Exterior penalty methods transform constrained optimization problems into unconstrained ones by adding penalty terms to the . These methods start with infeasible solutions and gradually move towards feasibility, making them useful for problems with complex constraints or when finding initial feasible points is challenging.
The key components include penalty terms for constraint violations, a controlling their importance, and an unconstrained formulation. As the penalty parameter increases, solutions converge to the original constrained problem's solution. Careful selection of penalty functions and update strategies is crucial for method performance.
Exterior Penalty Methods for Optimization
Concept and Purpose
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Transform constrained optimization problems into unconstrained problems by adding penalty terms to the objective function
Discourage solutions violating constraints by imposing increasingly severe penalties as increases
Start with an infeasible solution and gradually move towards feasibility during optimization
Utilize continuous and differentiable penalty functions allowing use of standard unconstrained optimization algorithms
Prove particularly useful for problems with complex constraint structures (nonlinear constraints)
Allow comprehensive exploration of potential solutions by balancing search space exploration and constraint satisfaction
Enable solving problems where finding an initial feasible point presents challenges (highly constrained problems)
Key Components and Formulation
Add penalty terms to objective function for each constraint violation
Construct penalty function combining original objective function and sum of penalty terms
Use quadratic functions of constraint violations as penalty terms ensuring continuous differentiability
Introduce penalty parameter (μ or ρ) controlling relative importance of penalty terms
Formulate unconstrained problem as minf(x)+μ∗P(x)
f(x) represents original objective function
μ denotes penalty parameter
P(x) signifies sum of penalty terms
Increase penalty parameter progressively causing solutions to converge to original constrained problem solution
Select penalty function and parameter update strategy carefully to influence method performance (quadratic penalty, adaptive penalty parameter)
Penalty Terms in Optimization
Types and Characteristics
Employ quadratic penalty terms for inequality constraints Pi(x)=max(0,gi(x))2
Utilize squared penalty terms for equality constraints Pj(x)=hj(x)2
Ensure penalty terms remain continuously differentiable to maintain smoothness
Design penalty terms to increase rapidly as constraints become violated (steep gradients near constraint boundaries)
Incorporate problem-specific knowledge into penalty term design (logarithmic barriers for positivity constraints)
Balance penalty term magnitude with original objective function to prevent domination
Consider alternative penalty functions for specific problem types (absolute value penalty, exponential penalty)
Penalty Parameter Strategies
Start with relatively small initial penalty parameter value to explore search space
Increase penalty parameter systematically between optimization iterations (multiply by constant factor)
Implement adaptive penalty parameter update strategies based on constraint violation progress