Consider the nonlinear optimization model stated below.Min 2X2 ? 18X + 2XY + Y2 ? 14Y + 51s.t. X + 4Y ? 6(a)Find the minimum solution to this problem.32 Changed: Your submitted answer was incorrect. Your current answer has not been submitted. at (X, Y) = Incorrect: Your answer is incorrect. (b)If the right-hand side of the constraint is increased from 6 to 7, how much do you expect the objective function to change?Based on the dual value on the constraint X + 4Y ? 6, we expect the optimal objective function value to decrease by -8 Changed: Your submitted answer was incorrect. Your current answer has not been submitted. .(c)Resolve the problem with a new right-hand side of the constraint of 7. How does the actual change compare with your estimate?If we resolve the problem with a new right-hand-side of 7 the new optimal objective function value is 40 Incorrect: Your answer is incorrect. , so the actual change is a decrease of 3 Changed: Your submitted answer was incorrect. Your current answer has not been submitted. rather than what we expected in part (b).

to this problem We solve this using the Karush-Kuhn-Tucke (KKT) conditions, a standard method for nonlinear optimization with inequality constraints. Step 1: Set up the Lagrangian [[X.Y.X)- 2X4 - 18X +2XY +93 -14 +51 4 XX + 4 -6) Here, A 2 0 is the Lagrange multiplier for the nequality constraint Step 2: Compute partial derivatives Set gradients to zero for stationarity: LaTex Copy cod \\frac(\\partial \\mathcal(L}}{\\partial Y) = 2x + 2Y - 14 + 4\\lambda = 0 \\tag(2} ] LaTex copy code (Constraint: ) x + 47 \\leq 6 \\text{Complementary Slackness: } \\lambda (X + 4Y - 6) = 0 \\tag(4) ] since the Cons constraint is active at the minimum (as we will confirm), we set: LaTex copy code X + 4Y - 6 \\Rightarrow X - 6 - 4Y Step 3: Solve the system Substitute Eq (5) into Eq (1) and (2) From (1) LaTex copy code \\Rightarrow 24 - isY - 18 + 24 4 \\lambd . \\Rightarrow \\lambda - -6 . 14 \\tag(6) From (2 LaTex Copy code \\Rightarrow - 2 - 6Y + 4\\lanbe \\fractay . 1)(2) \\tazt7) Step 4: Equate (6) and (7) LaTex Copy code 6 + 149 - Mrac(3Y + 1)12) Rightarrow 257 - 13 \\Rightarrow Y " \\frac(13}(25) \\tag(8) From Eq (5) O Copy code \\frac(52)(25) - \\frac(98)(25) \\caz(9} Step 5: Compute the objective function f(X, Y) - 2X2 - 18X +2XY + Y? - 14Y + 51 Plug in X = , Y = 1 Break into parts: . X = 12 = 2X? = 105 . -18X = -18 . = =1394 . 2XY = 2 . * - 1 = 3545 . -14Y = -14 . # =32 Now sum: f = (10206 +218+160) - (17618182) +51 = \\frac(21925)(625) - \\frac(1946)(25) + 51 = 35.08 - 77.84 + 51 = \\boxed(8.24) ] Final Answer for (a): (x, Y) = 25' 25 ) = 8.24 V (b) Estimate change in HS changes from 6 to 7 Use the Lagrange multiplier > as the shadow price (dual value). From equation (6): A - -6+148 - -6+14-1 --6+ 49 - 18-[2.25] So, increasing RHS by 1 unit is estimated to decrease the objective by: 1.28 V (c) Resolve with new RHS = 7 and compare Now solve the problem again with: LaTex Copy cod * + 4Y - 7 \\Rightarrow x - 7 - 4Y Repeat substitution into the Lagrangian system as in part (a). From stationarity (same steps as above), we get: . Y= =0.8 . X = 7 - 4 .0.8 =7-3.2- 1 - 3.8 Compute new objective: X = 3.8. Y = 0.8 . -18X = -68.4 . 2XY = 6.08 . Yz = 0.64 . -14Y = -11.2 Now: f = 28.88 - 68.4 + 6.08 + 0.64 - 11.2 + 51 = [7.0 Compare: . Previous value: 8.24 New value: 7.0 * Actual change: 8.24 - 7.0 = -1.24 Estimated change from part (b): -1.28 Conclusion: The estimate was very close to the actual value. V Final Answers Summary Part Answer (a) Minimum at /98 13 \\ bbjective value 854 (b) Estimated decrease in objective (from A): -1.28 New optimal value: -1.24 . very close to estimateObjective Function Contours with Constraint Lines 2.00 --- Constraint: X + 4Y = 6 50 -= = Constraint: X + 4Y = 7 Feasible Region (RHS=6) 1.75 Optimal (RHS=6) . Optimal (RHS=7) 44 1.50 38 1.25 32 > 1.00 26 0.75 20 0.50 14 0.25 8 0.00 o 1 2 3 4