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A multi‐dimensional global optimization method has been developed. This method uses the curves of IRⁿ called α‐dense. A characterization of α‐dense curves is given in terms of γ‐stochastically independent functions as well as a constructive method to generate them by means of only one function φ called γ‐uniformly distributed has been developed. A very large class of functions which generate α‐dense curves is discussed. This class contains the γ‐uniformly distributed functions, the periodic functions and even functions which are not periodic, but which fulfil some properties.

To use α‐dense curves for solving an optimization problem with constraints involving integer variables.

α‐dense curves are curves in Rⁿ depending on a single variable able to approximate a compact K⊂Rⁿ with precision α. It is proposed α‐dense curves allowing to obtain all integer points of a compact domain in Rⁿ. This transformation allows to transform the functional into a new function depending on a single variable. Then we can calculate the global optimum of the functional.

Alienor method invented by Y. Cherruault allows to find global minimum of n‐continuous variables functions. Here, α‐dense curves are extended to problems involving integer variables. The curves pass through all points having integer coordinates and belonging to the compact domain. By this method integer programming (nonlinear) problems arising in operational research have been easily and exactly solved.

It is the first time the technique based on α‐dense curves to optimization problems with integer variables are extended. This approach is totally original and allows to solve very easily and fastly nonlinear optimization problems.

This paper seeks to present an original method for transforming multiple integrals into simple integrals.

This can be done by using α‐dense curves invented by Y. Cherruault and A. Guillez at the beginning of the 1980s.

These curves allow one to approximate the space Rⁿ (or a compact of Rⁿ) with the accuracy α. They generalize fractal curves of Mandelbrobdt. They can be applied to global optimization where the multivariables functional is transformed into a functional depending on a single variable.

Applied to a multiple integral, the α‐dense curves using Chebyshev's kernels permit one to obtain a simple integral approximating the multiple integral. The accuracy depends on the choice of α.

The paper presents an original method for transforming integrals into simple integrals.

The theoretic calculation time associated to every α‐dense curve into a fixed H of Rⁿ is inversely proportional to the discretization step depending on the length of the curve and, more directly, of the derivatives of its coordinate functions. For a given degree of density α, it is interesting to seek curves into H which may minimize the theoretic calculation time and then to solve the practical problem of computing approximations for global optimization of a given continuous function defined in H, by means of its restriction over a family of curves with the same degree of density into the cube H.

The purpose of this paper is to describe a general method for solving all problems arising in industrial processes and more generally in operational research.

The paper's aim is to present a new method based on α‐dense curves first developed at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. This technique allows to solve all problems of operational research in a simple way. For instance, industrial problems leading to optimization or optimal control problems can be easily and precisely solved by this very general technique. The main idea consists in expressing n variables by means of a single one.

This new method, based on “alpha‐dense curves” allows to express n variables in function of a single variable. One of the most important applications is related to global optimization. Multivariable optimization problems can be quickly and easily solved, even for great numbers of variables and for integer or boolean variables. Every problem (linear or nonlinear) coming from operational research or from industry becomes simple to solve in a very short time on micro‐calculators.

This method is deduced from the original works of Yves Cherruault et al. of MEDIMAT laboratory. The reducing transformations were initiated at the beginning of the 1980s by Yves Cherruault and Arthur Guillez. Then they were generalized by the notion of α‐dense curves. A lot of applications were derived covering entirely the operational research and a part of functional analysis.

The Alienor method offers a powerful approximation technique for the optimization of continuous multivariable functions defined on a compact set H of Rⁿ. Its computational efficiency is completed by the fact that it gave rise to the theory of space‐densifying curves. Presents a survey of these curves, analysing their most important properties and characteristics. Finally, the concept of theoretic calculation time (t.c.t.) associated with each curve suggests an interesting geometric problem on the existence of a curve with minimal t.c.t.

Follows on from previous papers by the authors, where TWONE, a method for solving global optimization problems in two dimensions was presented. Presents an improved version. If Ω is a compact connect subset of R² looks for the minimum of a function f defined on Ω with values in R.² TWONE method considers the restriction of f to an α‐dense curve in R². That implies the resolution of a one dimensional problem. Gives some α‐dense curves in R² with α as small as we want. Has biomedical and biocybernetic implications.

The purpose of this paper is to present an efficient algorithm to solve multi‐objective linear programming (MOLP) problem.

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.

A determinist algorithm for solving general MOLP problem contributes to research in the decision‐makers area.

Some improvements could probably be obtained. In future work, other scalarized functions will be used and this algorithm's complexity will be studied.

The new algorithm can be advantageously compared with other methods To illustrate this new approach, an example is studied.

A new algorithm is given which guarantees all efficient solutions are easily obtained in most cases.

In this article the aim is to propose a new form to densify parallelepipeds of R^N by sequences of α‐dense curves with accumulated densities.

This will be done by using a basic α‐densification technique and adding the new concept of sequence of α‐dense curves with accumulated density to improve the resolution of some global optimization problems.

It is found that the new technique based on sequences of α‐dense curves with accumulated densities allows to simplify considerably the process consisting on the exploration of the set of optimizer points of an objective function with feasible set a parallelepiped K of R^N. Indeed, since the sequence of the images of the curves of a sequence of α‐dense curves with accumulated density is expansive, in each new step of the algorithm it is only necessary to explore a residual subset. On the other hand, since the sequence of their densities is decreasing and tends to zero, the convergence of the algorithm is assured.

The results of this new technique of densification by sequences of α‐dense curves with accumulated densities will be applied to densify the feasible set of an objective function which minimizes the quadratic error produced by the adjustment of a model based on a beta probability density function which is largely used in studies on the transition‐time of forest vegetation.

A sequence of α‐dense curves with accumulated density represents an original concept to be added to the set of techniques to optimize a multivariable function by the reduction to only one variable as a new application of α‐dense curves theory to the global optimization.

The reducing transformation and global optimization technique called Alienor has been developed in the 1980s by Cherruault and Guillez. These methods are based on the approximating properties of α ‐dense curves. The aim of this work is to give a very large class of functions generating α ‐dense curves in a hyper‐rectangle of Rⁿ.