Sensitivity in Optimization

Mathematical Modeling · Axiom Academy

LESSON Sensitivity in Optimization Understanding how optimal solutions respond to changes in parameters and constraints 1. What is Sensitivity Analysis? Sensitivity analysis examines how the optimal solution and optimal value of an optimization problem change when we perturb the problem's parameters. Consider the general optimization problem: The parameter p affects either the objective function, the constraints, or both. Sensitivity analysis asks: if we change p slightly, how does the optimal value change? 2. Shadow Prices and Lagrange Multipliers For constrained optimization problems, the Lagrange multipliers have a powerful economic interpretation as shadow prices . Consider the problem of maximizing f(x) subject to g(x) = b: The Lagrange multiplier tells us how much the optimal value would improve if we could relax the constraint by one unit. This is the "shadow price" of the constraint. The Envelope Theorem provides a powerful way to compute sensitivity without re-solving the optimization problem. It states that the derivative of the optimal value function with respect to a parameter equals the partial derivative of the Lagrangian: This is remarkable: we only need to take a partial derivative, holding x fixed at its optimal value, without computing how the optimal x itself changes! 4. Sensitivity in Linear Programming

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.