Balira O. Konfe, Yves Cherruault, Blaise Some and Titem Benneouala
To introduce Optimization‐Preserving‐Operators (O‐P‐Os), which are operators that are defined on classes of real functions that depend on a single variable, and allow us to…
Abstract
Purpose
To introduce Optimization‐Preserving‐Operators (O‐P‐Os), which are operators that are defined on classes of real functions that depend on a single variable, and allow us to eliminate local optima and to preserve global optima.
Design/methodology/approach
Outline a new method to build O‐P‐Os. These are introduced as O‐P‐O* and lead to a new approach for solving global optimization problems.
Findings
It was found that classical discretization methods for obtaining optimum of one variable function was too time‐consuming. The simple method introduced provided solutions to the test functions chosen as examples. The solutions were provided in a short time.
Research limitations/implications
Provides new tools for mathematical programming and in particular the global optimization problems. The O‐P‐O* introduced innovative technique for solving such problems.
Practical implications
O‐P‐O* produces solutions to global optimization problems in a much improved time. The algorithm derived, and the steps for its operation proved on implementation, the efficiency of the new method. This was demonstrated by numerical results for selected functions obtained using microcomputer systems.
Originality/value
Provides new way of solving global optimization problems.
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Balira O. Konfe, Yves Cherruault, Blaise Some and Titem Benneouala
This paper presents an efficient algorithm for solving general constrained optimization problems that arise in operational research (OR).
Abstract
Purpose
This paper presents an efficient algorithm for solving general constrained optimization problems that arise in operational research (OR).
Design/methodology/approach
An unified approach is accomplished by converting the constrained optimization problem into an unconstrained one and by using Alienor method coupled to the new optimization preserving operator* (OPO*) technique for the resolution.
Findings
A new algorithm for solving general constrained optimization problems with continuous objective function contributes to research in this area and in particular, to applications to OR.
Research limitations/implications
Some improvements could probably be obtained at calculation time. We will in future work, develop an adaption of these methods and techniques to optimization problems with mixed variables or with integer and Boolean variables.
Practical implications
The new algorithm can be advantageously compared with other methods such as generalized reduced gradient. Small‐sized numerical examples are given.
Originality/value
A new algorithm is given which guarantees a global optimal solution is easily obtained in all cases.
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J.C. Mazza, Y. Cherruault, G. Mora, B. Konfé and T. Benneouala
To use a new method based on α‐dense curved for solving problems of operational research.
Abstract
Purpose
To use a new method based on α‐dense curved for solving problems of operational research.
Design/methodology/approach
The method of global optimization (called Alienor) is used for solving problems involving integer or mixed variables. A reducing transformation using α‐dense curves allows to transforms a n‐variables problem into a problem of a single variable.
Findings
Extends the basic method valid for continuous variables to problems involving integer, Boolean or mixed variables. All problems of operational research, linear or nonlinear, may be easily solved by or technique based on α‐dense curves (filling a n‐dimensional space). Industrial problems can be quickly solved by this technique obtaining the best solutions.
Originality/value
This method is deduced from the original works of Y. Cherruault and colleagues about global optimization and α‐dense curves. It proposes new techniques for solving operational research problems.
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Keywords
T. Benneouala and Y. Cherruault
To show the usefulness of the Alienor method when applied to the global optimization problems that depend on large number of variables.
Abstract
Purpose
To show the usefulness of the Alienor method when applied to the global optimization problems that depend on large number of variables.
Design/methodology/approach
The approach is to use reducing transformations. The first is due to Cherruault and the second to Mora.
Findings
It was found that the Alienor method was very efficient and reliable in solving global optimization problems of many variables. Results produced to confirm this conclusion.
Research limitations/implications
The numerical results presented showed that the Alienor method was suitable for finding global minimum even in the case of a very large number of variables. The research provides a new methodology for solving such problems.
Practical implications
No other method, we believe, can obtain such results in so short a time for hundreds or even thousands of variables.
Originality/value
The new approach relies on the originality of both the Cherruault and the Mora transformations and their earlier invention of the Alienor method.
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Mahamat Maimos, Yves Cherruault, Balira O. Konfe and Ange‐gar S. Nkokolo Massamba
The purpose of this paper is to present an efficient algorithm to solve multi‐objective linear programming (MOLP) problem.
Abstract
Purpose
The purpose of this paper is to present an efficient algorithm to solve multi‐objective linear programming (MOLP) problem.
Design/methodology/approach
This new approach consists to convert the constrained multicriteria problem into an unconstrained global optimization problem. Then, the Alienor method coupled to the optimization preserving operators* (OPO*) technique is used to solve the transformed problem.
Findings
A determinist algorithm for solving general MOLP problem contributes to research in the decision‐makers area.
Research limitations/implications
Some improvements could probably be obtained. In future work, other scalarized functions will be used and this algorithm's complexity will be studied.
Practical implications
The new algorithm can be advantageously compared with other methods To illustrate this new approach, an example is studied.
Originality/value
A new algorithm is given which guarantees all efficient solutions are easily obtained in most cases.
Details
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A recursive scheme for the ALIENOR method is proposed as a remedy for the difficulties induced by the method. A progressive focusing on the most promising region, in combination…
Abstract
Purpose
A recursive scheme for the ALIENOR method is proposed as a remedy for the difficulties induced by the method. A progressive focusing on the most promising region, in combination with a variation of the density of the alpha-dense curve, is proposed.
Design/methodology/approach
ALIENOR method is aimed at reducing the space dimensions of an optimization problem by spanning it by using a single alpha-dense curve: the curvilinear abscissa along the curve becomes the only design parameter for any design space. As a counterpart, the transformation of the objective function in the projected space is much more difficult to tackle.
Findings
A fine tuning of the procedure has been performed in order to identity the correct balance between the different elements of the procedure. The proposed approach has been tested by using a set of algebraic functions with up to 1,024 design variables, demonstrating the ability of the method in solving large scale optimization problem. Also an industrial application is presented.
Originality/value
In the knowledge of the author there is not a similar paper in the current literature.
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Balira O. Konfe, Yves Cherruault and Blaise Some
To propose a new method for solving constrained global optimization problems using a method that consists of transforming a constrained global optimization problem into an…
Abstract
Purpose
To propose a new method for solving constrained global optimization problems using a method that consists of transforming a constrained global optimization problem into an unconstrained one without using any penalty coefficients.
Design/methodology/approach
Use of an unconstrained global optimization method such as the Alienor method which has been adapted for several variables.
Findings
Use of the adapted Alienor method allowed the solution of the transformed problem with little difficulty.
Research limitations/implications
Transforms the original objective function into a new one involves the introduction of some extra parameters. Cannot guarantee the convergence to a global solution of the original problem. The simple described approach, provides new possibilities.
Practical implications
No further parameters introduced in this new approach, and no conditions or hypotheses are imposed on the objective function or on the constraints.
Originality/value
New method of transforming a constrained problem into an unconstrained one, with use of proven Alienor method adapted to several variables.
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Esther Claudine Bitye Mvondo, Yves Cherruault and Jean‐Claude Mazza
The purpose of this paper is to use α‐dense curves for solving some Diophantine equations, such as Pythagorean triples, Linear Diophantine equations, the Pell Fermat equation, the…
Abstract
Purpose
The purpose of this paper is to use α‐dense curves for solving some Diophantine equations, such as Pythagorean triples, Linear Diophantine equations, the Pell Fermat equation, the Mordell equation for positive values.
Design/methodology/approach
The paper's aim is to present the applications in Number Theory of a new method based on α‐dense curves first developed at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. The α‐dense curves generalize the space filling curves (Peanocurves,…) and fractal curves. This technique can be used for solving all problems of operational research in a simple way. The main idea consists in expressing n variables by means of a single one.
Findings
Apply the method to Number Theory. One of the most important applications is related to global optimization. Multivariable optimization problems coming from operational research or from industry can be quickly and easily solved.
Originality/value
The paper presents a new method based on α‐dense curves for solving Diophantine equations.