Feasible region

In mathematical optimization and computer science, a feasible region, feasible set, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potentially including inequalities, equalities, and integer constraints.

The feasible set of the problem is separate from the objective function, which states the criterion to be optimized and which in the above example is

In many problems, the feasible set reflects a constraint that one or more variables must be non-negative.

In linear programming problems, the feasible set is a convex polytope: a region in multidimensional space whose boundaries are formed by hyperplanes and whose corners are vertices.

Constraint satisfaction is the process of finding a point in the feasible region.

Convex feasible sets arise in many types of problems, including linear programming problems, and they are of particular interest because, if the problem has a convex objective function that is to be minimized, it will generally be easier to solve in the presence of a convex feasible set and any local optimum will also be a global optimum.

In linear programming problems with n variables, a necessary but insufficient condition for the feasible set to be bounded is that the number of constraints be at least n + 1 (as illustrated by the above example).

If the feasible set is unbounded, there may or may not be an optimum, depending on the specifics of the objective function.

For example, if the feasible region is defined by the constraint set {x ≥ 0, y ≥ 0}, then the problem of maximizing x + y has no optimum since any candidate solution can be improved upon by increasing x or y; yet if the problem is to minimize x + y, then there is an optimum (specifically at (x, y) = (0, 0)).

In optimization and other branches of mathematics, and in search algorithms (a topic in computer science), a candidate solution is a member of the set of possible solutions in the feasible region of a given problem.

Constraint satisfaction is the process of finding a point in the feasible set.

[3] In calculus, an optimal solution is sought using the first derivative test: the first derivative of the function being optimized is equated to zero, and any values of the choice variable(s) that satisfy this equation are viewed as candidate solutions (while those that do not are ruled out as candidates).

First, it might give a minimum when a maximum is being sought (or vice versa), and second, it might give neither a minimum nor a maximum but rather a saddle point or an inflection point, at which a temporary pause in the local rise or fall of the function occurs.

In the simplex method for solving linear programming problems, a vertex of the feasible polytope is selected as the initial candidate solution and is tested for optimality; if it is rejected as the optimum, an adjacent vertex is considered as the next candidate solution.

A problem with five linear constraints (in blue, including the non-negativity constraints). In the absence of integer constraints the feasible set is the entire region bounded by blue, but with integer constraints it is the set of red dots.
A closed feasible region of a linear programming problem with three variables is a convex polyhedron .
A bounded feasible set (top) and an unbounded feasible set (bottom). The set at the bottom continues forever towards the right.
A series of linear programming constraints on two variables produce a region of possible values for those variables. Solvable two-variable problems will have a feasible region in the shape of a convex simple polygon if it is bounded. In an algorithm that tests feasible points sequentially, each tested point is in turn a candidate solution.