6.3 Optimization Techniques: Gradient Descent and Newton's Method

Optimization problems aim to find the best solution within a set of constraints. Key components include the objective function, decision variables, and constraints that define the feasible region. Understanding these elements is crucial for effectively formulating and solving optimization challenges.

Gradient-based techniques are powerful tools for solving optimization problems. Methods like gradient descent and Newton's method use derivative information to iteratively improve solutions. Factors like learning rate and algorithm choice significantly impact convergence speed and solution quality.

Optimization Problem Formulation

Components of optimization problems

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• Objective function mathematically expresses goal to minimize or maximize ($f(x)$ where $x$ is variable vector)
• Decision variables represent unknown quantities to determine optimal values
• Constraints restrict decision variables (equality $g(x) = 0$, inequality $h(x) \leq 0$)
• Feasible region encompasses all solutions satisfying constraints
• Optimal solution yields best objective function value within feasible region

Gradient-based Optimization Techniques

Gradient descent for unconstrained functions

• Gradient calculation involves partial derivatives of objective function (gradient vector $\nabla f(x) = [\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, ..., \frac{\partial f}{\partial x_n}]$)
• Update rule iteratively improves solution ($x_{k+1} = x_k - \alpha \nabla f(x_k)$, $\alpha$ is learning rate, $k$ is iteration)
• Stopping criteria determine convergence (max iterations, gradient magnitude threshold, objective function change threshold)

Learning rate in convergence

• Learning rate defines step size in steepest descent direction
• Small learning rate causes slow convergence
• Large learning rate leads to overshooting or divergence
• Adaptive learning rates adjust during optimization (Adagrad, RMSprop, Adam)
• Convergence guarantees exist for convex optimization problems

Newton's method vs gradient descent

• Newton's method update rule: $x_{k+1} = x_k - [H_f(x_k)]^{-1} \nabla f(x_k)$, $H_f(x_k)$ is Hessian matrix
• Hessian matrix contains second partial derivatives $H_f(x) = [\frac{\partial^2 f}{\partial x_i \partial x_j}]$
• Newton's method achieves quadratic convergence rate vs linear for gradient descent
• Higher computational cost per iteration for Newton's method
• Newton's method more sensitive to initial conditions
• Quasi-Newton methods approximate Hessian matrix (BFGS, L-BFGS)

Implementation of optimization algorithms

• Algorithm structure includes initialization, main loop for updates, termination conditions
• Numerical considerations address finite difference approximations, ill-conditioned Hessian matrices
• Line search techniques improve convergence (exact line search, backtracking line search)
• Performance evaluation examines convergence plots, computational time
• Constraint handling employs penalty methods, barrier methods