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KKT is karush kuhn tucker
Question 5 [15 marks] (Chapters 5, 6, 7 and 11) Consider...
(45 Points) Consider the constrained optimization problem: min f(x1, x2) = 2x} + 9x2 + 9x2...
(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...
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn...
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn be an (n x n) matrix and be R. Consider the problem 1 (P2) min2+ s.t. xe R" 1Ax-bil2 1 where & > O is fixed and Il IIl denot es the 2-norm. Call g.(x)=l|2 the objective function of problem (P2) 1Ax-bl2 i) [3 marks] Compute the gradient of g, and use it to show that the solution xi of this problem verifies (I+EATA)(x)...
Question 8 (Chapters 6-7) 12+2+2+3+2+4+4-19 marks] Let 0メS C Rn and fix E S. For a E R consider the following optimization problem: (Pa) min a r, and define the set K(S,x*) := {a E Rn : x. is a...
Question 8 (Chapters 6-7) 12+2+2+3+2+4+4-19 marks] Let 0メS C Rn and fix E S. For a E R consider the following optimization problem: (Pa) min a r, and define the set K(S,x*) := {a E Rn : x. is a solution of (PJ) (a) Prove that K(S,'). Hint: Check 0 (b) Prove that K(S, r*) is a cone. (c) Prove that K(S,) is convex d) Let S C S2 and fix eS. Prove that K(S2, ) cK(S, (e) Ifx. E...
x y 5 6 6 9 7 11 8 13 9 14 10 15 11 15...
x y 5 6 6 9 7 11 8 13 9 14 10 15 11 15 12 13 a) Generate a model for y as a function of x b) Is this model useful? Justify your conclusion based on i) R2 adjusted, ii) Hypothesis test for model coefficient, iii) overall model adequacy test and iv) regression assumptions c) If needed, modify model as appropriate and generate the new model. *Complete all parts of the problem please, be as detailed with...
Question 7 (Chapters 6-7) 2+2+2+3+2+4+4-19 mark Let 0メs c Rn and fix r' E S. For a R" consider the followi...
Question 7 (Chapters 6-7) 2+2+2+3+2+4+4-19 mark Let 0メs c Rn and fix r' E S. For a R" consider the following optimization problem: (Pa) min ar res and define the set K(S,) (aER z" is a solution of (Pa)) (e) If z' e int(S), prove that K(S, (0) (1) If possible, find a set S CR" and s* E S such that K(S,) (g) Let SB, 0.1] (rR l2l3 1) (the closed (, unit ball) and consider (1,0)7. Prove that...
hi i need answer from part d Question 2 (48 marks) Consider a firm which produces a good, y, using two factors of...
hi i need answer from part d
Question 2 (48 marks) Consider a firm which produces a good, y, using two factors of production, xi and x2 The firm's production function is Note that (4) is a special case of the production function in Question 1, in which α-1/2 and β-14. Consequently, any properties that the production function in Q1 has been shown to possess, must also be possessed by the production function defined in (4). The firm faces exogenously...
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" -...
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" - 6y' + 9y = 5e3x is given by yp = Cre3x where C is an undetermined constant. (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the "method of undetermined series coefficients". (iii) Most of the material in Lecture Notes from Week 3 to Week 5, inclusive, can be extended or generalized to higher-order ODES...
2. Consider the following four consumers (C1,C2,C3,C4) with the following utility functions: Consumer Utility Function C1...
2. Consider the following four consumers (C1,C2,C3,C4) with the following utility functions: Consumer Utility Function C1 u(x,y) = 2x+2y C2 u(x,y) = x^3/4y^1/4 C3 u(x,y) = min(x,y) C4 u(x,y) = min(4x,3y) On the appropriate graph, draw each consumer’s indifference curves through the following points: (2,2), (4,4), (6,6) and (8,8), AND label the utility level of each curve. Hint: Each grid should have 4 curves on it representing the same preferences but with different utility levels. 3. In the following parts,...
(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...
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn be an (n x n) matrix and be R. Consider the problem 1 (P2) min2+ s.t. xe R" 1Ax-bil2 1 where & > O is fixed and Il IIl denot es the 2-norm. Call g.(x)=l|2 the objective function of problem (P2) 1Ax-bl2 i) [3 marks] Compute the gradient of g, and use it to show that the solution xi of this problem verifies (I+EATA)(x)...
Question 8 (Chapters 6-7) 12+2+2+3+2+4+4-19 marks] Let 0メS C Rn and fix E S. For a E R consider the following optimization problem: (Pa) min a r, and define the set K(S,x*) := {a E Rn : x. is a solution of (PJ) (a) Prove that K(S,'). Hint: Check 0 (b) Prove that K(S, r*) is a cone. (c) Prove that K(S,) is convex d) Let S C S2 and fix eS. Prove that K(S2, ) cK(S, (e) Ifx. E...
Question 7 (Chapters 6-7) 2+2+2+3+2+4+4-19 mark Let 0メs c Rn and fix r' E S. For a R" consider the following optimization problem: (Pa) min ar res and define the set K(S,) (aER z" is a solution of (Pa)) (e) If z' e int(S), prove that K(S, (0) (1) If possible, find a set S CR" and s* E S such that K(S,) (g) Let SB, 0.1] (rR l2l3 1) (the closed (, unit ball) and consider (1,0)7. Prove that...
hi i need answer from part d
Question 2 (48 marks) Consider a firm which produces a good, y, using two factors of production, xi and x2 The firm's production function is Note that (4) is a special case of the production function in Question 1, in which α-1/2 and β-14. Consequently, any properties that the production function in Q1 has been shown to possess, must also be possessed by the production function defined in (4). The firm faces exogenously...
Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" - 6y' + 9y = 5e3x is given by yp = Cre3x where C is an undetermined constant. (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the "method of undetermined series coefficients". (iii) Most of the material in Lecture Notes from Week 3 to Week 5, inclusive, can be extended or generalized to higher-order ODES...
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