nonstrained

Nonstrained is an adjective meaning not constrained or free from external restrictions. In technical contexts it is typically treated as a variant spelling of unconstrained or non-constrained, though contemporary usage generally prefers unconstrained. In optimization, a nonstrained problem refers to one in which the decision variables may take any values in their ambient domain, with no explicit constraints on x.

Formally, an unconstrained optimization problem is to minimize or maximize an objective function f(x) over x in a space such as R^n, with no constraints on x. Solutions are usually sought where the gradient vanishes, i.e., ∇f(x*) = 0, with second-order conditions used to classify the point (minimum, maximum, or saddle). If f is convex, every stationary point is a global minimum; if f is concave, it is a global maximum. In non-convex settings, multiple local extrema and saddle points can occur, making global optimization more challenging.

Compared with constrained optimization, which restricts feasible x to satisfy one or more constraints (equalities or inequalities), unconstrained methods are typically simpler. Common approaches include gradient descent, Newton’s method, and other gradient-based algorithms. When constraints exist, methods such as Lagrange multipliers or Karush-Kuhn-Tucker conditions are employed.

Spelling and usage notes: while nonstrained appears in some texts, it is relatively uncommon. Most writers use unconstrained or non-constrained to describe problems without restrictions on the decision variables. The term may still be encountered in older literature or in informal contexts.

See also: unconstrained optimization, constrained optimization, gradient methods, Hessian.