In a nonlinear programming model, the Lagrange multiplier reflects the appropriate change in the _________________ due to a marginal change in the right-hand side of a constraint.

Please choose the option that best fit the empty space above.

Question

In a nonlinear programming model, the Lagrange multiplier reflects the appropriate change in the _________________ due to a marginal change in the right-hand side of a constraint.

Please choose the option that best fit the empty space above.

BlackTom AI · Accepted Answer

A Lagrange multiplier equals the marginal change in the optimal objective value — the shadow price.

1) Relevant knowledge points and concepts
- Lagrange multipliers (dual variables) attach to constraints in constrained optimization and quantify sensitivity of the optimum to constraint changes.
- Shadow price is defined as the rate of change of the optimal objective value with respect to a marginal change in the right-hand side (RHS) of a constraint.
- The envelope theorem (or direct differentiation of the Lagrangian) connects the multiplier to that marginal change.

2) Complete reasoning process step by step
Consider a maximization problem with objective Z(x) subject to one constraint g(x) ≤ b. Form the Lagrangian:
L(x, λ) = Z(x) + λ (b − g(x))
First-order conditions at an interior optimum: ∂L/∂x = ∇Z(x*) − λ∇g(x*) = 0 and ∂L/∂λ = b − g(x*) = 0.

3) Intermediate calculation steps (sensitivity)
Differentiate the Lagrangian with respect to the RHS b while holding optimal x* and λ* according to the envelope theorem:
∂L/∂b = λ.
Because at the optimum the Lagrangian evaluated equals the optimal objective (adjusted by the constraint term that is zero at feasibility), this derivative gives the sensitivity of the optimal objective value Z* to b:
dZ*/db = λ.
Thus the Lagrange multiplier λ equals the change in optimal Z per unit change in the RHS.

4) Why the correct answer is correct
By definition and derivation, the Lagrange multiplier is the shadow price: it measures how much the optimal objective (Z) will increase (for a maximization) if the RHS of the constraint is increased marginally.

5) Why other choices are incorrect
- "coefficient": Lagrange multipliers are not coefficients of the constraint expression; those are problem data.
- "left-hand side of the constraint": that is the function of decision variables (g(x)), not what the multiplier measures.
- "the 'Z' (from the objective function)": while the multiplier measures change in Z, the multiplier itself is called the shadow price — the best single-word fit asked for is "shadow price."
- "None of the above": incorrect because a standard interpretation is that the multiplier is the shadow price.

6) Conclusion
Therefore the correct fill is "shadow price," which matches the provided answer.

In a nonlinear programming model, the Lagrange multiplier reflects the appropriate change in the _________________ due to a marginal change in the right-hand side of a constraint. Please choose the option that best fit the empty space above.Single choice

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