# Feasible direction

Given a feasible point $LaTeX: x$, a direction vector, say $LaTeX: d$, is feasible if there exists $LaTeX: s > 0$ such that $LaTeX: x + sd$ is feasible. The method of feasible directions is designed to choose a feasible direction at each iteration, along which (as $LaTeX: s$ becomes positive) the objective value improves. Such directions exist for continuous mathematical programs (where the line segment $LaTeX: [x, x + sd]$ is feasible for some $LaTeX: s > 0$) unless $LaTeX: x$ is a local optimum. (Note: with nonlinear constraints, there is typically no feasible direction according to this (classical) definition. See the generalized reduced gradient method.)