The q-Gradient Vector for Unconstrained Continuous Optimization Problems

Soterroni, Aline C.; Galski, Roberto Luiz; Ramos, Fernando M.

doi:10.1007/978-3-642-20009-0_58

Cited by 18 publications

(17 citation statements)

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“…The Steepest Descent method, for example, sets d k = −∇F (x k ) as the search direction and the step length α k is usually determined by a line search technique that minimizes the objective function along the direction d k . In the q-G method, the search direction is the negative of the q-gradient of the objective function −∇ q F (x) defined in [1], [2] as…”

Section: The Q-g Methodsmentioning

confidence: 99%

“…Recently, based on the concepts of Jackson's derivative and simulated annealing, a generalization of the classical Steepest Descent method, called the q-gradient (q-G) method, has been proposed for solving unconstrained continuous global optimization problems [1]- [3]. The main idea behind this new method is the use of the q-gradient vector of the objective function as the search direction.…”

Section: Introductionmentioning

confidence: 99%

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Satellite constellation design using the q-G global optimization method

Soterroni

Galski

Ramos

2015

2015 Third World Conference on Complex Systems (WCCS)

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The field of optimization is interdisciplinary in nature, and it is an indispensable tool in various fields of science and engineering. Computational intelligence-based techniques, such as neural networks, simulated annealing, stochastic machines, genetic algorithms and others, have been proven to be effective in solving global optimization problems. In this work, the q-Gradient (q-G) method is applied for solving a satellite constellation design problem for two regional navigation satellite systems, one for covering the Brazilian territory and the other for South America. The q-G method is a global optimization algorithm based on the concepts of q-calculus and simulated annealing. The main idea behind the q-G method is the use of the q-gradient vector of the objective function as the search direction. The q-gradient vector is a generalization of the classical gradient vector based on the concept of Jackson's derivative and its use provides the algorithm an effective mechanism for escaping from local minima. The algorithm has three free parameters, and it is implemented so that the search process gradually shifts from global exploration in the beginning to almost local search in the end. The q-G method was applied on six 10-D multimodal test functions, and its performance was compared with eleven Evolutionary Algorithms (EAs). The q-G method performed well against the EAs arriving in forth position in a direct comparison with them. For the satellite constellation design problem, we are using the basic design assumption of using four satellites in geosynchronous orbits for covering both Brazil and South America. This optimization problem has 12 design variables (12-D) and includes calculations of the Geometric Dilution of Precision (GDOP) as the main metric for the design of the satellite constellations. Comparison of the results with a previous study for the Brazilian constellation indicates that the q-G method is able to solve this problem and that there is considerable room for improvement with the use of an additional satellite for the South America case study.

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Section: The Q-g Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Satellite constellation design using the q-G global optimization method

Soterroni

Galski

Ramos

2015

2015 Third World Conference on Complex Systems (WCCS)

Self Cite

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“…for q ∈ (q, 1) ∪ (1,q -1 ). The q-partial derivative of a function f : R n → R at x ∈ R n with respect to x i , where scalar q ∈ (0, 1), is given as [34]…”

Section: Preliminariesmentioning

confidence: 99%

“…A q-version of the steepest descent method was first developed in the field of optimization to solve single objective nonlinear unconstrained problems. The method was able to escape from many local minima and reach the global minimum [34]. The q-LMS (Least Mean Square) algorithm is proposed by employing the q-gradient to compute the secant of the cost function instead of the tangent [28].…”

Section: Introductionmentioning

confidence: 99%

On q-BFGS algorithm for unconstrained optimization problems

et al. 2020

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Variants of the Newton method are very popular for solving unconstrained optimization problems. The study on global convergence of the BFGS method has also made good progress. The q-gradient reduces to its classical version when q approaches 1. In this paper, we propose a quantum-Broyden–Fletcher–Goldfarb–Shanno algorithm where the Hessian is constructed using the q-gradient and descent direction is found at each iteration. The algorithm presented in this paper is implemented by applying the independent parameter q in the Armijo–Wolfe conditions to compute the step length which guarantees that the objective function value decreases. The global convergence is established without the convexity assumption on the objective function. Further, the proposed method is verified by the numerical test problems and the results are depicted through the performance profiles.

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“…A Figura 3 ilustra o comportamento do método da máxima descida (trajetória em azul) e de um método baseado em q-gradiente (trajetória em vermelho) em que a direção de buscaé a direção contráriaà direção do vetor q-gradiente para q i gerados segundo uma distribuição de probabilidade log-normal com desvio padrão variável e tamanho do passo obtido via seçãoáurea (veja [8]). …”

Section: Comportamento De Métodos Baseados Em Q-gradienteunclassified

O uso do vetor q-gradiente como direção de busca em métodos de otimização contínua

Soterroni¹,

Ramos²,

Galski³

et al. 2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo: O q-cálculo surgiu da generalização de séries, funções, números especiais, entre outros, por meio de um parâmetro multiplicativo q e tal que, no limite q → 1, retomam suas respectivas versões clássicas. Com base nos trabalhos de Euler e Heine, o reverendo inglês Frank Hilton Jackson desenvolveu, noínicio do século XX, o q-cálculo de forma sistemática com destaque para a reintrodução do conceito de q-derivada, que passou a ser mais conhecida como derivada de Jackson, e a criação da q-integral. Nasúltimas décadas, o q-cálculo tem conectado matemáticos e físicos em aplicações de mecânica estatística, teoria dos números e análise combinatória. Este trabalho mostra como o conceito de q-derivada pode ser usado naárea de otimização contínua por meio do vetor q-gradiente, uma generalização do vetor gradiente clássico no contexto do q-cálculo.Palavras-chave: q-cálculo, q-derivada, derivada de Jackson, q-gradiente IntroduçãoGeneralizações no contexto do q-cálculo remontam aos trabalhos de Fermat, Euler, Heine e Gauss, mas foi no início do século XX que o reverendo inglês Frank Hilton Jackson contribuiu para o desenvolvimento do q-cálculo de forma sistemática (veja [1]). Dentre uma série de generalizações de funções, séries e números especiais, F. H. Jackson reintroduziu o operador q-derivada 1 , amplamente conhecido como derivada de Jackson, e criou o conceito de q-integral definida (veja [2, 3, 4, 5]).Seja f (x) uma função diferenciável de umaúnica variável. A derivada clássica de f em relação a xé dada por1 O operador q-derivada tambémé conhecido como operador q-diferença, derivada de Jackson ou, simplesmente, q-derivada.

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The q-Gradient Vector for Unconstrained Continuous Optimization Problems

Cited by 18 publications

References 5 publications

Satellite constellation design using the q-G global optimization method

Satellite constellation design using the q-G global optimization method

On q-BFGS algorithm for unconstrained optimization problems

O uso do vetor q-gradiente como direção de busca em métodos de otimização contínua

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