The external point penalty function method and the internal point penalty function method are both commonly used methods in optimization problems. The main difference between them lies in how to deal with constraints.
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External point penalty function method:
- Idea: Introduce constraints into the objective function, and penalize solutions that do not satisfy the constraints by introducing a penalty term into the objective function.
- Method: Convert the constraints into a penalty function, usually by introducing a penalty function, and adding the degree of constraint violation to the original objective function in the form of a penalty term, so that the original The constraints of the problem become part of the optimization problem.
- Features: Will convert solutions that violate constraints into feasible solutions, but may make the optimization problem more complex because additional penalty terms need to be introduced in the objective function .
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Interior point function method:
- Idea: By staying within the constraints during the iteration process, we can gradually approach the feasible solution.
- Method: By adjusting the parameters of the penalty function (usually multipliers or other methods) to ensure that the constraints remain satisfied during the iterations, and to be as close to feasible as possible during the iterations Move forward in the direction of solution.
- Features: The interior point penalty function method focuses more on keeping the constraints satisfied and searching for the optimal solution within the constraints, so it is usually not introduced directly in the objective function. The penalty term is used to approximate the feasible solution through algorithm iteration.
In general, the external point penalty function method focuses more on integrating constraints into the objective function and punishing solutions that do not satisfy the constraints, while the internal point penalty function method focuses more on searching for the optimal solution within the constraints and gradually approaching the feasible solution through iteration solution without directly modifying the objective function.
The origin of exterior points and interior points
The origin of this term comes mainly from the geometric understanding of a set of constraints (such as equality constraints or inequality constraints). Here we take two-dimensional space as an example to explain:
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External points: Refers to points that do not satisfy the constraints, that is, points that fall outside the set defined by the constraints. In two-dimensional space, if there is a constraint that is a closed area (such as a circle), the outer points are points that fall outside this area.
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Inner point: It refers to the point that satisfies the constraint condition, that is, the point that falls within the set defined by the constraint condition. In two-dimensional space, if there is a constraint that is a closed area (such as a circle), the interior point is the point that falls within this area.
In the penalty function method,External point penalty function method will penalize those points that fall outside the constraints by introducing a penalty term Pull the points back within the constraints. The interior point penalty function method focuses on staying within the constraints, gradually approaching the feasible solution through algorithm iteration, and searching for the optimal solution within the constraints. The solution is therefore called an interior point.