cognitive/knowledge_base/mathematics/optimization_theory.md

title	type	status	created	complexity	processing_priority	tags	semantic_relations
Optimization Theory	concept	stable	2024-03-15	advanced	1	mathematics optimization variational_methods machine_learning	type links foundation_for control_theory machine_learning active_inference variational_inference type links implements calculus linear_algebra probability_theory type links relates statistical_physics information_theory free_energy_principle dynamical_systems

Optimization Theory

Overview

Optimization Theory provides the mathematical foundation for finding the best solutions to problems under given constraints. It forms a crucial bridge between theoretical principles and practical implementations in control theory, machine learning, and cognitive systems, particularly in understanding how systems minimize free energy and maximize performance.

Mathematical Foundation

Unconstrained Optimization

Gradient Descent

x_{k+1} = x_k - \alpha_k \nabla f(x_k)

where:

• x_k is current point
• \alpha_k is step size
• \nabla f is gradient

Newton's Method

x_{k+1} = x_k - [\nabla^2 f(x_k)]^{-1}\nabla f(x_k)

where:

• \nabla^2 f is Hessian matrix

Constrained Optimization

Lagrangian

L(x,\lambda) = f(x) + \sum_i \lambda_i g_i(x)

where:

• f(x) is objective function
• g_i(x) are constraints
• \lambda_i are Lagrange multipliers

KKT Conditions

\begin{align*}
\nabla_x L(x^*,\lambda^*) &= 0 \\
g_i(x^*) &\leq 0 \\
\lambda_i^* g_i(x^*) &= 0 \\
\lambda_i^* &\geq 0
\end{align*}

Implementation

Gradient-Based Optimizer

class GradientOptimizer:
    def __init__(self,
                 learning_rate: float = 0.01,
                 momentum: float = 0.9,
                 nesterov: bool = False):
        """Initialize gradient optimizer.
        
        Args:
            learning_rate: Learning rate
            momentum: Momentum coefficient
            nesterov: Whether to use Nesterov momentum
        """
        self.lr = learning_rate
        self.momentum = momentum
        self.nesterov = nesterov
        self.velocity = None
    
    def step(self,
            params: np.ndarray,
            grads: np.ndarray) -> np.ndarray:
        """Perform optimization step.
        
        Args:
            params: Current parameters
            grads: Parameter gradients
            
        Returns:
            new_params: Updated parameters
        """
        if self.velocity is None:
            self.velocity = np.zeros_like(params)
        
        # Update velocity
        self.velocity = (self.momentum * self.velocity +
                        self.lr * grads)
        
        if self.nesterov:
            # Nesterov update
            update = (self.momentum * self.velocity +
                     self.lr * grads)
        else:
            # Standard update
            update = self.velocity
        
        return params - update

Second-Order Optimizer

class NewtonOptimizer:
    def __init__(self,
                 damping: float = 1e-4,
                 max_iter: int = 100):
        """Initialize Newton optimizer.
        
        Args:
            damping: Hessian damping factor
            max_iter: Maximum iterations
        """
        self.damping = damping
        self.max_iter = max_iter
    
    def step(self,
            params: np.ndarray,
            grad_fn: Callable,
            hess_fn: Callable) -> np.ndarray:
        """Perform Newton optimization step.
        
        Args:
            params: Current parameters
            grad_fn: Gradient function
            hess_fn: Hessian function
            
        Returns:
            new_params: Updated parameters
        """
        grad = grad_fn(params)
        hess = hess_fn(params)
        
        # Add damping
        hess += self.damping * np.eye(len(params))
        
        # Compute Newton direction
        try:
            direction = np.linalg.solve(hess, -grad)
        except np.linalg.LinAlgError:
            # Fallback to gradient descent
            direction = -grad
        
        return params + direction

Constrained Optimizer

class ConstrainedOptimizer:
    def __init__(self,
                 constraints: List[Callable],
                 penalty_weight: float = 1.0,
                 weight_growth: float = 2.0):
        """Initialize constrained optimizer.
        
        Args:
            constraints: List of constraint functions
            penalty_weight: Initial penalty weight
            weight_growth: Penalty weight growth factor
        """
        self.constraints = constraints
        self.weight = penalty_weight
        self.growth = weight_growth
        
        # Initialize base optimizer
        self.base_optimizer = GradientOptimizer()
    
    def augmented_lagrangian(self,
                           x: np.ndarray,
                           lambda_: np.ndarray) -> float:
        """Compute augmented Lagrangian.
        
        Args:
            x: Current point
            lambda_: Lagrange multipliers
            
        Returns:
            L: Augmented Lagrangian value
        """
        # Objective
        L = self.objective(x)
        
        # Constraint terms
        for i, g in enumerate(self.constraints):
            c = g(x)
            L += lambda_[i] * c + 0.5 * self.weight * c**2
        
        return L
    
    def step(self,
            x: np.ndarray,
            lambda_: np.ndarray) -> Tuple[np.ndarray, np.ndarray]:
        """Perform optimization step.
        
        Args:
            x: Current point
            lambda_: Current multipliers
            
        Returns:
            new_x: Updated point
            new_lambda: Updated multipliers
        """
        # Minimize augmented Lagrangian
        def grad_L(x):
            return self.grad_augmented_lagrangian(x, lambda_)
        
        x_new = self.base_optimizer.step(x, grad_L(x))
        
        # Update multipliers
        lambda_new = np.array([
            max(0, lambda_[i] + self.weight * g(x_new))
            for i, g in enumerate(self.constraints)
        ])
        
        # Update penalty weight
        self.weight *= self.growth
        
        return x_new, lambda_new

Applications

Physical Systems

Energy Minimization

• Mechanical equilibrium
• Potential energy
• Variational principles
• Path optimization

Control Systems

• Optimal control
• Model predictive control
• Trajectory optimization
• Resource allocation

Machine Learning

Neural Networks

• Loss minimization
• Weight optimization
• Architecture search
• Hyperparameter tuning

Probabilistic Models

• Maximum likelihood
• Variational inference
• Expectation maximization
• Bayesian optimization

Cognitive Systems

Active Inference

• Free energy minimization
• Belief updating
• Policy optimization
• Learning and inference

Decision Making

• Utility maximization
• Risk minimization
• Multi-objective optimization
• Sequential decision making

Advanced Topics

Convex Optimization

• Linear programming
• Quadratic programming
• Semidefinite programming
• Interior point methods

Stochastic Optimization

• Stochastic gradient descent
• Evolutionary algorithms
• Simulated annealing
• Particle swarm optimization

Distributed Optimization

• Consensus algorithms
• Decomposition methods
• Federated learning
• Multi-agent optimization

Best Practices

Problem Formulation

1. Identify objectives
2. Define constraints
3. Choose variables
4. Select algorithm

Implementation

1. Initialize properly
2. Handle constraints
3. Monitor convergence
4. Validate solutions

Validation

1. Optimality conditions
2. Constraint satisfaction
3. Numerical stability
4. Solution quality

Common Issues

Technical Challenges

1. Local minima
2. Ill-conditioning
3. Constraint violations
4. Numerical instability

Solutions

1. Multiple initializations
2. Regularization
3. Constraint relaxation
4. Robust algorithms

Related Documentation

• control_theory
• statistical_physics
• machine_learning
• active_inference
• variational_inference
• dynamical_systems

Variational Methods

Free Energy Optimization

class FreeEnergyOptimizer:
    """
    Optimize variational free energy.
    
    Theory:
        - [[variational_methods]]
        - [[free_energy_principle]]
    Mathematics:
        - [[information_theory]]
        - [[natural_gradients]]
    """
    def __init__(self,
                 model: GenerativeModel,
                 learning_rate: float = 0.01):
        self.model = model
        self.lr = learning_rate
        self.history = []
    
    def minimize_free_energy(self,
                           observation: np.ndarray,
                           initial_belief: np.ndarray,
                           n_iterations: int = 100) -> np.ndarray:
        """Minimize variational free energy."""
        belief = initial_belief.copy()
        
        for i in range(n_iterations):
            # Compute free energy and gradient
            F, grad = self.compute_free_energy_gradient(
                observation, belief
            )
            
            # Natural gradient update
            belief = self.natural_gradient_update(belief, grad)
            
            # Track progress
            self.history.append(F)
            
            # Check convergence
            if self._check_convergence():
                break
        
        return belief

Policy Optimization

class PolicyOptimizer:
    """
    Optimize policies using expected free energy.
    
    Theory:
        - [[active_inference_pomdp]]
        - [[expected_free_energy]]
    Mathematics:
        - [[path_integral]]
        - [[information_theory]]
    """
    def __init__(self,
                 model: ActiveInferenceModel,
                 temperature: float = 1.0):
        self.model = model
        self.temperature = temperature
    
    def optimize_policy(self,
                       belief_state: np.ndarray,
                       policies: List[np.ndarray]) -> np.ndarray:
        """Optimize policy selection."""
        # Compute expected free energy for each policy
        G = np.array([
            self.compute_expected_free_energy(belief_state, pi)
            for pi in policies
        ])
        
        # Softmax policy selection
        p = self.softmax_policy_selection(G)
        
        return p

Gradient Methods

Natural Gradient Descent

class NaturalGradientOptimizer:
    """
    Natural gradient optimization.
    
    Theory:
        - [[natural_gradients]]
        - [[information_geometry]]
    Mathematics:
        - [[fisher_information]]
        - [[riemannian_geometry]]
    """
    def __init__(self,
                 learning_rate: float = 0.1,
                 damping: float = 1e-4):
        self.lr = learning_rate
        self.damping = damping
    
    def compute_natural_gradient(self,
                               params: np.ndarray,
                               grad: np.ndarray,
                               fisher: np.ndarray) -> np.ndarray:
        """Compute natural gradient direction."""
        # Add damping to Fisher matrix
        fisher_damped = fisher + self.damping * np.eye(fisher.shape[0])
        
        # Solve linear system for natural gradient
        natural_grad = np.linalg.solve(fisher_damped, grad)
        
        return natural_grad

Stochastic Optimization

class StochasticOptimizer:
    """
    Stochastic optimization methods.
    
    Theory:
        - [[stochastic_optimization]]
        - [[monte_carlo_methods]]
    Mathematics:
        - [[stochastic_approximation]]
        - [[variance_reduction]]
    """
    def __init__(self,
                 learning_rate: float = 0.01,
                 batch_size: int = 32):
        self.lr = learning_rate
        self.batch_size = batch_size
    
    def optimize_batch(self,
                      params: np.ndarray,
                      data_batch: np.ndarray) -> np.ndarray:
        """Perform stochastic optimization step."""
        # Compute batch gradient
        grad = self.compute_batch_gradient(params, data_batch)
        
        # Apply variance reduction
        grad = self.reduce_variance(grad)
        
        # Update parameters
        params = self.update_parameters(params, grad)
        
        return params

Dynamic Programming

Value Iteration for POMDPs

class ValueIterator:
    """
    Value iteration for POMDPs.
    
    Theory:
        - [[dynamic_programming]]
        - [[bellman_equation]]
    Mathematics:
        - [[markov_decision_process]]
        - [[value_functions]]
    """
    def __init__(self,
                 discount: float = 0.95,
                 tolerance: float = 1e-6):
        self.gamma = discount
        self.tol = tolerance
    
    def iterate_values(self,
                      belief_states: np.ndarray,
                      transition_model: np.ndarray,
                      reward_model: np.ndarray) -> np.ndarray:
        """Perform value iteration."""
        # Initialize value function
        V = np.zeros(len(belief_states))
        
        while True:
            # Backup old values
            V_old = V.copy()
            
            # Update values
            for i, b in enumerate(belief_states):
                V[i] = self._compute_optimal_value(
                    b, transition_model, reward_model, V_old
                )
            
            # Check convergence
            if np.max(np.abs(V - V_old)) < self.tol:
                break
        
        return V

Documentation Links

• ../../docs/research/research_documentation_index
• ../../docs/guides/implementation_guides_index
• ../../docs/api/api_documentation_index
• ../../docs/examples/usage_examples_index

References

• bertsekas - Dynamic Programming and Optimal Control
• amari - Information Geometry and Its Applications
• boyd - Convex Optimization
• sutton - Reinforcement Learning