Letter to the Editor—A Note on Some Classical Methods in Constrained Optimization and Positively Bounded Jacobians

Using a variation of the classical Implicit Function Theorem and the Chain Rule, Cottle in “Nonlinear Programs with Positively Bounded Jacobians,” SIAM 14 , 147–158 (1966) considers the problem \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$(I)\quad \min z^TW(z),\quad \mbox{subject to}\ W(z) \geq 0,\ z\geq 0,$$\end{document} where W is a continuously differentiable vector function defined on N -space, and proves that if W has a positively bounded Jacobian, then ( I ) has a solution z̄ satisfying z̄ T W ( z̄ ) = 0, when the problem is nondegenerate. The authors here point out by geometric interpretations and examples that: (1) there exist nondegenerate problems having optimal solutions that have no equivalent positively bounded representations, (2) degeneracy may be a common occurrence for problems of this form, and (3) the class of problems of the above form has yet to be studied for the general case where the infimum is finite, not necessarily assumed, and not necessarily zero.