Fejes, Pál

Hunya, Péter

Varga, Károly

English

Hunya Péter

Varga Károly

Fejes Pál

University of Szeged

Some remarks on the paper of K. Varga and P. Fejes B y P . H U N Y A In practical applications of the iterative unconstrained optimization methods (for example in the gradient method discussed in [1]) a difficult problem is to find the initial estimation of the solution. Generally m points are randomly chosen from the region containing the optimum place and that which represents the mini-mum (maximum) value of function is considered as the initial estimation. Follow-ing this strategy the sequence generated by the iterative process converges with a probability generally less then 1. By the modified method proposed in [1] one step of the iteration is performed for all the m points before the selection and this may improve the probability of the convergency. However the verification of this property is heuristic, it is based on a number of experimental calculations with various type of functions. The upper and lower limits of the convergency are also given in [1]. In this paper a generalization of the modified method is described and the probability of the convergency is discussed in detail. (The problem of minimiza-tion is exemined, since all considerations are analogous in the case of maximi-zation.) Let us consider the continuous function of real values F(x) defined on the complete metric space S. Let T be a subset of S with nonzero measure and let us suppose that F takes its minimum on T. Let y=Mx be a mapping of T into T with the property F(Mx)^F{x). (1) There exists obviously a 7 , conv?i0 subset of T for the elements x of which the sequence x = M ° x , M1x, M2x, ...,M"x, ... is convergent and lim Mnx = x, (2) n~* CO where F(x) = m i n F ( x ) ; that is, the iteration process generated by M converges to the solution of the optimiza-tion problem. To simlify the considerations we suppose the uniqueness of x, how-ever, this fact is not essential in the following proofs. Suppose further that e(Tcom)^0 (3) 62 P. Hunya (g(t) means the measure of the set ¡ c 5 ) and e ( T a l j ^ o (4) where Tmin is a subset of Tcom defined by Tmin = (x\ F(x) < inf F(y)). (5) y U J c o n v Now let us consider the following procedure: a) let the points xt, x2, ..., xm be chosen from Tindependently, with homogene-ous distribution on T; b) then form the sequence Mnx1,Mnx2,...,Mnxm for n ^ O and c) let a point x"—M"xk* be chosen for which F(Mnxk,,) = min F(M"Xi). We shall prove that by the previous conditions for n — °° the probability P= = P(x" £ r c o n v ) converges to the limit P while P„SPSP (6) where P„ is a monotonic increasing sequence i. e. Pn^Pn + l (7) and lim Pn = P. (7a) /J— oo In other words: by increasing the number of the iterative steps before selection the probability of the convergency approximates its upper limit with an arbitrary degree of accuracy. (The dependence on m is not considered in this paper.) For the proof we define the following events: 1) An denotes the event that among the points M"x1, ...,M"xm there exists element of Tconv A n = { 3 i ( M » X i i T ^ ) } . (n = 0 , 1 , 2 , . . . ) (8) As a consequence of the definition of 7"conv we have A„=At(=A), (9) since it is obvious that MkxeTconv*>Mk+1x£Tcony for k=0, 1,2, . . . . 2) B„ denotes that the selected point x" is element of Tconv , that is, the iterational process converges: B n = {-x"€ Tconv}- (10) 3) C„ denotes the event that x" is element of T"min C„ = {x"€r m i n } . (11) Some remarks on the paper of K. Varga and P. Fejes 63 ; 4) And finally Dn denotes the event that at least one of the points M"x1, M"x2, .... ..., Mnxm belongs to r m i n Dn = {3i(MnXi£Tm-m)}. (12) From the definitions 1)—4) immediately follows that for all «. DnoCn=>Bn=>AnoA, (13) consequently, the probabilities of A„, Bn, C„, D„, A satisfy the relations P{Dn)=P(Cn)^P(Bn)^P(A„)=P(A). (14) For ii—O (14) contains as a special case one of the results of [1] for the values p0 = P(C0)= i-(\-Qminr F=P(A0)= I - ( l - e c o n v y . Using the notations P,= P(C„)= P(D„) and P=P(A)=P(An) (14) proves (6) also. As a consequence of the condition (1) and the definition 4) we have Dn > A H ! . (15) and this implies the inequality (7). . Let us consider now the sequence JD0, D1} ...,D„, ... . We shall prove that 2 A- - A. (16) ¡=i From the continuity of the function F(x) follows that for the elements x of r c o n v we can find a natural number n(x), such that n>n(x) implies M"x£Tmin Let Tl denote the subset of Tcony for the elements of which n(x)=i ( /=0 , 1,2, ...) and define the event A' as follows A> = { 3 j ( x j e r } . . By the definition above A{ => D{. On the other hand if there exists Xj (l^j^m) such that M'xj£ Tmin is true, then XjdTi also holds; that is, Z>; implies A'. Thus we have A' = Di. it is obvious that A = ^ A l so we get (16). 1 = 0 Because of (15) and (16) one of the basic limit theorems of the probability theory ([2], §2.2) can be applied to the sequence D0, Dlt ..., Dn, ... , therefore Y\mP{D„) = P(A) n~* CO ' . so using the notations introduced previously we get (7a) lim P„ = P. Несколько замечаний к работе К. Варга и Р. Фейеш В итерационных методах безусловной оптимизации при случайно, выбранном началь-ном значении, получается последовательность сходящаяся к решению задачи, только с веро-ятностью Р (обычно меньшей единицы). Вероятность сходимости может увеличиться, если из т случайно выбранных точек, считаем начальной ту точку, в которой достигается мини-64 P. Hunya: Some remarks on the paper of K. Varga and P. Fejes мум (максимум). Дальнейшее улучшение получается при выпольнении п шагов итерационной процедуры перед выбором исходной точки. Доказывается, что при довольно общих условиях, нижний предел вероятности Р явля-ется монотонной функцией от л, и с ростом п, Р сходится к своему верхнему пределу, обеспечи-вая этим увеличение вероятности конвергенции. LABORATORY OF CYBERNETICS JÓZSEF ATTILA UNIVERSITY SZEGED, HUNGARY References [1] VARGA, К., P. FEJES, Combined application of drawing and steepest descent in generating initial estimates for subsequent optimization, Acta Cybernet., v. 2, 1973, pp. 53—59. [2] PRÉKOPA, A., Valószínűségelmélet, Budapest, 1962. (Received June 28, 1972) 

Some remarks on the paper of K. Varga and P. Fejes

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Some remarks on the paper of K. Varga and P. Fejes

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