Module constrained 

Constrained optimization methods

Constrained Optimization

This module implements advanced constrained optimization algorithms for handling equality and inequality constraints in optimization problems.

Mathematical Background

Constrained optimization solves problems of the form:

minimize f(x)
subject to:
  g_i(x) ≤ 0,  i = 1, ..., m  (inequality constraints)
  h_j(x) = 0,  j = 1, ..., p  (equality constraints)
  x ∈ X                       (bound constraints)

Key Algorithms

• Penalty Methods: Transform constrained problems into unconstrained ones
• Barrier Methods: Use barrier functions to enforce inequality constraints
• Lagrange Multipliers: Direct handling of KKT conditions
• Augmented Lagrangian: Combine penalties with Lagrange multipliers
• Sequential Quadratic Programming (SQP): Newton-like methods for constraints

KKT Conditions

For a local minimum x*, the Karush-Kuhn-Tucker conditions must hold:

∇f(x*) + Σᵢ λᵢ ∇gᵢ(x*) + Σⱼ μⱼ ∇hⱼ(x*) = 0
gᵢ(x*) ≤ 0,  λᵢ ≥ 0,  λᵢ gᵢ(x*) = 0  (complementary slackness)
hⱼ(x*) = 0

Structs

ConstrainedConfig

Configuration for constrained optimization algorithms

ConstrainedOptimizer

Constrained optimization solver

ConstrainedResult

Results from constrained optimization

Enums

PenaltyMethod

Penalty method types

Traits

ConstrainedObjective

Trait for constrained objective functions