6.4 Constrained Optimization and Linear Programming

Constrained optimization is all about finding the best solution while working within limits. It's like trying to make the most delicious cake with a limited budget and specific ingredients. We'll explore how to set up these problems and solve them.

Advanced techniques like KKT conditions and linear programming take optimization to the next level. These methods help tackle more complex real-world problems, from maximizing profits to optimizing resource allocation. We'll learn how to implement and interpret these powerful tools.

Constrained Optimization Fundamentals

Constrained optimization problem formulation

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• Constrained optimization maximizes or minimizes objective function subject to constraints on variables
• Components include objective function, decision variables, constraints
• Equality constraints $h(x) = 0$ and inequality constraints $g(x) \leq 0$ mathematically express limitations
• Constraints can be linear (straight lines), nonlinear (curves), or box (upper and lower bounds)
• Examples: maximizing profit subject to budget constraints, minimizing travel time with speed limits

Lagrange multipliers in optimization

• Lagrange multipliers scalar variables associated with constraints incorporate limitations into objective function
• Lagrangian function $L(x, \lambda) = f(x) + \sum_{i=1}^m \lambda_i h_i(x)$ transforms constrained problem to unconstrained
• Partial derivatives of Lagrangian set to zero yield system of equations for solving
• Multipliers interpret sensitivity of optimal solution to constraint changes (shadow prices)
• Applications include resource allocation, portfolio optimization

Advanced Optimization Techniques

KKT conditions for problem-solving

• KKT conditions generalize Lagrange multipliers for inequality constraints
• Four conditions: stationarity, primal feasibility, dual feasibility, complementary slackness
• Formulate Lagrangian with inequality constraints, derive and solve KKT conditions
• KKT multipliers provide economic interpretation in resource allocation (marginal values)
• Necessary but not always sufficient for optimality, limitations in non-convex problems

Linear programming with simplex method

• Linear programming optimizes linear objective function subject to linear constraints
• Standard form: maximize objective, convert inequalities to equalities using slack variables
• Basic feasible solutions correspond to vertices of feasible region (geometric interpretation)
• Simplex algorithm:
  1. Choose initial basic feasible solution
  2. Compute reduced costs
  3. Determine entering and leaving variables
  4. Update solution
  5. Repeat until optimality reached
• Results provide optimal solution, objective value, shadow prices (sensitivity analysis)

Implementation of optimization techniques

• Algorithms include interior point methods, sequential quadratic programming, gradient projection
• Implement using built-in functions (MATLAB, Python, R) or develop custom solutions
• Handle constraints through penalty methods (add violation costs) or barrier methods (interior approach)
• Analyze results: verify feasibility, check optimality, perform sensitivity analysis
• Interpret mathematical solution for practical insights, identify active constraints
• Address scaling issues, numerical instabilities, non-convex problems (local vs global optima)