A penalty function is a mathematical function that is used to impose a penalty or cost when a certain condition is violated. In optimization problems, penalty functions are often used to add constraints to the objective function. For example, if the objective function needs to be minimized subject to the constraint that a certain parameter must be greater than a specific value, a penalty function can be used to penalize the objective function when the parameter falls below that value. This encourages the optimization algorithm to find a solution that satisfies the constraint while minimizing the objective function. The penalty function is typically designed to be continuous and differentiable to facilitate its use in optimization algorithms.
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