Low Order-Value Optimization and applications |
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Authors: | R Andreani J M Martínez L Martínez F S Yano |
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Institution: | 1.Department of Applied Mathematics, IMECC-UNICAMP,State University of Campinas,Campinas,Brazil;2.Institute of Chemistry,State University of Campinas,Campinas,Brazil;3.Institute Pasteur,Paris,France;4.Itaú Bank,S?o Paulo,Brazil |
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Abstract: | Given r real functions F
1(x),...,F
r
(x) and an integer p between 1 and r, the Low Order-Value Optimization problem (LOVO) consists of minimizing the sum of the functions that take the p smaller values. If (y
1,...,y
r
) is a vector of data and T(x, t
i
) is the predicted value of the observation i with the parameters , it is natural to define F
i
(x) = (T(x, t
i
) − y
i
)2 (the quadratic error in observation i under the parameters x). When p = r this LOVO problem coincides with the classical nonlinear least-squares problem. However, the interesting situation is when
p is smaller than r. In that case, the solution of LOVO allows one to discard the influence of an estimated number of outliers. Thus, the LOVO
problem is an interesting tool for robust estimation of parameters of nonlinear models. When p ≪ r the LOVO problem may be used to find hidden structures in data sets. One of the most successful applications includes the
Protein Alignment problem. Fully documented algorithms for this application are available at www.ime.unicamp.br/~martinez/lovoalign.
In this paper optimality conditions are discussed, algorithms for solving the LOVO problem are introduced and convergence
theorems are proved. Finally, numerical experiments are presented. |
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Keywords: | Order-Value Optimization Algorithms Convergence Robust estimation of parameters Hidden patterns |
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