consisting of all the $n \times n$ symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set $\mathcal Q$ of bipartite quantum correlations, as projection of an affine section of it. We have two main results concerning the structure of the completely positive semidefinite cone, namely about its interior and about its closure. On the one hand we construct a hierarchy of polyhedral cones covering the interior of $\mathcal{CS}_+$, which we use for computing some variants of the quantum chromatic number by way of a linear program. On the other hand we give an explicit description of the closure of the completely positive semidefinite cone by showing that it consists of all matrices admitting a Gram representation in the tracial ultraproduct of matrix algebras.

scholarly journals A Characterization of Polynomial Density on Curves via Matrix Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1231
Author(s):  
Carmen Escribano ◽  
Raquel Gonzalo ◽  
Emilio Torrano
Keyword(s):  
Matrix Algebra ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Point Of View ◽  
Alternative Proof ◽  
General Context ◽  
Positive Semidefinite Matrices ◽  
Jordan Curves ◽  
Point Evaluations

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

scholarly journals Focusing on the Golden Ball Metaheuristic: An Extended Study on a Wider Set of Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
E. Osaba ◽  
F. Diaz ◽  
R. Carballedo ◽  
E. Onieva ◽  
A. Perallos
Keyword(s):  
Genetic Algorithms ◽  
Combinatorial Optimization ◽  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Optimization Problems ◽  
Traveling Salesman ◽  
Routing Problem ◽  
Combinatorial Optimization Problems ◽  
Golden Ball ◽  
The One

Nowadays, the development of new metaheuristics for solving optimization problems is a topic of interest in the scientific community. In the literature, a large number of techniques of this kind can be found. Anyway, there are many recently proposed techniques, such as the artificial bee colony and imperialist competitive algorithm. This paper is focused on one recently published technique, the one called Golden Ball (GB). The GB is a multiple-population metaheuristic based on soccer concepts. Although it was designed to solve combinatorial optimization problems, until now, it has only been tested with two simple routing problems: the traveling salesman problem and the capacitated vehicle routing problem. In this paper, the GB is applied to four different combinatorial optimization problems. Two of them are routing problems, which are more complex than the previously used ones: the asymmetric traveling salesman problem and the vehicle routing problem with backhauls. Additionally, one constraint satisfaction problem (the n-queen problem) and one combinatorial design problem (the one-dimensional bin packing problem) have also been used. The outcomes obtained by GB are compared with the ones got by two different genetic algorithms and two distributed genetic algorithms. Additionally, two statistical tests are conducted to compare these results.

scholarly journals Reinforcement Learning for Route Optimization with Robustness Guarantees

2021 ◽  
Author(s):  
Tobias Jacobs ◽  
Francesco Alesiani ◽  
Gulcin Ermis
Keyword(s):  
Reinforcement Learning ◽  
Optimization Problems ◽  
Route Optimization ◽  
Research Trend ◽  
Learning To Learn ◽  
Combinatorial Optimization Problems ◽  
One Step ◽  
Full Solution ◽  
Original Objective ◽  
The One

Application of deep learning to NP-hard combinatorial optimization problems is an emerging research trend, and a number of interesting approaches have been published over the last few years. In this work we address robust optimization, which is a more complex variant where a max-min problem is to be solved. We obtain robust solutions by solving the inner minimization problem exactly and apply Reinforcement Learning to learn a heuristic for the outer problem. The minimization term in the inner objective represents an obstacle to existing RL-based approaches, as its value depends on the full solution in a non-linear manner and cannot be evaluated for partial solutions constructed by the agent over the course of each episode. We overcome this obstacle by defining the reward in terms of the one-step advantage over a baseline policy whose role can be played by any fast heuristic for the given problem. The agent is trained to maximize the total advantage, which, as we show, is equivalent to the original objective. We validate our approach by solving min-max versions of standard benchmarks for the Capacitated Vehicle Routing and the Traveling Salesperson Problem, where our agents obtain near-optimal solutions and improve upon the baselines.

Semidefinite optimization

Acta Numerica ◽  
2001 ◽  
Vol 10 ◽  
pp. 515-560 ◽  
Author(s):  
M. J. Todd
Keyword(s):  
Combinatorial Optimization ◽  
Differential Inclusion ◽  
Computational Methods ◽  
Duality Theory ◽  
Optimization Problems ◽  
Symmetric Matrix ◽  
Positive Semidefinite ◽  
Semidefinite Optimization ◽  
Combinatorial Optimization Problems ◽  
The Stability

Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps ‘or’ would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.

Cones of Hermitian matrices and trigonometric polynomials

2011 ◽  
Author(s):  
Mihály Bakonyi ◽  
Hugo J. Woerdeman
Keyword(s):  
Semidefinite Programming ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Positive Semidefinite ◽  
Trigonometric Polynomials ◽  
Hermitian Matrices ◽  
Positive Semidefinite Matrices ◽  
Entropy Maximization ◽  
The Real ◽  
Basic Properties

This chapter studies cones in the real Hilbert spaces of Hermitian matrices and real valued trigonometric polynomials. Based on an approach using such cones and their duals, it establishes various extension results for positive semidefinite matrices and nonnegative trigonometric polynomials. In addition, it shows the connection with semidefinite programming and includes some numerical experiments. The discussions cover cones and their basic properties, cones of Hermitian matrices, cones of trigonometric polynomials, determinant and entropy maximization, and semidefinite programming. Exercises and notes are provided at the end of the chapter.

MULTIDIMENSIONAL CALIBRATION OF CRUDE OIL AND REFINED PRODUCTS VIA SEMIDEFINITE PROGRAMMING TECHNIQUES

2019 ◽  
Vol 22 (01) ◽  
pp. 1850056
Author(s):  
CAROLINA EFFIO SALDIVAR ◽  
JOSÉ HERSKOVITS ◽  
JUAN PABLO LUNA ◽  
CLAUDIA SAGASTIZÁBAL
Keyword(s):  
Crude Oil ◽  
Semidefinite Programming ◽  
Interior Point Method ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrices ◽  
Out Of Sample ◽  
Joint Dynamics ◽  
Programming Techniques ◽  
Space Formulation ◽  
Heating Oil

To describe the joint dynamics of prices of crude oil and refined products we extend two-factor models to a multidimensional setting. The new model captures directly the general correlation structure between the different commodities in the form of certain covariance matrix. Since the associated state-space formulation makes use of such correlations, the feasible set of the resulting estimation problem includes the cone of positive semidefinite matrices. Tractability is ensured by means of an interior point method, specially tailored for nonlinear semidefinite programming problems. For different sets of historical prices of crude oil, heating oil, and gasoline, the empirical out-of-sample forecasts obtained with the approach proposed in this work systematically provide an excellent fit to data.

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