Minimizescalar

Minimizescalar refers to the problem of finding the minimum value of a scalar-valued function, or the point at which that minimum is attained. In mathematical optimization, a scalar-valued function f defined on a domain D, with outputs in the real numbers, is minimized by identifying a point x* in D such that f(x*) ≤ f(x) for all x in D. The minimum value is denoted as m = f(x*), and the set of points achieving it is called the set of minimizers, often written as argmin f.

Minimization problems can be univariate (one decision variable) or multivariate (several variables). For univariate problems, a stationary point occurs where the derivative f'(x) = 0, with the second-derivative test used to distinguish minima from maxima or inflection points. In multivariate problems, stationary points satisfy the gradient ∇f(x*) = 0, and the Hessian matrix informs the nature of the stationary point (positive definiteness implies a local minimum).

Two key concepts are global and local minima. If f is convex on D, any local minimum is also global. Strong convexity ensures a unique minimizer. When constraints are present, the problem becomes constrained minimization, solved by methods such as Lagrange multipliers, projection techniques, or interior-point methods.

In practice, minimization of a scalar objective is central to fields such as machine learning, economics, physics, and engineering. When multiple objectives exist, scalarization techniques convert them into a single scalar objective to enable standard minimization methods.