--- title: "Feasible region" source: "https://systems-analysis.info/eng/Feasible_region" wiki: "systems-analysis.info/eng" article: "Feasible_region" language: "en" categories: - "Category:English" - "Category:Mathematical modeling" - "Category:Operations research" - "Category:Science" - "Category:Terminology" revision_id: 144 wiki_created_at: 2026-09-06T22:17:57Z wiki_modified_at: 2026-09-06T22:17:57Z downloaded_at: 2026-09-07T22:21:19Z --- # Feasible region **Feasible Region** (also **feasible set**) — in [operations research](https://systems-analysis.info/eng/Operations_research "Operations research"), [optimization](https://systems-analysis.info/eng/Optimization "Optimization"), and [mathematical modeling](https://systems-analysis.info/eng/Mathematical_modeling "Mathematical modeling"), is the set of all possible solutions (sets of variable values) that satisfy **all** the constraints of a problem. The feasible region constitutes the subspace within which the optimal solution is sought. Any solution outside this region is considered infeasible. ## Definition and Formulation The feasible region is formed by the intersection of the sets defined by each individual constraint of the problem. [Constraints](https://systems-analysis.info/eng/Constraints "Constraints") can be represented as: - **Inequalities:** These establish upper or lower bounds for the values of variables or their combinations (e.g., "consumption of resource A must not exceed 100 units," "the quantity of products manufactured must be at least 50 units"). - **Equalities:** These require a condition to be met exactly (e.g., "the total volume of shipments must be equal to 1000 tons," "the balance of incoming and outgoing flows is zero"). - **Variable sign conditions:** Often, variables are required to be non-negative, integer, or belong to a specific discrete set. A point (or a vector of variable values) belongs to the feasible region if and only if it simultaneously satisfies all these constraints. ## Geometric Interpretation The feasible region often has a clear geometric interpretation, especially in problems with a small number of variables: - **In two-dimensional space (2 variables):** Each linear inequality constraint defines a half-plane. The feasible region is the intersection of these half-planes, forming a convex polygon (which may be unbounded or empty). - **In three-dimensional space (3 variables):** Each linear inequality constraint defines a half-space. The feasible region is the intersection of these half-spaces, forming a convex polyhedron. - **In multi-dimensional space:** A feasible region defined by linear constraints is a convex polytope. In the case of non-linear constraints, the feasible region can have a more complex shape and may not be convex. ## Role in Optimization The feasible region plays a fundamental role in [optimization](https://systems-analysis.info/eng/Optimization "Optimization"): 1. **Defining the Search Space:** The optimal solution to the problem (if one exists) is **always** located within the feasible region or on its boundary. [Optimization](https://systems-analysis.info/eng/Optimization "Optimization") algorithms search for the extremum of the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") specifically within this region. 2. **Verifying the Existence of Solutions:** If the feasible region is an empty set (i.e., the [constraints](https://systems-analysis.info/eng/Constraints "Constraints") are contradictory), then the problem has no feasible solutions and, consequently, no optimal solution. 3. **Influence on the Optimal Solution:** The shape and size of the feasible region directly affect the possibility of reaching an extremum of the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") and the value of that extremum. ## Properties of the Feasible Region (in Linear Programming) In [linear programming](https://systems-analysis.info/eng/Linear_programming "Linear programming") (LP) problems, where all [constraints](https://systems-analysis.info/eng/Constraints "Constraints") and the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") are linear, the feasible region has important properties: - **Convexity:** If two points belong to the feasible region, the entire line segment connecting them also belongs to the region. This property ensures that the optimal solution (if it exists and is unique) will be located at one of the vertices of the feasible region's polytope. - **Closedness:** The feasible region includes its boundaries (due to non-strict inequalities ≤, ≥, and equalities). A feasible region can be: - **Bounded:** It has finite dimensions. - **Unbounded:** It extends infinitely in one or more directions. - **Empty:** It contains no points. ## External links - Feasible region — Wikipedia ## See also - [Operations research](https://systems-analysis.info/eng/Operations_research "Operations research") - [Optimization](https://systems-analysis.info/eng/Optimization "Optimization") - [Mathematical model](https://systems-analysis.info/eng/Mathematical_model "Mathematical model") - [Constraint](https://systems-analysis.info/eng/Constraints "Constraints") - [Optimal solution](https://systems-analysis.info/eng/Optimal_solution_(optimization) "Optimal solution (optimization)") - [Objective function](https://systems-analysis.info/eng/Objective_function "Objective function") - [Linear programming](https://systems-analysis.info/eng/Linear_programming "Linear programming") - [Nonlinear programming](https://systems-analysis.info/eng/Nonlinear_programming "Nonlinear programming") ## Literature - *Taha, Hamdy A.* Operations Research: An Introduction. — Pearson. (10th ed., 2017) - *Hillier, Frederick S.; Lieberman, Gerald J.* Introduction to Operations Research. — McGraw-Hill Education. (11th ed., 2021)