Linear Number Of Variables Research Articles

A nonlinear programming model with implicit variables for packing ellipsoids

The problem of packing ellipsoids is considered in the present work. Usually, the computational effort associated with numerical optimization methods devoted to packing ellipsoids grows quadratically with respect to the number of ellipsoids being packed. The reason is that the number of variables and constraints of ellipsoids' packing models is associated with the requirement that every pair of ellipsoids must not overlap. As a consequence, it is hard to solve the problem when the number of ellipsoids is large. In this paper, we present a nonlinear programming model for packing ellipsoids that contains a linear number of variables and constraints. The proposed model finds its basis in a transformation-based non-overlapping model recently introduced by Birgin et al. (J Glob Optim 65(4):709---743, 2016). For solving large-sized instances of ellipsoids' packing problems with up to 1000 ellipsoids, a multi-start strategy that combines clever initial random guesses with a state-of-the-art (local) nonlinear optimization solver is presented. Numerical experiments show the efficiency and effectiveness of the proposed model and methodology.

A new bound for 3-satisfiable MaxSat and its algorithmic application

Let F be a CNF formula with n variables and m clauses. F is 3-satisfiable if for any 3 clauses in F, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least 23 of its clauses can be satisfied by a truth assignment. We improve this result by showing that every 3-satisfiable CNF formula F contains a subset of variables U, such that some truth assignment τ will satisfy at least 23m+13mU+ρn′ clauses, where m is the number of clauses of F, mU is the number of clauses of F containing a variable from U, n′ is the total number of variables in clauses not containing a variable in U, and ρ is a positive absolute constant. Both U and τ can be found in polynomial time.We use our result to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MaxSat-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 23m+k clauses, where k is the parameter.

Dispersion in Disks

We present three new approximation algorithms with improved constant ratios for selecting n points in n disks such that the minimum pairwise distance among the points is maximized. A very simple O(nlog n)-time algorithm with ratio 0.511 for disjoint unit disks. An LP-based algorithm with ratio 0.707 for disjoint disks of arbitrary radii that uses a linear number of variables and constraints, and runs in polynomial time. A hybrid algorithm with ratio either 0.4487 or 0.4674 for (not necessarily disjoint) unit disks that uses an algorithm of Cabello in combination with either the simple O(nlog n)-time algorithm or the LP-based algorithm. The LP algorithm can be extended for disjoint balls of arbitrary radii in ℝ d , for any (fixed) dimension d, while preserving the features of the planar algorithm. The algorithm introduces a novel technique which combines linear programming and projections for approximating Euclidean distances. The previous best approximation ratio for dispersion in disjoint disks, even when all disks have the same radius, was 1/2. Our results give a positive answer to an open question raised by Cabello, who asked whether the ratio 1/2 could be improved.

Um algoritmo de planos-de-corte para o número cromático fracionário de um grafo

O número cromático fracionário χF(G) de um grafo G é um conhecido limite inferior para seu número cromático χ(G). Experimentos relatados na literatura mostram que usar χF(G), em lugar do tamanho da clique máxima, pode ser muito mais eficiente para orientar a busca em um algoritmo tipo branch-and-bound para determinação de χ(G). Uma dificuldade, porém, é tratar o modelo linear conhecido para χF(G), o qual apresenta um número exponencial de variáveis e demanda um caro processo de geração de colunas. Neste trabalho, examinamos uma formulação alternativa para obter um limite inferior para χF(G) que possui um número de variáveis linear no tamanho do grafo, porém um número exponencial de restrições. Utilizamos o método de planos-de-corte para lidar com esse inconveniente. Algumas heurísticas de separação são propostas, e experimentos computacionais mostram que valores muito próximos de χF(G), em muitos casos iguais, são encontrados em tempo inferior à implementação com geração de colunas.

A parallel algorithm for partially separable non-convex global minimization: Linear constraints

The global minimization of large-scale partially separable non-convex problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of a separable concave part, an unseparated convex part, and a strictly linear part, which are all coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. An important special class of problems which can be reduced to this form are the synomial global minimization problems. Such problems often arise in engineering design, and previous computational methods for such problems have been limited to the convex posynomial case. In the current work, a convex underestimating function to the objective function is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and convex underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results obtained on the four processor Cray 2, both sequentially and in parallel on all four processors, are also presented.

Guaranteed ε-approximate solution for indefinite quadratic global minimization

The global minimization of large-scale indefinite quadratic problems over a bounded polyhedral set using a parallel branch-and-bound approach is considered. The objective function consists of both a nonlinear part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A convex quadratic underestimating function to the indefinite quadratic objective function is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and convex quadratic underestimating problems over each subregion are solved in parallel. Branch-and-bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with the number of nonlinear variables varying from 25 to 125 and up to 400 linear variables. These results were obtained on a four-processor CRAY 2 using both sequential and parallel implementations of the algorithm. The computation times using a four-processor CRAY 2 ranged from less than one second to a maximum of 703 seconds with a relative stopping tolerance of 0.001. The average (ten problems) parallel solution time for a problem with 125 nonlinear variables and 200 purely linear variables (the largest problem solved in terms of the number nonlinear variables) was approximately 35 seconds. The parallel implementation is very efficient as shown by speedups greater than the number of processors in certain cases.

A parallel algorithm for constrained concave quadratic global minimization

The global minimization of large-scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of both a concave part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A linear underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables. These results were obtained on a four processor CRAY2 using both sequential and parallel implementations of the algorithm. The average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0.001. For a relative tolerance of 0.1, the average computation time appears to increase only linearly with the number of linear variables.