A Basis for Cones

Why do the human cones have the spectral sensitivities they do? We hypothesize that they may have evolved to their present form because their sensitivities are optimal in terms of their ability to recover the spectrum of incident light. As evidence in favor of this hypothesis, we compare the accuracy with which the incoming spectrum can be approximated by a three-dimensional linear model based on the cone responses and compare this to the optimal approximations defined by models based on principal components analysis, independent component analysis, non-negative matrix factorization and non-negative independent component analysis. We introduce a new method of reconstructing spectra from the cone responses and show that the cones are almost as good as these optimal methods in estimating the spectrum

Two new weak constraint qualifications and applications

We present two new constraint qualifications (CQs) that are weaker than the recently introduced relaxed constant positive linear dependence (RCPLD) CQ. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new CQ, which we call the constant rank of the subspace component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, such as local stability and the validity of an error bound. We also introduce an even weaker CQ, called the constant positive generator (CPG), which can replace RCPLD in the analysis of the global convergence of algorithms. We close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: sequential quadratic programming, augmented Lagrangians, interior point algorithms, and inexact restoration.Facultad de Ciencias ExactasDepartamento de Matemátic

TWO NEW WEAK CONSTRAINT QUALIFICATIONS AND APPLICATIONS

We present two new constraint qualifications (CQs) that are weaker than the recently introduced relaxed constant positive linear dependence (RCPLD) CQ. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new CQ, which we call the constant rank of the subspace component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, such as local stability and the validity of an error bound. We also introduce an even weaker CQ, called the constant positive generator (CPG), which can replace RCPLD in the analysis of the global convergence of algorithms. We close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: sequential quadratic programming, augmented Lagrangians, interior point algorithms, and inexact restoration.RONEX-Optimization (PRONEX-CNPq/FAPERJ) [E-26/171.510/2006-APQ1]Fapesp [2006/53768-0, 2009/09414-7, 2010/19720-5]CNPq [300900/2009-0, 303030/2007-0, 305740/2010-5, 474138/2008-9