Question 1

What are Lagrange multipliers?

Accepted Answer

Lagrange multipliers are a mathematical method for finding the local maxima and minima of a function subject to equality constraints. Named after Joseph-Louis Lagrange, this technique converts a constrained optimization problem into a system of equations. The Lagrange multiplier (λ) itself represents the rate of change of the optimal value with respect to changes in the constraint, providing economic and physical interpretation.

Question 2

How do you use Lagrange multipliers to solve optimization problems?

Accepted Answer

Step 1: Identify the objective function f(x,y,...) to maximize or minimize. Step 2: Identify the constraint g(x,y,...) = 0. Step 3: Form the Lagrange function: L(x,y,λ) = f(x,y) - λg(x,y). Step 4: Find all partial derivatives: ∂L/∂x = 0, ∂L/∂y = 0, ∂L/∂λ = 0. Step 5: Solve the system of equations to find critical points. Step 6: Evaluate f at critical points and boundary to determine max/min.

Question 3

What is the Lagrange multiplier formula?

Accepted Answer

The Lagrange multiplier method states that at a constrained optimum, the gradients of the objective function and constraint must be proportional: ∇f = λ∇g. This leads to the Lagrange function L(x,y,...,λ) = f(x,y,...) - λg(x,y,...). Setting all partial derivatives equal to zero: ∂L/∂x = ∂f/∂x - λ∂g/∂x = 0, ∂L/∂y = ∂f/∂y - λ∂g/∂y = 0, and ∂L/∂λ = -g(x,y,...) = 0. Solving these equations gives the optimal point and λ value.

Question 4

When should you use Lagrange multipliers?

Accepted Answer

Use Lagrange multipliers when: 1) You have an optimization problem with equality constraints (g(x,y) = c). 2) The constraint is not easily substitutable into the objective function. 3) You're working with multivariable functions. 4) You need to find optimal points on a constraint curve or surface. Don't use for inequality constraints (use KKT conditions instead) or unconstrained optimization (use gradient = 0 directly).

Question 5

What does the Lagrange multiplier λ represent?

Accepted Answer

The Lagrange multiplier λ has important interpretations: 1) Mathematically: λ measures how much the optimal value changes if you relax the constraint slightly. 2) Economically: λ is the 'shadow price' - the marginal value of relaxing the constraint. 3) Physically: λ represents the constraint force in mechanics. For example, if maximizing utility subject to a budget constraint, λ is the marginal utility of an extra dollar.

Question 6

How do you know if a critical point is a maximum or minimum?

Accepted Answer

To classify critical points from Lagrange multipliers: 1) Use the bordered Hessian matrix (second derivative test for constrained optimization). 2) Evaluate the objective function at all critical points and compare values. 3) Check the physical or geometric meaning of the problem. 4) For one constraint in 2D, analyze the curvature along the constraint curve. The method of Lagrange multipliers finds critical points but doesn't automatically classify them.

Question 7

Can Lagrange multipliers handle multiple constraints?

Accepted Answer

Yes! For multiple constraints g₁(x,y) = 0, g₂(x,y) = 0, etc., use multiple Lagrange multipliers: L(x,y,λ₁,λ₂) = f(x,y) - λ₁g₁(x,y) - λ₂g₂(x,y). Set all partial derivatives to zero, creating a system with more equations and unknowns. Each λᵢ represents the sensitivity to its corresponding constraint. This generalizes to any number of variables and equality constraints.

Question 8

What is the difference between Lagrange multipliers and KKT conditions?

Accepted Answer

Lagrange multipliers handle equality constraints (g(x) = 0), while Karush-Kuhn-Tucker (KKT) conditions extend the method to inequality constraints (g(x) ≤ 0). KKT conditions include: stationarity (∇f = Σλᵢ∇gᵢ), primal feasibility (constraints satisfied), dual feasibility (λᵢ ≥ 0), and complementary slackness (λᵢgᵢ = 0). For equality constraints only, KKT reduces to Lagrange multipliers. KKT is essential in nonlinear programming and optimization.

Question 9

What are real-world applications of Lagrange multipliers?

Accepted Answer

Lagrange multipliers are used in: Economics (utility maximization with budget constraints, cost minimization), Engineering (optimal design with resource constraints), Physics (constrained motion, least action principle), Machine Learning (support vector machines, regularized optimization), Operations Research (resource allocation), Portfolio Theory (optimal asset allocation), and Chemistry (equilibrium calculations). Any optimization problem with constraints can potentially use this method.

Question 10

How do you solve a Lagrange multiplier problem by hand?

Accepted Answer

Example: Maximize f(x,y) = xy subject to x + y = 10. Step 1: L = xy - λ(x + y - 10). Step 2: ∂L/∂x = y - λ = 0 → y = λ. Step 3: ∂L/∂y = x - λ = 0 → x = λ. Step 4: ∂L/∂λ = -(x + y - 10) = 0 → x + y = 10. Step 5: Since x = y, we have 2x = 10, so x = 5, y = 5. Step 6: Maximum value: f(5,5) = 25. The Lagrange multiplier is λ = 5.

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