Penalty Function Methods for Solving Optimization Problems

Penalty function methods are effective approaches for solving constrained optimization problems, with the core concept of transforming original constrained problems into unconstrained ones by introducing penalty terms. These methods are particularly suitable for optimization problems with inequality or equality constraints. Based on different penalty term application methods, they can be classified into two main types: interior point methods and exterior point methods.

Interior point methods are characterized by approaching optimal solutions from within the feasible region. They incorporate barrier functions into the objective function, ensuring the iterative process remains within the feasible domain. As penalty coefficients are adjusted, solutions gradually approach the optimal solution on the boundary. This method suits strictly feasible problems but requires initial points to be within the feasible region. In code implementation, logarithmic barrier functions like -log(-g(x)) are commonly used for inequality constraints, with careful handling of penalty parameter updates to maintain numerical stability.

Exterior point methods approach optimal solutions progressively from outside the feasible region. They penalize constraint violations, making solutions gradually satisfy constraints during iteration. Exterior methods are insensitive to initial point selection and have broader applicability, but may generate infeasible solutions in early iterations. Implementation typically involves quadratic penalty terms like ρ*max(0, g(x))² for inequality constraints, where ρ is a gradually increasing penalty parameter that controls constraint satisfaction.

Both methods require appropriate penalty coefficient settings to balance constraint violation and objective optimization. Interior point methods offer more stability but limited applicability, while exterior methods provide greater flexibility but potentially slower convergence. In practical applications, suitable methods can be selected based on problem characteristics, or combined for better results. Algorithm implementation often involves adaptive penalty parameter strategies, where parameters are updated based on constraint violation magnitudes during optimization cycles.

The advantage of penalty function methods lies in transforming complex constrained problems into sequences of unconstrained subproblems, facilitating the use of mature optimization algorithms. They have wide applications in engineering optimization, machine learning parameter training, and other fields. Modern implementations often integrate these methods with gradient-based optimizers like BFGS or Newton methods, where automatic differentiation can compute precise gradients of penalized objective functions for efficient convergence.