10.3 Penalty and barrier methods

Penalty and barrier methods transform constrained optimization problems into unconstrained ones. These techniques add terms to the objective function, either penalizing constraint violations or creating barriers to maintain feasibility during optimization.

Implementation involves iteratively solving unconstrained problems while adjusting penalty or barrier parameters. Penalty methods allow infeasible solutions, while barrier methods maintain feasibility. Each approach has unique advantages and challenges, with the choice depending on problem specifics.

Fundamentals of Penalty and Barrier Methods

Principles of penalty and barrier methods

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• Transform constrained optimization problems into unconstrained problems allowing use of unconstrained optimization techniques
• Penalty methods add penalty term to objective function for constraint violations permitting infeasible solutions during optimization (exterior approach)
• Barrier methods add barrier term to objective function preventing constraint violations maintaining feasibility throughout optimization (interior approach)
• Iterative process solves sequence of unconstrained problems gradually increasing penalty or decreasing barrier parameter

Formulation of penalty and barrier functions

• General constrained optimization problem includes objective function $f(x)$, equality constraints $h_i(x) = 0$, and inequality constraints $g_j(x) \leq 0$
• Penalty function formulation uses quadratic penalty $P(x, r_k) = f(x) + r_k [\sum_i h_i^2(x) + \sum_j \max(0, g_j(x))^2]$ where $r_k$ increases with iterations
• Barrier function formulation employs logarithmic barrier $B(x, \mu_k) = f(x) - \mu_k \sum_j \log(-g_j(x))$ where $\mu_k$ decreases with iterations

Implementation and Application

Application of exterior penalty method

1. Choose initial point $x_0$ and penalty parameter $r_0$
2. Solve unconstrained problem $\min_x P(x, r_k)$
3. Update penalty parameter $r_{k+1} > r_k$
4. Repeat until convergence

• Convergence criteria include small change in solution between iterations and constraint violation below threshold
• Challenges involve ill-conditioning as penalty parameter increases and balancing constraint satisfaction with objective improvement

Implementation of interior penalty method

1. Choose initial feasible point $x_0$ and barrier parameter $\mu_0$
2. Solve unconstrained problem $\min_x B(x, \mu_k)$
3. Update barrier parameter $\mu_{k+1} < \mu_k$
4. Repeat until convergence

• Feasible region considerations require strictly feasible initial point and maintaining feasibility throughout optimization
• Convergence criteria involve small change in solution between iterations and barrier parameter approaching zero

Penalty vs barrier method comparison

• Penalty methods can start from infeasible points and apply to both equality and inequality constraints but may produce infeasible final solutions and face numerical issues with large penalty parameters
• Barrier methods guarantee feasible solutions and often converge faster for inequality-constrained problems but require strictly feasible initial point and are limited to inequality constraints
• Computational considerations show penalty methods are easier to implement but may require more iterations while barrier methods have more complex implementation but often need fewer iterations
• Problem-specific considerations include nature of constraints (equality vs inequality), availability of feasible initial point, and desired solution accuracy