A downhill direction tells you which way to go — but how FAR? Step too little and you crawl; too far and you overshoot back uphill. Line search answers it: try a big step, and while it doesn’t reduce the value ENOUGH, halve it — backtracking until the decrease is genuine (the Armijo condition). It’s the trusty sidekick that makes gradient and Newton methods robust. Slide to backtrack the step down to a good one.
Given a descent direction d at point x, LINE SEARCH chooses the step length α. Exact minimization along the ray is costly, so inexact backtracking is used: start with α=1 and HALVE it until the Armijo (sufficient-decrease) condition holds, f(x+αd) ≤ f(x)+c₁α∇f·d — guaranteeing the step buys a decrease proportional to its size and slope (the Wolfe conditions add a curvature test). This makes [[the-gradient-descent|gradient descent]], Newton, and quasi-Newton (BFGS) globally convergent and removes the fragile fixed [[the-learning-rate|learning rate]]. A fail-loud self-check throws unless backtracking returns a step that satisfies Armijo and lowers f. ◆ real optimization, node-verified.
Backtracking guarantees SUFFICIENT decrease, not the true 1-D minimum; too-small c₁ or bad directions still crawl, and each trial costs a function evaluation. The Armijo condition and the halving rule are exact.