Convex optimization
Convex optimization — “minimizing convex functions over convex sets” — is the exact / structured branch the synthesis said was missing: where the founding metaheuristics tackle arbitrary landscapes with no guarantees, convexity buys certainty. Source: Wikipedia.
The decisive property
“Every point that is a local minimum is also a global minimum.” This single fact dissolves the problem every metaheuristic is built to fight (local-optima traps): convex problems are “solvable with polynomial-time algorithms,” whereas “general optimization is NP-hard.” Solved by interior-point, subgradient/bundle, and Newton methods (often gradient-based).
Why it matters here — the cleanest statement of the trade
Convex vs. metaheuristic is the sharpest face of the no-free-lunch-theorem in this wiki: structure ⇄ generality. “Convex problems trade generality for certainty; metaheuristics accept uncertainty for generality.” If you can prove your problem convex, you don’t want a swarm — you want an interior-point solver with a global guarantee. Metaheuristics exist precisely for the non-convex, black-box remainder where that structure is absent.
Related
gradient-descent · stochastic-gradient-descent · no-free-lunch-theorem · metaheuristic-optimization · exploration-vs-exploitation