Iterative Method like Steepest Descent for Positivity Constrained Optimization -- Literature tips

Question

Consider the positivity constrained optimization problem $$ \min_{\vec{x}:\, \vec{x} \ge 0} f\left(\vec{x}\right). $$ The first order conditions are that for index $i$ either $x_i=0$ or $\nabla_x f\left(\vec{x}\right)_i = 0$. I am thinking about the iterative method that starts with some random, strictly positive, estimate $\vec{x}^{(0)}$, then iteratively takes a step in the direction $$-\nabla_x f\left(\vec{x}^{(k)}\right) \odot \vec{x}^{(k)},$$ with scaling chosen by line search.

This isn't strictly "Steepest Descent", but is very similar. Is this a known method, and what is it generally called? Can somebody provide a reference or name?

Answer 1

A key question is how you adapt when the sequence approaches the boundary. It's not exactly what you describe, but a similar approach to what you are doing is the use of a barrier function. A barrier function is basically a sufficiently smooth penalty function added to the original objective function. The ideal barrier function is zero or near zero away from the boundary but blows up rapidly as you get too close for comfort to the boundary.