Let's proceed step by step as requested. --- ### **Step 1: State the Karush-Kuhn-Tucker (KKT) Conditions** #### **Given Problem:** **Minimize:** \( f(X, Y) = 3000X + 2300Y + 600 - 1700 \frac{XY}{X+Y+1} + 4300X^2 + 3300Y^2 \) **Subject to:** \[ \begin{align*} g_1(X, Y) &= X + Y - 42 + 0.5 \frac{XY}{X+Y+1} \leq 0 \\ g_2(X, Y) &= 200 - X^2 - Y^2 \leq 0 \\ g_3(X) &= -X \leq 0 \\ g_4(X) &= X - 12 \leq 0 \\ g_5(Y) &= -Y \leq 0 \\ g_6(Y) &= Y - 26 \leq 0 \\ \end{align*} \] #### **KKT Conditions:** Let the Lagrange multipliers for the constraints be \(\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_6 \geq 0\). 1. **Stationarity:** \[ \nabla f(X, Y) + \lambda_1 \nabla g_1(X, Y) + \lambda_2 \nabla g_2(X, Y) + \lambda_3 \nabla g_3(X) + \lambda_4 \nabla g_4(X) + \lambda_5 \nabla g_5(Y) + \lambda_6 \nabla g_6(Y) = 0 \] 2. **Primal Feasibility:** \[ g_i(X, Y) \leq 0 \quad \forall i \] 3. **Dual Feasibility:** \[ \lambda_i \geq 0 \quad \forall i \] 4. **Complementary Slackness:** \[ \lambda_i g_i(X, Y) = 0 \quad \forall i \] **Explanation:** The KKT conditions generalize the method of Lagrange multipliers to handle inequality constraints. They provide necessary conditions for a solution to be optimal, assuming differentiability and constraint qualifications. --- ### **Step 2: Solve or Interpret the KKT Conditions to Find Candidate Solutions** #### **Interpretation and Solution:** - The KKT conditions require checking which constraints are active at the solution (i.e., which are equalities) and assigning nonzero multipliers only to those. - For such a complex, nonlinear problem, the solution typically involves: - Guessing the set of active constraints (those satisfied with equality). - Solving the resulting system of equations (stationarity + active constraints). - Verifying which solution(s) are feasible and yield minimum cost. **Let's analyze the likely active constraints:** - The variable bounds will likely bind the solution, so let's check at boundaries: #### **Case 1: Try at the Lower Bounds (X=0, Y=0)** - \(X=0, Y=0\) - \(g_1 = 0 + 0 - 42 + 0 = -42 < 0\) (inactive) - \(g_2 = 200 - 0 - 0 = 200 > 0\) (violates the constraint) - So, this is infeasible. #### **Case 2: Try at Upper Bound of X (X=12, Y=?)** - Maximum X is 12. Let's see what Y can be, given \(X^2 + Y^2 \geq 200\). - Let’s try \(Y = 12\) (for symmetry): \(X^2 + Y^2 = 144 + 144 = 288 > 200\) (feasible) Compute \(g_1\): \(g_1 = 12 + 12 - 42 + 0.5 \frac{12 \times 12}{12+12+1} = 24 - 42 + 0.5 \frac{144}{25} = -18 + 0.5 \times 5.76 = -18 + 2.88 = -15.12 < 0\) (inactive) - Try \(Y = 26\): \(X^2 + Y^2 = 144 + 676 = 820 > 200\) (feasible) \(g_1 = 12 + 26 - 42 + 0.5 \frac{12 \times 26}{12+26+1} = 38 - 42 + 0.5 \frac{312}{39} = -4 + 0.5 \times 8 = -4 + 4 = 0\) (active) So, **at \(X=12, Y=26\):** - \(g_1 = 0\) (active) - \(g_2 = 820-200=620>0\) (feasible) - Bounds are satisfied. #### **Cost at (X=12, Y=26):** \[ \begin{align*} f(12, 26) &= 3000(12) + 2300(26) + 600 - 1700 \frac{12 \times 26}{12+26+1} + 4300(12^2) + 3300(26^2) \\ &= 36{,}000 + 59{,}800 + 600 - 1700 \frac{312}{39} + 4300(144) + 3300(676) \\ &= 36{,}000 + 59{,}800 + 600 - 1700(8) + 619{,}200 + 2{,}230{,}800 \\ &= 36{,}000 + 59{,}800 + 600 - 13{,}600 + 619{,}200 + 2{,}230{,}800 \\ &= (36{,}000 + 59{,}800 + 600 + 619{,}200 + 2{,}230{,}800) - 13{,}600 \\ &= 2{,}946{,}400 - 13{,}600 = 2{,}932{,}800 \end{align*} \] **Explanation:** By checking which constraints are active (those that are zero at the solution), we can try candidate solutions at the boundary values. Here, both variable bounds and the first constraint are active at \((X=12, Y=26)\), and all other constraints are satisfied. --- ### **Final Answer** **The KKT conditions require stationarity, primal/dual feasibility, and complementary slackness. For this problem, a candidate solution is \(X=12, Y=26\) where the first constraint and upper bounds are active. The minimum cost at this point is:** \[ \boxed{2,\!932,\!800} \] **This is a candidate optimal point based on KKT analysis and checking constraint activity. For exact optimality, further check all KKT multipliers, but this is the likely solution given the structure.**