Optimisation is the hunt for a best feasible choice
Mathematical optimisation—also called mathematical programming—picks a best element from allowed alternatives by some criterion. Discrete problems seek integers, permutations, or graphs; continuous ones search real-valued sets, sometimes with constraints or many peaks. Every quantitative field meets the pattern.
In the general setup one maximises or minimises a real-valued function f over a set A of candidates. An optimal solution is a feasible point that makes f as large or as small as allowed. Maximisation becomes minimisation by flipping the sign of f, so texts often state only minimisation problems. Physics may call f an energy; machine learning tracks a cost whose lowest value marks preferred parameters. A is usually carved from Euclidean space by equalities and inequalities; its members are feasible or candidate solutions.
A local minimum beats every nearby feasible point inside some radius; a global minimum beats every feasible point anywhere. Without convexity, many local minima can exist. In a convex minimisation problem an interior local minimum is global, but nonconvex problems can hide several locals that are not global. Most commercial solvers for nonconvex models cannot certify the difference and may return a local answer as if it solved the original task. Global optimisation builds deterministic algorithms that guarantee the true optimum of a nonconvex problem in finite time.
Notation packs the ask: min of x² + 1 over the reals equals 1 at x = 0; max of 2x over the reals is unbounded. Arg-min asks for the argument that achieves the best value—for x² + 1 on (−∞, −1] the answer is x = −1 because x = 0 is infeasible. Names for f shift by culture: objective, loss, cost, utility, fitness, or energy. Solution methods have interested mathematicians for centuries because the same skeleton models so many real problems.
From operations research and economics to engineering and computer science, the craft is choosing the search space, writing constraints honestly, and knowing whether your algorithm promises a local dip or the true summit. Convexity is the quiet dividing line between “any local win is global” and “check again.”
Source: Mathematical optimization
