Roberto Quirino do Nascimento

A Branch-and-Bound Algorithm for the Close-Enough Traveling Salesman Problem

This paper addresses the close-enough traveling salesman problem. In this problem, rather than visiting the vertex (customer) itself, the salesman must visit a specific region containing such vertex. To solve this problem, we propose a simple yet effective exact algorithm, based on branch-and-bound and second order cone programming. The proposed algorithm was tested in 824 instances suggested in the literature. Optimal solutions are obtained for open problems with up to a thousand vertices. We consider instances both in two- and three-dimensional space.

DC-GRASP: directing the search on continuous-GRASP

Several papers in the scientific literature use metaheuristics to solve continuous global optimization. To perform this task, some metaheuristics originally proposed for solving combinatorial optimization problems, such as Greedy Randomized Adaptive Search Procedure (GRASP), Tabu Search and Simulated Annealing, among others, have been adapted to solve continuous global optimization problems. Proposed by Hirsch et al., the Continuous-GRASP (C-GRASP) is one example of this group of metaheuristics. The C-GRASP is an adaptation of GRASP proposed to solve continuous global optimization problems under box constraints. It is simple to implement, derivative-free and widely applicable method. However, according to Hedar, due to its random construction, C-GRASP may fail to detect promising search directions especially in the vicinity of minima, which may result in a slow convergence. To minimize this problem, in this paper we propose a set of methods to direct the search on C-GRASP, called Directed Continuous-GRASP (DC-GRASP). The proposal is to combine the ability of C-GRASP to diversify the search over the space with some efficient local search strategies to accelerate its convergence. We compare the DC-GRASP with the C-GRASP and other metaheuristics from literature on a set of standard test problems whose global minima are known. Computational results show the effectiveness and efficiency of the proposed methods, as well as their ability to accelerate the convergence of the C-GRASP.

Proposta de mensuração de risco baseado em utilidade

The measurement of risk presented in Markowitz’ work was based on the variance of returns. In spite of penalizing deviation greater than the unity, this methodology does not differentiate positive from negative deviation. This study proposes a measurement of risk in which a utility function is incorporated to describe the characteristics of a rational investor. The disappointment of the investor with the loss is greater than the satisfaction with the gain. This new component, in addition to supplying other interpretations for volatility, seems to be a more effective alert instrument than the usual methodology.

A Continuous Strategy to Solve a Class of Discrete Optimization Problems

Abstract In this work we develop a method to solve a class of discrete optimization problems. This class covers linear, quadratic, convex, and discrete geometric programming problems. The methodology consists in inserting additional geometric constraints where any viable solution is also discrete. Moreover, we also adopt a methodology for solution of signomial geometric programming problems and solve the problem. We present some examples of facility location problems and the results obtained.

Problemas de programação geométrica signomial via otimização DC

Neste trabalho os Problemas de Programacao Geometrica Signomial, nao convexos, sao escritos em sua forma posinomial e posteriormente reescritos na forma padr˜ao (CDC) de um problema de diferenca de funcoes convexas (DC), definidos por DCG (DC geometrico). Tal metodologia foi utilizada pois permite que se obtenha a solucao global para o problema. Com base na t´tecnica de otimizacao global, Otimizacao DC, foi proposto um algoritmo e foram resolvidos problemas classicos da literatura de Programacao Geometrica onde os resultados obtidos foram iguais e em alguns casos melhores do que os j´a existentes. Esta melhora na solucao se deve ao fato de que, em geral, os resultados encontrados na literatura sao obtidos atraves da implementacao de metaheuristicas.

The discrete ellipsoid covering problem: A discrete geometric programming approach

The ellipsoid covering problem consists of covering an ellipsoid with spheres whose radii belong to a discrete set. The discrete nature of the radii of the spheres is one of the difficulties inherent to solving this problem. The other difficulty is to ensure that every point of the ellipsoid is covered by at least one sphere. Despite these difficulties, a good reason for studying this problem is its application in configuring gamma ray machines, used for treatment of brain tumors. This is a semi-infinite nonlinear discrete problem. To solve it, we present a weak version and model it as a discrete signomial geometric programming problem. To obtain convex constraints, we apply the condensation technique to approximate the model in combination with a continuous model applied to the discrete part. Thus, we use a primal dual interior point method to solve the problem. During model construction, we determine the smallest number of spheres that can be used to cover the ellipsoid using an auxiliary model that is similar to the model of the Knapsack problem. Finally, we present computational results for experiments.

Modelo de priorização de investimentos em saneamento básico utilizando programação linear com base em indicadores ambientais

In this article a contribution concerning the development of a hierarchization of investments in sanitation model is presented. The technique of the linear programming is used, based on an environmental health indicator, the ISA/JP1 (Silva, 2006). The aim is the maximization of the environmental health conditions in the communities to be benefited. In this way, it is attempted to interfere in the allocation of the benefits related to the sub-indicator variables corresponding to basic sanitation. The model was applied in five periurban communities of Lower Gramame River Basin, on the Southern Littoral of Paraiba State, Brazil. In this case, the proposed benefits would result in significant changes, even reaching the salubrious condition, according to values of available investments.