Optimization

class OptimizationProblem:
    def __init__(self, objective_fn, constraints=None):
        self.objective_fn = objective_fn
        self.constraints = constraints or []
        
    def is_feasible(self, x, tol=1e-6):
        """Check if point satisfies constraints"""
        if not self.constraints:
            return True
            
        return all(
            abs(h(x)) <= tol for h in self.constraints.get('equality', [])
        ) and all(
            g(x) <= tol for g in self.constraints.get('inequality', [])
        )

def unconstrained_minimize(f, x0, method='gradient_descent'):
    """Solve unconstrained optimization problem"""
    if method == 'gradient_descent':
        optimizer = GradientDescent()
    elif method == 'newton':
        optimizer = NewtonMethod()
    else:
        raise ValueError("Unsupported method")
    
    return optimizer.optimize(f, x0)

def constrained_minimize(f, constraints, x0, method='barrier'):
    """Solve constrained optimization problem"""
    if method == 'barrier':
        optimizer = BarrierMethod()
    elif method == 'lagrange':
        optimizer = LagrangeMultipliers()
    else:
        raise ValueError("Unsupported method")
    
    return optimizer.optimize(f, constraints, x0)

class OptimalityConditions:
    @staticmethod
    def check_first_order(f, grad_f, x, tol=1e-6):
        """Check first-order necessary conditions"""
        grad = grad_f(x)
        return np.linalg.norm(grad) <= tol
    
    @staticmethod
    def check_second_order(f, grad_f, hess_f, x, tol=1e-6):
        """Check second-order sufficient conditions"""
        grad = grad_f(x)
        hess = hess_f(x)
        
        # Check first-order conditions
        if np.linalg.norm(grad) > tol:
            return False
        
        # Check positive definiteness of Hessian
        try:
            np.linalg.cholesky(hess)
            return True
        except np.linalg.LinAlgError:
            return False

class ConvergenceAnalysis:
    @staticmethod
    def linear_convergence(sequence):
        """Check for linear convergence"""
        ratios = []
        for i in range(2, len(sequence)):
            ratio = (sequence[i] - sequence[-1]) / \
                   (sequence[i-1] - sequence[-1])
            ratios.append(ratio)
        return np.mean(ratios)
    
    @staticmethod
    def quadratic_convergence(sequence):
        """Check for quadratic convergence"""
        ratios = []
        for i in range(2, len(sequence)):
            ratio = (sequence[i] - sequence[-1]) / \
                   (sequence[i-1] - sequence[-1])**2
            ratios.append(ratio)
        return np.mean(ratios)

class FunctionAnalysis:
    @staticmethod
    def compute_gradient(f, x, eps=1e-8):
        """Compute numerical gradient"""
        n = len(x)
        grad = np.zeros(n)
        
        for i in range(n):
            ei = np.zeros(n)
            ei[i] = eps
            grad[i] = (f(x + ei) - f(x - ei)) / (2 * eps)
        
        return grad
    
    @staticmethod
    def compute_hessian(f, x, eps=1e-8):
        """Compute numerical Hessian"""
        n = len(x)
        hess = np.zeros((n, n))
        
        for i in range(n):
            for j in range(n):
                ei = np.zeros(n)
                ej = np.zeros(n)
                ei[i] = eps
                ej[j] = eps
                
                hess[i,j] = (f(x + ei + ej) - f(x + ei - ej) - \
                            f(x - ei + ej) + f(x - ei - ej)) / \
                           (4 * eps**2)
        
        return hess

class ModelTraining:
    def __init__(self, model, loss_fn, optimizer):
        self.model = model
        self.loss_fn = loss_fn
        self.optimizer = optimizer
        
    def train(self, X, y, epochs=100, batch_size=32):
        """Train model using optimization"""
        history = []
        
        for epoch in range(epochs):
            # Mini-batch training
            for i in range(0, len(X), batch_size):
                batch_X = X[i:i+batch_size]
                batch_y = y[i:i+batch_size]
                
                # Forward pass
                pred = self.model(batch_X)
                loss = self.loss_fn(pred, batch_y)
                
                # Backward pass and optimization
                grads = self.model.backward(loss)
                self.optimizer.step(grads)
                
            # Record metrics
            history.append({
                'epoch': epoch,
                'loss': loss.item()
            })
        
        return history

class HyperparameterTuning:
    def __init__(self, model_fn, param_space):
        self.model_fn = model_fn
        self.param_space = param_space
        
    def optimize(self, X, y, n_trials=50):
        """Optimize hyperparameters"""
        best_params = None
        best_score = float('inf')
        
        for _ in range(n_trials):
            # Sample parameters
            params = {
                k: np.random.uniform(v[0], v[1])
                for k, v in self.param_space.items()
            }
            
            # Train and evaluate model
            model = self.model_fn(**params)
            score = self.evaluate(model, X, y)
            
            # Update best parameters
            if score < best_score:
                best_score = score
                best_params = params
        
        return best_params, best_score
    
    def evaluate(self, model, X, y):
        """Evaluate model performance"""
        # Implement cross-validation or validation split
        return np.mean([
            model.fit(X_train, y_train).score(X_val, y_val)
            for X_train, X_val, y_train, y_val in cross_validate(X, y)
        ])