Low Order-Value Optimization and applications

Given r real functions F1(x),...,Fr(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 (y1,...,yr) is a vector of data and T(x, ti) is the predicted value of the observation i with the parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in I\!\!R^n$$\end{document} , it is natural to define Fi(x) =  (T(x, ti) − yi)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.