Summary: J Glob Optim (2009) 43:122
Low Order-Value Optimization and applications
R. Andreani · J. M. Martínez · L. Martínez · F. S. Yano
Received: 10 December 2007 / Accepted: 28 January 2008 / Published online: 12 February 2008
© Springer Science+Business Media, LLC. 2008
Abstract 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 x IRn, 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