Optimointialue

Optimointialue, or the feasible region, is the subset of the decision variable space that satisfies all constraints of an optimization problem. In a typical formulation, decision variables x belong to a space such as R^n, there is an objective function f(x) to be minimized or maximized, and a set of constraints that x must satisfy (for example Ax ≤ b, Cx = d, gi(x) ≤ 0, hj(x) = 0). The optimointialue consists of all points x that meet these requirements.

It is important to distinguish the optimointialue from the domain of the objective function. The feasible region is determined by the constraints, while the domain of definition of f(x) refers to where the objective function is well-defined. In some problems, the feasible region is the intersection of the constraint set with a predefined domain D for the variables.

Properties of the optimointialue depend on the problem structure. If all constraints are linear, the optimointialue is a convex polyhedron. If the constraints are convex, the region is typically convex; closed and bounded constraints yield a compact optimointialue. The geometry of the optimointialue influences the choice of optimization algorithms and convergence behavior.

Examples include linear programming, where the optimointialue is a polyhedron defined by linear inequalities, and nonlinear programming, where the feasible region can have curved boundaries. In unconstrained optimization, the optimointialue is the entire decision space, usually R^n, unless otherwise restricted.