Question

(45 Points) Consider the constrained optimization problem: min f(x1, x2) = 2x} + 9x2 + 9x2 - 6x1x2 – 18x1 X1 X2 Subject to 4x1 – 3x2 s 20 X1 + 2x2 < 10 -X1 < 0, - x2 < 0 a) Is this problem convex? Justify your answer. (5 Points) b) Form the Lagrange function. (5 Points) c) Formulate KKT conditions. (10 Points) d) Recall that one technique for finding roots of KKT condition is to check all permutations of the switching conditions. Find an optimal solution (x*) via this approach. Is this solution a global minimum? (10 Points) e) Compute the objective function and identify each constraint as active or inactive at the optimal solution. (5 Points) f) Solve this problem using graphical optimization to check your results. Remember to show the feasible region and the contour of objective function on the graph. (10 Points)

Answer 1

. Given en formation constrained optimization problem min (1,42) = 2x, ² + 9x2 + 2x2 - 6x, 12-1842 subject to 4x2-3%, 520 4, +212 = 0 -2, 2o, 2220 19 first of all We plot' objective function aq well as constraints. 1. Marco UL! fig (1) figca) figis & fig (2) that are objective function as well as constraints are convex functions. Since the problem is minimization problem It is called Convex optimization problem. , (6) The Lagrange function, for the problem is given by foor &=(2,12] ER², define Lord) e 2x, +9 x 2² - 6+, 22-18+, +242 +21 (4x, ~ 3x-20) tie (9 +244-10)

1 162,A) = 2x + 942 - 6x, x2 + x, (4), + 2₂-18) +*2 (-37, +2/2+0)–(20), +0dd); (e) for the Lagrangian function : 1 Lex,d) = f(x)+ 1 g(x) The kkt conditions are given by xt is a (e4/ กฤตา : If there fristas a unique x* such that 9xL (**, ** )=0 AAL (e*, &*)=0 (or) feuivalently, vx 1 (**, 4*) -0 * g(**) = 6 for given minimization problem, we have wx4(x*.7*)-( 4x,*-67,*+ 41,*+1*-18, 180,*-6%,*-34+24,*+9) 1,* (4x,* - 38,*-20) +0 -> de* (24*+*,*-10) -o → C1,* , 12") so ccan ③ to 4 D to @given the pelet conditions for problem, Please "upuott", thank you.