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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.

Gives a new method for defining and calculating multiple integrals. More precisely proposes that it is possible to define a multiple integral by means of a simple integral. This can be performed by using α‐dense curves in Rⁿ, already introduced for global optimization using the ALIENOR method.

Classical multidimensional global optimization methods are difficult to implement in high dimensions. To show that the Alienor method coupled with the Brent algorithm can avoid this difficulty.

Use is made of the Alienor method and the Brent algorithm to obtain algorithms that were applied to test functions having several local minima.

Interesting results concerning the number of evaluation points were obtained. It was shown that this coupling can be improved if α‐dense curves of minimal length were used.

Multidimensional global optimization problems have proven to be difficult to implement in high dimensions. This research continues the search for improved methods by coupling existing established methods such as Alienor with others such as the Brent algorithm.

A new coupled method has been developed and algorithms obtained to tackle such global optimization problems. The coupling is unique and the algorithms are tested numerically on selected functions.

Aims to present study of the coupling of the Alienor method with the algorithm of Piyavskii‐Shubert for global optimization applications.

The Alienor method allows us to transform a multivariable function into a function of a single variable for which it is possible to use an efficient and rapid method for calculating the global optimum. This simplification is based on the use of the established Alienor methodology.

The Alienor method allows us to transform a multidimensional problem into a one‐dimensional problem of the same type. It was then possible to use the Piyavskii‐Shubert method based on sub‐estimators of the objectives function. The obtained algorithm from coupling the two methods was found to be simple and easy to implement on any multivariable function.

This method does not require derivatives and the convergence of the algorithm is relatively rapid if the Lipschitz constant is small.

The classical multidimensional global optimization methods involve great difficulties for their implementation to high dimensions. The coupling of two established methods produces a practical easy to implement technique.

New method couples two established ones and produces a simple and user‐friendly technique.

To introduce an algorithm to solve inequalities defined by real functions on a certain compact set D of a general metric space (E, D). The device is based on α‐dense curves.

The solution of inequalities using α‐dense curves and also an approach to a global optimization technique, similarly obtained to that of the inequalities.

A new method is presented. The algorithm for solving inequalities is described which is based on a proven result. Inequalities of n‐variable dependence are reduced by transformation using α‐dense curves.

The research is a continuation of studies that resulted in a new method called Alienor for solving global optimization problems associated with multi‐variable functions.

Based on earlier research by the authors α‐dense curves have been used which allow the transformation of a n‐variables global optimization problem into a one‐variable global one. This paper gives a new method for quickly solving the one‐variable problem.

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.

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.

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.

In the knowledge of the author there is not a similar paper in the current literature.

Some results concerning the existence of α‐dense curves with minimal length are given. This type of curves used in the reducing transformation called Alienor was invented by Cherruault and Guillez. They have been applied to global optimization in the following way: a multivariable optimization problem is transformed in an optimization problem depending on a single variable. Then this idea was extended by Cherruault and his team for obtaining general classes of reducing transformations having minimal properties (length of the α‐dense curves, minimization of the calculus time, etc.).

This paper presents an efficient algorithm for solving general constrained optimization problems that arise in operational research (OR).

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.

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.

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.

The new algorithm can be advantageously compared with other methods such as generalized reduced gradient. Small‐sized numerical examples are given.

A new algorithm is given which guarantees a global optimal solution is easily obtained in all cases.

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.

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.

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.

Provides new tools for mathematical programming and in particular the global optimization problems. The O‐P‐O* introduced innovative technique for solving such problems.

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.

Provides new way of solving global optimization problems.

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.