Let's solve this in **2 steps** as requested, explaining each step and providing the final answer at the end. --- ## **Step 1: State the KKT Conditions Explicitly** ### **Objective Function:** Minimize \( f(X, Y) = 3000X + 2300Y + 600 - \frac{1700XY}{X + Y + 1} + 4300X^2 + 3300Y^2 \) ### **Constraints (convert all to ≤ form):** 1. \( g_1(X, Y) = X + Y - 42 + 0.5 \frac{XY}{X + Y + 1} \leq 0 \) 2. \( g_2(X, Y) = 20 - X^2 - Y^2 \leq 0 \) (since \( X^2 + Y^2 \geq 20 \)) 3. \( g_3(X) = -X \leq 0 \) 4. \( g_4(X) = X - 12 \leq 0 \) 5. \( g_5(Y) = -Y \leq 0 \) 6. \( g_6(Y) = Y - 26 \leq 0 \) ### **Lagrangian:** Let \( \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_6 \) be the multipliers for each constraint. \[ L(X, Y, \lambda) = f(X, Y) + \lambda_1 g_1(X, Y) + \lambda_2 g_2(X, Y) + \lambda_3 g_3(X) + \lambda_4 g_4(X) + \lambda_5 g_5(Y) + \lambda_6 g_6(Y) \] ### **KKT Conditions:** 1. **Stationarity:** \( \nabla_X L = 0 \), \( \nabla_Y L = 0 \) 2. **Primal Feasibility:** All constraints satisfied. 3. **Dual Feasibility:** \( \lambda_i \geq 0 \) for all \( i \) 4. **Complementary Slackness:** \( \lambda_i g_i(X, Y) = 0 \) for all \( i \) **Explanation:** The KKT conditions ensure that any optimal solution must satisfy these derivative (stationarity), feasibility, and multiplier slackness conditions. --- ## **Step 2: Interpret KKT and Find Candidate Solutions** ### **Stationarity (Partial Derivatives):** Set the partial derivatives of the Lagrangian with respect to \( X \) and \( Y \) to zero: \[ \frac{\partial L}{\partial X} = \frac{\partial f}{\partial X} + \lambda_1 \frac{\partial g_1}{\partial X} - 2\lambda_2 X - \lambda_3 + \lambda_4 = 0 \] \[ \frac{\partial L}{\partial Y} = \frac{\partial f}{\partial Y} + \lambda_1 \frac{\partial g_1}{\partial Y} - 2\lambda_2 Y - \lambda_5 + \lambda_6 = 0 \] Where (for reference): - \( \frac{\partial f}{\partial X} = 3000 - \frac{1700Y(Y + 1)}{(X + Y + 1)^2} + 8600X \) - \( \frac{\partial f}{\partial Y} = 2300 - \frac{1700X(X + 1)}{(X + Y + 1)^2} + 6600Y \) - \( \frac{\partial g_1}{\partial X} = 1 + 0.5 \frac{Y(X+Y+1) - XY}{(X+Y+1)^2} \) - \( \frac{\partial g_1}{\partial Y} = 1 + 0.5 \frac{X(X+Y+1) - XY}{(X+Y+1)^2} \) ### **Active Constraints:** - At optimality, some constraints will be "active" (hold as equalities), others "inactive" (with zero multiplier). - Typically, for bounded variables (like \( X \) and \( Y \)), the optimum often lies on the boundary or at the intersection of two or more active constraints. ### **How to proceed:** - **Guess which constraints are active:** For example, \( X^2 + Y^2 = 20 \) or variable bounds. - **Solve the stationarity equations** with the active constraints as equalities, and check which multipliers are positive. **Explanation:** The above system of equations can be solved numerically for candidate solutions, or interpreted to check which combinations of active constraints and variables might yield a local minimum. --- ## **Final Answer (Summary):** **The KKT conditions for this problem are explicitly:** - The gradients of the Lagrangian with respect to \( X \) and \( Y \) must be zero (stationarity), - All constraints must be satisfied (primal feasibility), - All Lagrange multipliers must be non-negative (dual feasibility), - The product of each multiplier and its constraint must be zero (complementary slackness). **To find candidate solutions:** - Examine which constraints are likely active (e.g., at the bounds or where \( X^2 + Y^2 = 20 \)), - Solve the resulting system using the KKT equations above, typically with numerical methods, - Evaluate \( f(X, Y) \) at these candidates to find the minimum. **In practice:** The optimal solution will be found by checking the KKT conditions at all combinations where constraints are active (especially at boundaries), and the candidate yielding the lowest cost is the minimizer. For this nonlinear problem, this is typically done using numerical optimization software. --- **If you want to see a worked numerical example or a specific boundary checked, let me know!**