Optimal solution (optimization)
An optimal solution — in operations research, Optimization, and Decision Theory — is a feasible solution (i.e., one that satisfies all the problem's 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 problems.
Essence and Characteristics
An optimal solution has two key characteristics:
- Feasibility: It must satisfy all constraints imposed on the variables of the model. In other words, an optimal solution always belongs to the Feasible Region.
- 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.
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. An optimal solution is a point (or points) within this region where the 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 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 is parallel to one of the active 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 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 is chosen incorrectly, or important 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
See also
- Operations Research
- Optimization
- Mathematical Model
- Objective Function
- Constraints
- Feasible Region
- Criterion
- Decision Theory
- Multi-objective Optimization
- 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)