Optimin

Optimin is a term used in mathematics and computer science to denote the optimum minimum of an objective function in a minimization problem. It may refer to the minimum value itself, the set of points that achieve that value, or both. The concept applies to unconstrained problems as well as those with constraints.

Notation and existence: If x lies in a feasible set C and f: C -> R is the objective, the minimum value is f* = inf_{x in C} f(x). If a point x* in C attains this value, x* is a minimizer (often written as argmin f). When the minimum is attained, f* is the minimum; if not, f* is the infimum and there may be no optimin in C.

Existence and properties: A minimum exists, for example, when C is compact and f is continuous (Weierstrass). In convex problems, a convex f over a convex C guarantees that any local minimum is global; if f is strictly convex, the minimizer is unique.

Algorithms and methods: Locating the optimin uses optimization methods such as gradient descent, Newton's method, or interior-point techniques for convex problems. For global minima in nonconvex settings, approaches include branch-and-bound, simulated annealing, and evolutionary algorithms.

Examples: Minimize f(x) = (x-2)^2 over R. The optimin value is f* = 0, achieved at x* = 2. Another example: minimize f(x,y) = (x-3)^2 + (y+1)^2 subject to x+y=4; the optimin occurs at (x,y) = (3,1) with f* = 1.

See also: minimum, global minimum, local minimum, argmin, optimization, convex optimization. References: standard texts on optimization theory and numerical methods.