Lagrange multipliers and optimality
SIAM Review
Testing Unconstrained Optimization Software
ACM Transactions on Mathematical Software (TOMS)
A D. C. Optimization Algorithm for Solving the Trust-Region Subproblem
SIAM Journal on Optimization
On the Constant Positive Linear Dependence Condition and Its Application to SQP Methods
SIAM Journal on Optimization
Large-Scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients
Computational Optimization and Applications
A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization
SIAM Journal on Optimization
Mesh Adaptive Direct Search Algorithms for Constrained Optimization
SIAM Journal on Optimization
Augmented Lagrangian methods under the constant positive linear dependence constraint qualification
Mathematical Programming: Series A and B
Continuous optimization methods for structure alignments
Mathematical Programming: Series A and B
On Augmented Lagrangian Methods with General Lower-Level Constraints
SIAM Journal on Optimization
Low order-value approach for solving VaR-constrained optimization problems
Journal of Global Optimization
A New Sequential Optimality Condition for Constrained Optimization and Algorithmic Consequences
SIAM Journal on Optimization
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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 $$x \in I\!\!R^n$$ , it is natural to define F i (x) = (T(x, t i ) 驴 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 驴 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.