Suppose we have a $f(x)$ to be minimized (we only know that $f(x)$ is three-differentiable), and a feasible, convex set of $S$ such that all $x$ belong to $S$. Using line search method, how we can mathematically prove that a descending direction pointing from previous $x$ towards a new $x$ within the $S$ exists?And, how we can find the best $a\geq 0$ such that $x + ad$ belongs to $S$? I mean if we have a direction, how can we optimize the $d$ to have the least iterations?1.I'm neither a beginner nor an expert, somewhere in between.2.I need all the iterates to be feasible.Many thanks in advance.
Using Line Search Method for Constrained Optimization
constrained optimization;nonlinear programming