optimization

What I do

• Solve linear and nonlinear programming problems
• Apply gradient descent and Newton methods
• Work with constrained optimization (Lagrange multipliers)
• Analyze convex optimization problems
• Apply dynamic programming
• Use integer and combinatorial optimization

When to use me

When maximizing/minimizing functions subject to constraints.

Key Concepts

• Linear Programming: Minimize c^Tx subject to Ax ≤ b, x ≥ 0
• Gradient Descent: x_{k+1} = x_k - α∇f for unconstrained minimization
• Lagrange Multipliers: ∇f = λ∇g for constraint g(x) = 0
• KKT Conditions: Necessary conditions for constrained optima
• Convexity: f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y) ensures global optima
• Dynamic Programming: Optimal substructure + memoization