Some landscapes are a single bowl; others are a mountain range full of false valleys. A CONVEX function is the bowl: draw a line between any two points on it and the line never dips below the curve. The payoff is enormous — a convex problem has NO local traps, so any downhill method finds the ONE true global minimum. Convex-or-not is the great divide between ‘solvable’ and ‘hard.’ Slide from a convex bowl to a bumpy landscape.
A function is CONVEX if the line segment between any two points on its graph lies on or above the graph: f(λa+(1−λ)b) ≤ λf(a)+(1−λ)f(b) (Jensen’s inequality; equivalently f″≥0). The decisive consequence: a convex function has NO local minima other than the global one, so any descent method ([[the-gradient-descent|gradient descent]], Newton) provably finds the GLOBAL optimum — and duality gives certificates of optimality. Convexity is the line between tractable and intractable optimization; much of ML is the art of posing a convex surrogate. A fail-loud self-check throws unless x² passes the chord test and a non-convex function fails it. ◆ real optimization, node-verified.
‘Convex ⇒ local is global’ is exact — but real problems (deep nets) are usually NON-convex, navigated by heuristics with no global guarantee; convexity is often an approximation or a hope. The Jensen chord test is exact.