Adilson Elias Xavier

The hyperbolic smoothing clustering method

The minimum sum-of-squares clustering problem is considered. The mathematical modeling of this problem leads to a min-sum-min formulation which, in addition to its intrinsic bi-level nature, has the significant characteristic of being strongly nondifferentiable. To overcome these difficulties, the resolution, method proposed adopts a smoothing strategy using a special C^~ differentiable class function. The final solution is obtained by solving a sequence of low dimension differentiable unconstrained optimization subproblems which gradually approach the original problem. The use of this technique, called hyperbolic smoothing, allows the main difficulties presented by the original problem to be overcome. A simplified algorithm containing only the essentials of the method is presented. For the purpose of illustrating both the reliability and the efficiency of the method, a set of computational experiments was performed, making use of traditional test problems described in the literature

Optimal Covering of Plane Domains by Circles Via Hyperbolic Smoothing

We consider the problem of optimally covering plane domains by a given number of circles. The mathematical modeling of this problem leads to a min–max–min formulation which, in addition to its intrinsic multi-level nature, has the significant characteristic of being non-differentiable. In order to overcome these difficulties, we have developed a smoothing strategy using a special class C∞ smoothing function. The final solution is obtained by solving a sequence of differentiable subproblems which gradually approach the original problem. The use of this technique, called Hyperbolic Smoothing, allows the main difficulties presented by the original problem to be overcome. A simplified algorithm containing only the essential of the method is presented. For the purpose of illustrating both the actual working and the potentialities of the method, a set of computational results is presented.

Hyperbolic smoothing and penalty techniques applied to molecular structure determination

Abstract This work considers the problem of estimating the relative positions of all atoms of a protein, given a subset of all the pair-wise distances between the atoms. This problem is NP-hard, and the usual formulations are nonsmoothed and nonconvex, having a high number of local minima. Our contribution is an efficient method that combines the hyperbolic smoothing and the penalty techniques that are useful in obtaining differentiability and reducing the number of local minima.

The Euclidean Steiner tree problem in Rn: A mathematical programming formulation

A nonconvex mixed-integer programming formulation for the Euclidean Steiner Tree Problem (ESTP) in Rn is presented. After obtaining separability between integer and continuous variables in the objective function, a Lagrange dual program is proposed. To solve this dual problem (and obtaining a lower bound for ESTP) we use subgradient techniques. In order to evaluate a subgradient at each iteration we have to solve three optimization problems, two in polynomial time, and one is a special convex nondifferentiable programming problem.

An evaluation of the bihyperbolic function in the optimization of the backpropagation algorithm

The backpropagation algorithm is one of the most used tools for training artificial neural networks. However, this tool may be very slow in some practical applications. Many techniques have been discussed to speed up the performance of this algorithm and allow its use in an even broader range of applications. Although the backpropagation algorithm has been used for decades, we present here a set of computational results that suggest that by replacing bihyperbolic functions the backpropagation algorithm performs better than the traditional sigmoid functions. To the best of our knowledge, this finding was never previously published in the open literature. The efficiency and discrimination capacity of the proposed methodology are shown through a set of computational experiments, and compared with the traditional problems of the literature.

Modelagens min-max-min para o problema de localização de estações de rádio base

We report a new proposal to solve the base station location problem. This proposal is related to a non-differentiable min-max-min problem with multi-level nature. In order to overcome these difficulties and, thus, to be able to use more robust and efficient optimization tools, such as Gradient and Newton methods, we have developed a smoothing strategy using a special smoothing function of class C¥. Then, the final solution is obtained by solving a sequence of differentiable sub-problems which gradually approaches the original problem. The use of this technique, called Hyperbolic Smoothing, permits to overcome the main difficulties that arise from the original problem. Besides the initial min-max-min modelling, we propose three variant methods through which other issues of the base station location problem are also taken into account. An algorithm containing the essentialities of the method is also presented together a set of computational results.

Solving the continuous multiple allocation p-hub median problem by the hyperbolic smoothing approach

Hub-and-spoke (HS) networks constitute an important approach for designing transportation and telecommunications systems. The continuous multiple allocation -hub median problem consists in finding the least expensive HS network, locating a given number of hubs in planar space and assigning traffic to them, given the demands between each origin-destination pair and the respective transportation costs, where each demand centre can receive and send flow through more than one hub. The specification of the problem corresponds to a strongly non-differentiable min-sum-min formulation. The proposed method overcomes this difficulty with the hyperbolic smoothing strategy, which has been proven able to solve large instances of clustering problems quite efficiently. The solution is ultimately obtained by solving a sequence of differentiable unconstrained low-dimension optimization subproblems. The consistency of the method is shown through a set of computational experiments with large hub-and-spoke problems in continuous space with up to 1000 cities.

A hyperbolic smoothing approach to the Multisource Weber problem

The Multisource Weber problem, also known as the continuous location-allocation problem, or as the Fermat-Weber problem, is considered here. A particular case of the Multisource Weber problem is the minimum sum-of-distances clustering problem, also known as the continuous

Flying elephants: a general method for solving non-differentiable problems

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