--- title: "Optimal solution (optimization)" source: "https://systems-analysis.info/eng/Optimal_solution_(optimization)" wiki: "systems-analysis.info/eng" article: "Optimal_solution_(optimization)" language: "en" categories: - "Category:English" - "Category:Modeling" - "Category:Operations research" - "Category:Science" - "Category:Terminology" revision_id: 288 wiki_created_at: 2026-09-06T22:20:05Z wiki_modified_at: 2026-09-06T22:20:05Z downloaded_at: 2026-09-07T22:22:28Z --- # Optimal solution (optimization) An **optimal solution** — in operations research, [Optimization](https://systems-analysis.info/eng/Optimization "Optimization"), and Decision Theory — is a feasible solution (i.e., one that satisfies all the problem's [constraints](https://systems-analysis.info/eng/Constraints "Constraints")) that provides an extreme value (maximum or minimum, depending on the problem formulation) for the objective function. Finding an optimal solution is the primary goal of most [Optimization](https://systems-analysis.info/eng/Optimization "Optimization") problems. ## Essence and Characteristics An optimal solution has two key characteristics: 1. **Feasibility:** It must satisfy all [constraints](https://systems-analysis.info/eng/Constraints "Constraints") imposed on the variables of the model. In other words, an optimal solution always belongs to the Feasible Region. 2. **Optimality with respect to the objective function:** Among all feasible solutions, it provides the best (maximum or minimum) value for the objective function, which formalizes the [optimality criterion](https://systems-analysis.info/eng/Criterion "Criterion"). Not every feasible solution is optimal, but every optimal solution must be feasible. ## Relationship with the Feasible Region The Feasible Region represents the set of all alternatives (sets of variable values) that satisfy the problem's [constraints](https://systems-analysis.info/eng/Constraints "Constraints"). An optimal solution is a point (or points) within this region where the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") reaches its extremum. If the feasible region is empty, the problem has no feasible solutions and, consequently, no optimal solutions. ## Role of the Objective Function and Constraints - [Constraints](https://systems-analysis.info/eng/Constraints "Constraints") define the set of **possible** solutions (the Feasible Region). - The Objective Function determines which of these possible solutions is the **best** (optimal). Without an objective function, it is impossible to determine which feasible solution is optimal. Without constraints, a problem may be trivial or may not have a finite optimal solution (for example, maximizing a linear function without constraints). ## Uniqueness of the Optimal Solution An optimal solution is **not always unique**. In some problems (for example, in Linear Programming, if the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") is parallel to one of the active [constraints](https://systems-analysis.info/eng/Constraints "Constraints")), there can be an infinite number of optimal solutions that share the same objective function value. However, the value of the objective function at the optimum point(s) is always unique (if an optimum exists). ## Methods for Finding Solutions Various mathematical methods are used in operations research to find optimal solutions, depending on the type of model: - Simplex method (for Linear Programming) - Gradient Descent methods and other numerical methods (for Nonlinear Programming) - [Branch and bound](https://systems-analysis.info/eng/Branch_and_bound "Branch and bound") method, cutting-plane methods (for Integer Programming) - Dynamic Programming methods ## Dependence on the Model It is important to understand that a solution is optimal **only within the framework of the adopted mathematical model**. If the model inadequately represents the real situation (e.g., the [objective function](https://systems-analysis.info/eng/Objective_function "Objective function") is chosen incorrectly, or important [constraints](https://systems-analysis.info/eng/Constraints "Constraints") or dependencies are not considered), then the formally derived optimal solution may prove to be ineffective or even erroneous in practice. ## Optimality in Multi-Criteria Problems In problems with multiple objective functions (Multi-objective Optimization), the concept of a single optimal solution is often replaced by the concept of **Pareto optimality**. A Pareto-optimal solution is a feasible solution where it is impossible to improve the value of one objective function without worsening the value of at least one other. ## External links - Optimization problem — 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") - [Objective Function](https://systems-analysis.info/eng/Objective_function "Objective function") - [Constraints](https://systems-analysis.info/eng/Constraints "Constraints") - [Feasible Region](https://systems-analysis.info/eng/Feasible_region "Feasible region") - [Criterion](https://systems-analysis.info/eng/Criterion "Criterion") - [Decision Theory](https://systems-analysis.info/eng/Decision_theory "Decision theory") - [Multi-objective Optimization](https://systems-analysis.info/eng/Multi-objective_optimization "Multi-objective optimization") - [Optimization criterion](https://systems-analysis.info/eng/Optimization_criterion "Optimization criterion") ## 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)