Gaspar Mora

Global optimization: A new variant of the Alienor method

Abstract The multidimensional Alienor global optimization method has been elaborated in the 1980s by Cherruault and Guillez. It has been used, among other possibilities, for identification of mathematical models. Many interesting results have been obtained [1–3]. In this paper, we present a new variant of Alienor. It uses an α-dense curve with a small length and it has a quite simple parametrical representation. This variant has been coupled with the Evtushenko algorithm for decreasing the time required to obtain the global minimum.

The theoretic calculation time associated to α‐dense curves

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.

An approximation method for the optimization of continuous functions of n variables by densifying their domains

Most of the known optimization methods for a given continuous function f defined on a compact set H = Πi=1,..,n[ai,bi] require strong conditions on f. In the early 1980s, Cherruault proposed a method, called ALIENOR which was able to reduce the multidimensional optimization problem to another one‐dimensional optimization: the optimization of the restriction fh* of f to some adequate α‐dense curve h into the domain H. The characterization, the generation of such curves as well as the theoretic calculation times associated with them, have been studied previously by the authors. Their consequences and the general problem concerning the error in the approximation to global minimum of f and the minimization of the error itself, that such reduction produces, will be the subject of this paper.

Global optimization‐preserving operators

In this paper, some kinds of operators, defined on a class of real functions depending on a single real variable, preserving some fundamental properties for solving global optimization problems are introduced. These operators allow the transfer of the search of the global extremum of a determined objective function, f into those of its transformed function, Tf, for which a global extremum is easy to obtain.

Optimization by space‐densifying curves as a natural generalization of the Alienor method

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.

The Critical Strips of the Sums

We give a partition of the critical strip, associated with each partial sum 1 + 2 𝑧 + ⋯ + 𝑛 𝑧 of the Riemann zeta function for Re 𝑧 − 1 , formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a large class of almost-periodic functions with bounded spectrum.

The existence of α‐dense curves with minimal length in a metric space

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.).

On the minimal length curve that densifies the square

This paper deals with the existence of a curve of minimal length, expressed in parametric coordinates, which densifies the square J2=≤ft [ −1,1\right ] × ≤ft [ −1,1\right ] \ with a given degree of density α. Nevertheless, the same problem has no solution if we consider the family of curves defined by means of the graphics of continuous and rectifiable functions f: J→ J. Their consequences on the approximation method to the global optimization are also derived.

Approximating multiple integrals via α‐dense curves

This paper is intended to provide a numerical method for computing integrals of several variables. The method is based on a intuitive geometric idea relative to the meaning of densifying a domain in Rn+1(n≥1) by a curve h(t), contained in that domain, say K, with a very small density α (this must be interpreted as the following property: for any point of K there exists a point of the curve at distance less or equal than α).Thus, the method states that any area, volume, etc, can be computed as the limit of the length of a certain curve, densifying that domain, multiplied by a power of its density. Therefore, the computation of a multiple integral of a nonnegative continuous function can be approached by a simple integral corresponding to the length of the curve h(t) and certain power of its density.

A new universal method for solving all problems of operational research

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.