Dimensionality Reducing by Alpha-Dense Curves: Application to Global Optimization, Multiple

1. Dimensionality Reducing by Alpha-Dense Curves: Application to Global Optimization, Multiple Gaspar Mora Departamento de Análisis Matemático Universidad de Alicante (Spain)

2. Abstract In Mathematics the word complexity, as synonym of difficulty, has not a clear meaning. However, it seems that it is very easy to understand that one mathematical problem involving functions of several variables is, in general, more complex than one problem with a univariable function. The Alpha-Dense Curves theory tries to reduce the complexity associated to the dimension and establishes an Approximation Theory on the domains, in the sense of the Hausdorff metric, instead of that of the functions. Some applications to optimize multivariable continuous functions, to approximate a multiple integral by a simple one, and to determine if a non-linear inequality has solution, will be the subjects which we shall deal with techniques based on Alpha-Dense Curves.

3. 1.Definitions and examples Definition Given a real number 0 , an  -dense curve (or curve of density  ) in a subset K of finite diameter in a metric space (E,d), is a continuous mapping whose image , now on denoted , is contained in K and the distance, , for any In others words, an  -Dense Curve in a compact set K is a Peano Continuum (a connected, locally connected and compact set) whose Hausdorff distance from K When =0, one has a Peano curve (also called space-filling curve) provided that the interior of K, it follows that if a curve  is -dense in K, then it is also -dense for any  >. Thus the minimal  verifying the properties is, strictly speaking, the density of the curve in K , which coincides with the Hausdorff distance

4. Example In the euclidean space , N2 for each positive integer m , the unit cube is densified by the curve defined by It is easy to check that has density in where is a positive constant depending on N. For instance, Therefore their densities go to 0 as m . The curve is said to be the Cosines Curve of order m .

5. ExampleIn , N2, for each positive integer m the unit cube is densified by the polynomial curve defined by where is the Chebyshev polynomial of the first kind of degree j, that is, The density of in the unit cube is ,where (N) is a positive constant that depends on N (for instance, , etc.). Hence, also the densities tend to 0 as m . The curve is called the Chebyshev Curve of order m .

6. DefinitionA subset K of (E,d) is said to be densifiable if it contains -dense curves for arbitrary  >0. As we just have seen, the unit cube is a densifiable set. This set has been densified by the Cosines and the Chebyshev Curves. Now, taking into account the transformation , ,..., ,..., defined by an  -dense curve in is transformed on a -dense curve in the cube , where Consequently, any cube is densifiable. Furthermore, as a cube is a Peano Continuum, the Hahn-Mazurkiewicz theorem implies that it is a continuous image of I. Hence any cube is in fact a Space-Filling Curve and so is, in particular, densifiable. Clearly, in a metric space (E,d) , the class of all densifiable sets contains the class of all Peano Continua . However, both classes are distinct. For instance, the set is densifiable but it is not a Peano Continuum, as it is well-known.

7. 2.Application of alpha -Dense Curves to Global Optimization 2.1.Convergence. As we know, given a real function f and a nonvoid set X , an usual form to present an Optimization Problem (O.P) is to find, or to approximate to, the value (finite or infinite) The function f is called the objective function and X the feasible set. Even for a differentiable objective function on a compact feasible set of the type in , , ,the minimization methods are not very efficient when N is large. Many problems, closely connected with the location of the global minimum, occur.These difficulties, in general, decrease when the objective function f is univariable. Thus, for f continuous on K , the multivariable optimization problem (1)is substituted by the univariable one    (2)where and is an -dense curve in K . The reduction of problem 1 to 2 has an error that we can estimate: Given a densifiable set K , let us denote by the class of all curves in K of density  .

8. Thus, for a given , the error in the approximation is Observe that since , always is . Furthermore, if and only if the -dense curve passes through one global minimizer, i.e., a point where f attains the global minimum. An estimation of the error is given in the following result Theorem For each -dense curve , the error is bounded where and is the modulus of continuity of f of order  . As corollary of the preceding theorem, the convergence of this reductional method is guaranted by the fact that the modulus of continuity tends to 0 as  goes to 0 . Whenever f is Lipschitzian, that is, it satisfies with 0<M, 0<1, the bound estimated is which directly depends on the density.

9. 2.2.Minimization-Preserving Operators (M.P.O.). When we apply the -Dense Curves method to the multivariable optimization problem (1)we substitute (1) by the univariable one being , and is an  -dense curve in K . Then the new objective function will have, possibly, those extreme points (local and global extrema) that corresponding to f added to the points for which the gradient vector of f (if there exists) is orthogonal to the tangent vector to . For instance, if K is a cube always we can take with tangent vector nonnull for all t. That is, speaking in a colloquial language, we could say that the substitution of problem (1) by problem (2), can introduce "noise" due to the many extrema of . To eliminate this "noise" we introduced a type of operators that we called Optimization-Preserving Operators(Mora, Cherruault and Benabidallah, 2003). Later, these operators were studied from the specific role devoted to minimization and then they were called Minimization-Preserving Operators (M.P.O.)(Mora, 2005).

10. Definition Let , be the spaces of continuous and bounded real functions , respectively on . and a subset of .We say that a mapping , , is a Minimization-Preserving Operator if and only if for any ,i) the minimizer set of , denoted , is contained in the minimizer set of g, denoted , ii) and have at least a global minimum in common to both functions and g Trivial examples of minimization-preserving operators in are: a) The exponential operator, b) The dilation operator, with. real numbers In general, any strictly increasing function defines a M.P.O. on the space by means of the composition law, that is, , for . .

11. Observe that in all the above examples , for all . In this case, these operators will be called trivial . Nevertheless, our objective is for defining a M.P.O. T such that be strictly included in for some functions g , such as the following example shows (Mora, 2005). Example Each arbitrary function >0 continuous on I, defines a nontrivial M.P.O. T on by the formula where is a global minimizer of g ( for instance, can be chosen as the minimum, in I, of all the global minimizers). The main results to reduce the mentioned noise are the following (Mora, 2005).

12. Theorem Let A be an arbitrary constant, the Heaviside function and an arbitrary bijective function of class with . Let M be positive, consider the numbers:and Thus, for each the formula for any x I and arbitrary , defines an M.P.O. on the closed ball on the space . Furthermore i) in the level set , constant, ii) all the minimizers, nonnull, of are in the level set .

13. The reiterated application of the above theorem generates a procedure for finding a global minimizer, as we can see in the next result. Corollary Under the hypotheses of the above theorem and reiterating its application, for each , with a set of minimizers proper and finite, we obtain an algorithm to find a global minimizer of g.

14. 3.Application of alpha-Dense Curves to Dimensionality Reducing Integrals Now, our purpose is to reduce directly the integral of a non-negative real function f, of class on the unit cube , , to a simple one on the interval [-1,1]. For that, we shall use an -Dense Curve that densifies with arbitrary small density each elementary cube of the region of quadrature. The key of the result that we are looking for is in the following facts: 1) For any c>0, the cube is densified by the Cosines Curve, that is, by the curve defined by 2) The volume of can be calculated by the formula where isthe length of given by the integralwhere is the tangent vector. Now, the dimension of the integral is reduced by means of the following result (Mora and Mora-Porta, 2005).

15. TheoremLet f be a real non-negative function of class on and the Cosines Curve of order m in . Thus the integral may be approximated, for large enough m, by where is the Chebyshev polynomial of the second kind of degree and . Analogously, if each elementary cube of the region of quadrature , , is densified by a Chebyshev Curve with one has the following result (Mora, Benavent and Navarro, 2002).

16. Theorem Let fbe a real nonnegative function of class on and the Chebyshev Curve of order m in .Thus the multiple integral can be approximated, for large enough m, by the simple integral where is de Chebyshev polynomial of the second kind of degree and If we denote by the error in the approximation to the multiple integral of f by using the Cosines Curve, one has An estimation of is given in the following result (Mora and Mora-Porta, 2005). TheoremFor the Cosines Curve of order m in the unit cube , the error in the integral reduction formula is an . RemarkThe constant in depends on the 1-norm of f defined bywhere is the gradient of f . For the Chebyshev Curve we think that it would have a similar result.

17. 4.Application of alpha-Dense Curves to nonlinear Mathematical Programming Given a real function f defined on a densifiable compact set K of a metric space (E,d), it is crucial in many applications, to know the set (possibly empty) An -Dense Curves approach to this problem would be: 1) Let us take an -dense curve in K, , which without loss of generality we can suppose nonconstant on any subinterval of I. Furthermore we can also suppose the existence of an increasing bijective function , only continuous at 0, and satisfying 2) Define on I the univariable function , and for each s such that consider the number where is the open ball centered at and radius r . Taking into account the above considerations, one has the following result (Mora, Cherruault and Ubeda, 2005).

18. TheoremLet K be a densifiable compact of a metric space (E,d) ,  an  -dense curve in K (satisfying the preceding assumptions 1,2) and f a real function defined on K , continuous on . Then, on the set , the recursive sequence . ( ) defined by satisfies: i) If for some , then either or . ii) If for all , , then the sequence converges to some . When the hypothesis of the continuity of f on the  -dense curve  is extended to the whole K, we have the following result. Corollary Let be a sequence of -dense curves in K, with as . Under the same hypotheses of the above theorem and adding the continuity of f on K, assume that for each curve there is a positive integer such that the corresponding recursive sequence . Then the set where is the set of all zeros of f.

19. 5.Application of the Alpha-Dense CurvesTheory Alpha-Dense Curves method has been applied to minimize, to integrate and to solve inequalities involving functions of a great number of variables. These functions appear when we model some processes where the experiments are not possible ,difficult or very expensive. The private enterprises that have supported research on this theory and its applications at the Laboratoire MEDIMATde l'Université Pierre et Marie Curie, Paris VI (France) are: • Renault, • DGA, • Pharmaceutical Laboratories: Boiron, Servier and Sandoz, • Banking Sector: Crédit Agricole.

20. The main concrete problems which have been studied following this method were: 1. Degranulation of Basophyls, Visual Perception for Artificial Perception. 2. Study on the gluclose-insuline system (a preliminar study on the conception of an artificial pancreas). 3. Study of the Pelagic Ecosystem of an upwelling location. 4. Competition model between two different cultures and diffussion microbes in water. 5. Industrial process for clarifying apple juice. 6. Difussion of Streptadivin and DTPA-Biotin in inmmunoscintigraphy of heart in two stages. 7. Determination of temperature in the interior of oil wells. 8. Analysis of stocks corresponding to agricultural and food products. 9. Models of the combustion by hydrogen, reduction of pollution and comfortability of cars. 10.Dispersion of materials in blood plasma. 11.Transport of anticancer substances applied to the brain surface. 12.The Nitroarginine and the problem of inhibition of the NOsynthase. In the U.N.T. and the U.A. (Spain), we are applying the  -Dense Curves method to parameter estimation of transition-time probability density in forest models.

21. 6.The Alpha-Dense Curves Theory from a theoretical point of view and its connections The -Dense Curves as mathematical theory is of interest in itself and has links with other subjects. 6.1. Peano Curves. In 1890, G.Peano demonstrated that the interval I=[0,1] could be mapped surjectively and continuously onto the square .These curves are called Peano Curves or also Space-filling Curves. Some of them are famous such as that of Hilbert (1891),Moore (1900) and Lebesgue (1904). However a general method for generating these curves remained unsettled. H.Steinhaus in 1936 solved the problem by means of a surpresively result : " if two continuous non-constant functions on I are stochastically independent with respect to Lebesgue measure, then they are the coordinate functions of a space-filling curve " . We have proved that the condition of stochastically independence (s.i.) is too strong to characterize space-filling curves. We pointed out that there are space-filling curves whose coordinate functions are not s.i. Indeed (Mora,G. and Mira,J.A., 2001), The coordinate functions of the Lebesgue curve are not stochastically independent. For the purpose to characterize the Space-Filling Curves we introduced a filling condition (f.c.) in terms of the quasi-stochastically independent concept.

22. Definition(Filling condition )We shall say that N measurable functions ,...,are quasi-stochastically independent (q.s.i.), with respect to the Lebesgue measure on R (denoted ) , if for any open sets . of R the condition implies Using this filling condition or q.s.i. concept, we obtained a characterization of space-filling curves by means of the following results (Mora,G. and Mira,J.A., 2001): TheoremLet ,..., be quasi-stochastically independent functions ( r.L.m.). Assume also that they are continuous and nonconstant.Thus the curve defined by ,..., fills the parallelepiped . Conversely Theorem Let ,..., be nonconstant continuous functions such that the curve ,..., fills the parallelepiped Thus ,..., are quasi-stochastically independent (r.L.m). In fact, the idea of filling condition was already, implicitly, handled when we gave a characterization of -dense curves in parallelepipeds of (Mora,G. and Cherruault, 1997). We also settled a characterization theorem on the filling condition in terms of Borel measures. Moreover, this result yields a method to construct the coordinate functions of a space-filling curve (Mora,G. and Mira,J.A., 2001).

23. TheoremAssume are Q.S.I continuous nonconstant functions and H .Then the set function on the class of all cubes contained in H , defines a Borel measure for which Reciprocally, any Borel measure  on a parallelepiped satisfying (1) defines N continuous nonconstant functions that are Q.S.I. We recently gave a characterization of the space-filling curves by uniform limits of -dense curves in the compact that is filled.The -dense curves must have densities tending to zero and coordinate functions with variation tending to infinity as  tends to zero (Mora, 2005). More precisely, the result is. Theorem A continuous mapping is a Peano curve filling if and only if is the uniform limit of a sequence of -dense curves in with densities , for which there is no constant M such that the variation , for all n, for some .

24. 6.2. -Dense Curves of the Minimal Length. In a densifiable compact K of a metric space (E,d), the existence of an -dense curve, for fixed >0, with a minimal length, has been solved. We have used the set that is, the set of all , -dense in K, satisfying for some and . When the set will be denoted , and we shall consider that as a subset of the space of all continuous mappings on I and valued in E, endowed with the metric Because of the Ascoli's theorem, the compactness of is proved in the following result. Lemma Let (E,d) be a metric space and K a densifiable compact. Then, is a compact in the space . for the metric . Now, the existence of -dense curves with minimal length follows (Ziadi,Cherruault and Mora, 2000). Theorem Let K be a densifiable compact of a metric space (E,d) and the set of all  -dense curves in K, >0. Assume contains at least a rectifiable curve, thus there exists such that ,where is the length of  .

25. 6.3.Functional equations generating -dense curves. A surpresive connection between -dense curves and functional equations emerged when we were trying to solve one of these. Such is the case, for instance, of the functional equations and whose continuous solutions can densify, with arbitary small density, compacts of the plane. Indeed, in this manner we have proved the following results.(Mora, Cherruault and Ziadi,2000) TheoremLet p<q be arbitrary integers and the vector subspace of all continuous solutions of the functional equation .Thus contains an infinite set of linear independent functions that densifies the rectangle with arbitrary small density. A similar result is obtained for the other functional equation. TheoremLet p<q be arbitrary integers, a zero of the complex function and the vector subspace of all continuous solutions of the functional equation .Thus contains an infinite set of linear independent functions that densifies with arbitrary small density the compact with .

26. Open problem with respect to the functional equations and -Dense Curves. We have just seen how special compacts have been densified by continuous solutions of the preceding functional equations. Then, except traslation and dilation of the compact set, our problem is: Problem In , given a densifiable compact K having by boundary a path, is there a functional equation such that its continuous solutions can densify it ? 6.4. The -Dense Curves in Topological Vector Spaces. Whenever E is a topological vector space,not necessarily metrizable, the concept of -density is substituted by that of the V-localization, V being a given 0-neighborhood. This concept is formalized by means of the following definition. Definition We say that a subset B of a topological vector space is V-localized by a curve, i.e., by a continuous mapping, , if and only if

27. DefinitionA subset B of a topological vector space is said to be localizable if and only if is V-localized by any 0-neighborhood V. A nontrivial example of localizable set in a nonmetrizable topological vector space is given in the following result. (Mora, 2005): The space, or briefly , of all real measurable (Lebesgue) functions on satisfying endowed with the weak topology , is not metrizable.In this space , the probability functions set where , is localizable. For the Banach spaces we have the following result about the localization concept and the dimensionality (G.Mora & J.A. Mira, 2003). Theorem In a Banach space E , the unit ball is localizable if and only if E is finite-dimensional. Let E be a normed space and B its unit ball. Each curve  in B , has a density (minimal) given by where denotes and is the metric induced by the norm. Then, considering the set of all curves contained in B, denoted , we can define the degree of densificability of B as the number

28. Observe that for any normed space and, from the preceding theorem, in a Banach space, if and only if the space has finite dimension.Then if the space has infinite dimension it follows that . Now, a natural question is: does it exist a normed space for each value between 0 and 1? The answer is given by the following result (Mora and Mira, 2003). Theorem For any normed space, the degree of densificability of its unit ball can take exactly two values, either 0 or 1. Now, we obtain a well-known classical theorem as corollary the Riesz theorem on the dimension: Corollary Any normed space locally compact is finite dimensional.

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