The existence theorem for the solution of a nonlinear least squares problem


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1 61 The existence theorem for the solution of a nonlinear least squares problem Dragan Jukić Abstract. In this paper we prove a theorem which gives necessary and sufficient conditions which guarantee the existence of the global minimum for a continuous real valued function bounded from below, which is defined on a noncompact set. The use of the theorem is illustrated by an example of the least squares problem. Key words: least squares, existence problem, exponential function Sažetak. Teorem o egzistenciji rješenja nelinearnog problema najmanjih kvadrata. U radu je naveden teorem koji daje nužan i dovoljan uvjet za egzistenciju globalnog minimuma neprekidne i odozdo omed ene realne funkcije definirane na skupu koji nije kompaktan. Korisnost teorema ilustrirana je na primjeru problema najmanjih kvadrata. Ključne riječi: problem najmanjih kvadrata, problem egzistencije, eksponencijalna funkcija 1. The least squares problem We are given a modelfunction t f(t; a), (1) and the data (p i, t i, f i ), i = 1,..., m, where a Λ R n is the vector of unknown parameters, t 1 < t 2 <... < t m are the abscissae and f 1,..., f m are the data s ordinates. The number p i > 0 is the weight of the ith datum. Usually we have The lecture presented at the Mathematical Colloquium in Osijek organized by Croatian Mathematical Society  Division Osijek, January 19, Faculty of Agriculture, Department of Mathematics, Trg Svetog Trojstva 3, HR Osijek, Croatia,
2 62 D. Jukić m n, i.e. usually the number of data is considerably bigger than the number of unknown parameters. In practice, the unknown parameter vector a Λ for the functionmodel (??) is usually determined either in the sense of ordinary least squares (c.f. [?], [?], [?], [?], [?], [?]), by finding a Λ such that S(a ) = inf a Λ S(a), S(a) = 1 2 p i [f i f(t i ; a)] 2 (2) (Figure??.a), or in the sense of total least squares (c.f. [?], [?], [?]) by finding (a, δ ) Λ R m such that F (a, δ ) = inf (a,δ) Λ R m F (a, δ), F (a, δ) = 1 2 (Figure??.b), where δ = (δ 1,..., δ m ) T R m. { p i [fi f(t i + δ i ; a)] 2 + δi 2 }, (3) y y (t i, f i ) (t i, f i ) r ε i i δ i a) t b) t Figure 1: Hence, in both cases one has an existence problem for a global minimum of a continuous function on some set. In the next section we prove a theorem giving necessary and sufficient conditions for the above problem to have a solution.
3 The existence theorem The existence theorem for the solution of the least squares problem Let Λ R n and let G : Λ R be a continuous function which is bounded from below, and denote G := inf λ Λ G(λ). One has the following problem: (P1) Does there exist a point λ Λ, such that G(λ ) = G? If the set Λ is compact, then, by continuity of G, the problem (P1) has a solution. Therefore, from now on we assume that the set Λ is not compact. This means that the set Λ is either unbounded or is not closed (or both). In the first case there exists a sequence (x k ) in Λ such that x k, and in the second case there exists a sequence (x k ) in Λ converging to a point x Cl Λ \ Λ. Denote by N(Λ) the set of all sequences (x k ) in Λ, such that x k or x k x Cl Λ \ Λ. Following Demidenko [?] (see Remark??), define the number G E := inf{lim inf G(x k ) : (x k ) N(Λ)} (4) which we call the infimum of the function G at the end. Since G G(λ) for all λ Λ, from the definition (??) one can easily prove the following inequality: Proposition 1. G G E. Example 1. Let G(λ) = λ 2. a) If Λ = R, then G = 0, G E =, b) If Λ = [0, ), then G = 0, G E =, c) If Λ = (0, ), then G = 0, G E = 0, d) If Λ = (, 1] (1, ), then G = 1, G E = 1. Note that the problem (P1) has no solution only in the case c). Proposition 2. If G < G E, then the problem (P1) has a solution. Proof. Let (λ k ) be a sequence in Λ such that G = lim G(λ k). By our k assumption G < G E, and therefore neither the sequence (λ k ), nor any of its subsequence, belong to the set N(Λ). This means that the sequence λ k does not tend to, and hence the sequence (λ k ) has a bounded subsequence. By the BolzanoWeierstrass theorem, the sequence (λ k ) has a convergent subsequence (λ ki ). Let λ ki λ. Then λ Λ, since otherwise, the sequence (λ ki ) would belong to the set N(Λ). By the continuity of the function G we obtain G = lim G(λ k i ) = G( lim λ ki ) = G(λ ). i i Theorem 1. The problem (P1) has a solution if and only if there exists a point λ Λ such that G(λ ) G E.
4 64 D. Jukić Proof. a) Suppose the problem (P1) has a solution. Then there exists a point λ Λ such that G(λ ) = G. By Proposition?? we obtain G(λ ) G E. b) Suppose the problem (P1) has no solution. Then G(λ) > G for all λ Λ. Furthermore, by Proposition??, G = G E. Hence, if the problem (P1) has no solution, then G(λ) > G E for all λ Λ. Remark 1. In [?], the infimum of the function G at the end, is defined as the number G E := inf{lim inf G(x k ) : (x k ) N (Λ)}, where N (Λ) is the set of all sequences (x k ) in Λ, such that either x k or x k x Cl Λ \ Int Λ. Since Cl Λ \ Λ Cl Λ \ Int Λ, one has G E G E. Furthermore, it is easy to show that Theorem?? holds also if G E is replaced by G E. Example 2. We illustrate the application of Theorem?? on the following simple nonlinear ordinary least squares problem. Let the data (p i, t i, f i ), i = 1,..., m, m 3, be such that t 1 < t 2 <... < t m, and f i > 0, i = 1,..., m. For the modelfunction we take the exponential function f(t; λ) = e λt, λ R. We consider the existence problem for the global minimum of the functional S : R R given by S(λ) = p i [f i e λti ] 2. Let I = {1,..., m}, I 0 := {i : t i = 0}. Note that the set I 0 is either empty or contains only one point. Let us show that in the set R there exists a point of the global minimum for the functional S. a) If some of the numbers t 1,..., t m are strictly positive and some are strictly negative, then obviously S E =. By Theorem?? there exists a point of the global minimum for the functional S on R. b) If 0 t 1, then S E = i I\I 0 p i f 2 i + i I 0 p i (f i 1) 2. Let (see Figure??.a) λ ln f i = min = ln f k. i I\I 0 t i t k Geometrically, (see Figure??.a), λ is the slope of the line through the origin and the point (t k, ln f k ), and none of the other points (t i, ln f i ), i I \ I 0, lies below this line. Since f k = e λ t k and f i e λ t k, i k, we have S(λ ) = 1 2 p i [f i e λ t i ] 2 = 1 2 p i [f i e λ t i ] i I\I 0 i k i k p i fi p i (f i 1) 2 < S E. i I 0 By Theorem?? in this case there aslo exists a point of the global minimum for the functional S on R.
5 The existence theorem 65 ln f i ln f i (t i, ln f i ) (t 1, ln f 1 ) 0 a) (t m, ln f m ) (t k, ln f k ) t (t k, ln f k ) b) (t i, ln f i ) 0 t Figure 2: c) If t m 0, then S E = 1 2 Figure??.b) i I\I 0 p i fi i I 0 p i (f i 1) 2. λ ln f i = max = ln f k. i I\I 0 t i t k Let (see Proceeding similarly as in b), one can show that there exists a point λ R such that S(λ ) < S E. Therefore, our least square problem always has a solution. References [1] A. Björck, Least Squares Methods, in Handbook of Numerical Analysis, Vol. I, P. G. Ciarlet and J. L. Lions, eds., Elsevier Science Publishers B.V., North  Holland, Amsterdam, 1990, [2] P. T. Boggs, R. H Byrd, R. B. Schnabel, A stable and efficient algorithm for nonlinear orthogonal distance regression, SIAM J. Sci. Statist. Comput. 8(1987), [3] E. Z. Demidenko, Optimizacia i regresija, Nauka, Moskva, [4] P. E. Gill, W. Murray, M. H. Wright, Practical Optimization, Academic Press, London, [5] D. Jukić, R. Scitovski, The existence of optimal parameters of the generalized logistic function, Appl. Math. Comput., to appear.
6 66 D. Jukić [6] H. Schwetlick, V. Tiller, Numerical methods for estimating parameters in nonlinear models with errors in the variables, Technometrics 27(1985), [7] R. Scitovski, A special nonlinear least squares problem, J. Comput. Appl. Math. 53(1994), [8] R. Scitovski, D. Jukić, A method for solving the parameter identification problem for ordinary differential equations of the second order, Appl. Math. Comput. 74(1996), [9] R. Scitovski, D. Jukić, Total least squares problem for exponential function, Inverse Problems, to appear.
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