Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices Kristóf Ábele-Nagy Eötvös Loránd University (ELTE); Corvinus University of Budapest (BCE) Sándor Bozóki Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI); Corvinus University of Budapest (BCE) 15 December, 2010 Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 1/39
Outline Pairwise comparison matrix Incomplete pairwise comparison matrix Eigenvalue optimization Cyclic coordinates Newton’s method in one variable Newton’s method in higher dimensions Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 2/39
Given n objects with weights w 1 , w 2 , w 3 , . . . , w n . The pairwise comparison matrix is defined as follows: w 1 w 1 w 1 1 . . . w 2 w 3 w n w 2 w 2 w 2 1 . . . w 1 w 3 w n w 3 w 3 w 3 1 . . . , w 1 w 2 w n . . . . ... . . . . . . . . w n w n w n . . . 1 w 1 w 2 w 3 where w ij > 0 , 1 w ij = , w ji w ij = w ik w kj . for any i, j, k indices. Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 3/39
In real decision situations, weights are unknown, but pairwise comparisons can be made: 1 a 12 a 13 . . . a 1 n a 21 1 a 23 . . . a 2 n a 31 a 32 1 . . . a 3 n A = , . . . . ... . . . . . . . . a n 1 a n 2 a n 3 . . . 1 where a ij > 0 , 1 a ij = . a ji for i, j = 1 , . . . , n. The aim is to determine the weight vector w = ( w 1 , w 2 , . . . , w n ) ∈ R n + . Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 4/39
In the Eigenvector Method ( EM ) the approximation w EM of w is defined by Aw EM = λ max w EM , where λ max denotes the maximal eigenvalue, also known as Perron eigenvalue, of A and w EM denotes the the right-hand side eigenvector of A corresponding to λ max . By Perron’s theorem, w EM is positive and unique up to a scalar multiplication. The most often used normalization is n w EM � = 1 . i i =1 Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 5/39
λmax − n Saaty defined the inconsistency ratio as CR = n − 1 , RI n where λ max is the Perron eigenvalue of the complete pairwise comparison matrix given by the decision maker, and RI n is defined as λ max − n n − 1 , where λ max is an average value of the Perron eigenvalues of randomly generated n × n pairwise comparison matrices. It is well known that λ max ≥ n and equals to n if and only if the matrix is consistent, i.e., the transitivity property holds. It follows from the definition that CR is a positive linear transformation of λ max . According to Saaty, larger value of CR indicates higher level of inconsistency and the 10%-rule ( CR ≤ 0 . 10) separates acceptable matrices from unacceptable ones. Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 6/39
Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements) 1 a 12 − . . . a 1 n 1 /a 12 1 a 23 . . . − − 1 /a 23 1 . . . a 3 n A = . . . . . ... . . . . . . . . 1 /a 1 n − 1 /a 3 n . . . 1 Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 7/39
Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements) 1 a 12 x 1 . . . a 1 n 1 /a 12 1 a 23 . . . − 1 /x 1 1 /a 23 1 . . . a 3 n A = . . . . . ... . . . . . . . . 1 /a 1 n − 1 /a 3 n . . . 1 Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 8/39
Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements) 1 a 12 x 1 . . . a 1 n 1 /a 12 1 a 23 . . . x d 1 /x 1 1 /a 23 1 . . . a 3 n A = , . . . . ... . . . . . . . . 1 /a 1 n 1 /x d 1 /a 3 n . . . 1 Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 9/39
Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements) 1 a 12 x 1 . . . a 1 n 1 /a 12 1 a 23 . . . x d 1 /x 1 1 /a 23 1 . . . a 3 n A = , . . . . ... . . . . . . . . 1 /a 1 n 1 /x d 1 /a 3 n . . . 1 where x 1 , x 2 , . . . , x d ∈ R + . Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 10/39
Based on the idea above, Shiraishi, Obata and Daigo considered the eigenvalue optimization problems as follows. In case of one missing element, denoted by x , the λ max ( A ( x )) to be minimized: min x> 0 λ max ( A ( x )) . In case of more than one missing elements, arranged in vector x , the aim is to solve min x > 0 λ max ( A ( x )) . Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 11/39
Graph representation of a pairwise comparison matrix Given A incomplete pairwise comparison matrix of size n × n. Graph G = ( V, E ) is defined as follows: V = { 1 , 2 , . . . , n } E = { e ( i, j ) | a ij (and a ji ) are given and i � = j } Special case: all the comparisons are given, the corresponding graph is K n . Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 12/39
Graph representation of a pairwise comparison matrix Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 13/39
Theorem (B., Fülöp, Rónyai, 2010): The optimal solution of the eigenvalue minimization problem min x > 0 λ max ( A ( x )) . is unique if and only if the graph G corresponding to the incomplete pairwise comparison matrix is connected . If graph G corresponding to the incomplete pairwise comparison matrix is connected, then by using the exponential parametrization x 1 = e y 1 , x 2 = e y 2 , . . . x d = e y d , the eigenvalue minimization problem is transformed into a strictly convex optimization problem. Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 14/39
Example 1 2 x Q = . 1 / 2 1 4 1 /x 1 / 4 1 λ max ( Q ( x )) and, by using the exponential scaling x = e t , λ max ( Q ( e t )) are plotted. Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 15/39
Algorithms for solving the eigenvalue minimization problem min x > 0 λ max ( A ( x )) . cyclic coordinates with Matlab’s function fminbnd cyclic coordinates univariate Newton’s method multivariate Newton’s method Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 16/39
Method of cyclic coordinates 1 5 3 7 6 6 1 / 3 1 / 4 1 / 5 1 x 1 5 x 2 3 x 3 1 / 7 1 / 3 1 /x 1 1 x 4 3 x 5 6 x 6 1 / 7 1 / 5 1 /x 4 1 x 7 1 / 4 x 8 1 / 8 M ( x ) = 1 / 6 1 /x 2 1 / 3 1 /x 7 1 x 9 1 / 5 x 10 1 / 6 1 / 3 1 /x 5 4 1 /x 9 1 x 11 1 / 6 3 1 /x 3 1 / 6 1 /x 8 5 1 /x 11 1 x 12 4 7 1 /x 6 8 1 /x 10 6 1 /x 12 1 Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 17/39
Method of cyclic coordinates Let x ( k ) denote the value of x i in the k -th step of the i iteration, which has d (in the example, d = 12 ) substeps for each k . For k = 0 : Let the initial points be equal to 1 for every variable: x (0) := 1 ( i = 1 , 2 , . . . , d ) . i i − x k − 1 while i =1 , 2 ,...,d � x k max � > T i x ( k ) := i x i λ max ( M ( x ( k ) 1 , . . . , x ( k ) i − 1 , x i , x ( k − 1) , . . . , x ( k − 1) arg min )) , i = i +1 d 1 , 2 , . . . , d next k end while Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 18/39
Method of cyclic coordinates x i λ max ( M ( x ( k ) 1 , . . . , x ( k ) i − 1 , x i , x ( k − 1) , . . . , x ( k − 1) Focus on min )) i +1 d Matlab’s function fminbnd solves is directly and fast. Univariate Newton’s method can be also applied. Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 19/39
min x> 0 λ max ( A ( x )) Let x = e t and L ( t ) = λ max ( e t ) . ∂λ max ( x ) t n +1 = t n − L ′ ( t n ) ∂x L ′′ ( t n ) = t n − . ∂ 2 λ max ( x ) · e t n + ∂λ max ( x ) ( ∂x ) 2 ∂x and ∂ 2 λ max ( x ) By Harker, formal derivatives ∂λ max ( x ) are ∂x ( ∂x ) 2 known. Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 20/39
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