The simplest search: always step toward higher ground; stop when every neighbour is lower. It’s fast and needs no gradient — but it’s GREEDY, so it climbs the nearest hill and calls it a day, blind to the taller peak next door. It’s the cautionary tale of local optima that annealing and restarts exist to escape. Slide the starting point and watch which peak it gets stuck on.
Hill climbing is greedy local search: from the current point, move to a neighbour that improves the objective; halt at a LOCAL optimum where none does. It is memory-light and gradient-free, but it cannot escape a local optimum, a plateau, or a ridge — where it started determines where it stops. Remedies include random restarts, stochastic hill climbing, and [[the-simulated-annealing|simulated annealing]]. A fail-loud self-check throws unless two different starts on a multi-peak landscape reach different local optima. ◆ real local search, node-verified.
Deterministic steepest-ascent shown; the ‘stuck at a local optimum, start-dependent’ behaviour is exact and is the whole point — it motivates the exploration strategies of the other spheres here.