Question

2. (40 marks) Consider the problem of finding lower and upper bounds for the function f(x) = x1^3 + x2^3 + x3^3 . (a) Find the gradient vector and Hessian matrix of f, and state the circumstances under which the Hessian is positive definite or negative definite. (b) Consider the following constrained optimisation problem min(x1^3 + x2^3 + x3^3) subject to(x1 + x2 + x3)^2 = 96 x1^2 + x2^2 + x3^2 = 48 i. Explain why finding a solution to this minimization problem can also provide a solution to the corresponding maximisation problem. ii. Find an expression for the Lagrangian. iii. State the KKT conditions, and find all solutions for x and the Lagrange multipliers. iv. Are there any local or global maxima? You may use MATLAB or R to support your answer for this part if you like. HINT: x1^3 + x2^3 + x3^3 = x1^3 + 3/2(x2 + x3)(x2^2 + x3^2) - 1/2(x2 + x3)^3. (c) Repeat (b) for the modified problem min(x1^3 + x2^3 + x3^3) subject to(x1 + x2 + x3)^2 ? 96 x1^2 + x2^2 + x3^2 ? 48

          2. (40 marks) Consider the problem of finding lower and upper bounds for the function
f(x) = x1^3 + x2^3 + x3^3 .
(a) Find the gradient vector and Hessian matrix of f, and state the circumstances under which the Hessian is positive definite or negative definite.
(b) Consider the following constrained optimisation problem
min(x1^3 + x2^3 + x3^3)
subject to(x1 + x2 + x3)^2 = 96
x1^2 + x2^2 + x3^2 = 48
i. Explain why finding a solution to this minimization problem can also provide a solution to the corresponding maximisation problem.
ii. Find an expression for the Lagrangian.
iii. State the KKT conditions, and find all solutions for x and the Lagrange multipliers.
iv. Are there any local or global maxima? You may use MATLAB or R to support your answer for this part if you like.
HINT: x1^3 + x2^3 + x3^3 = x1^3 + 3/2(x2 + x3)(x2^2 + x3^2) - 1/2(x2 + x3)^3.
(c) Repeat (b) for the modified problem
min(x1^3 + x2^3 + x3^3)
subject to(x1 + x2 + x3)^2 ? 96
x1^2 + x2^2 + x3^2 ? 48

2. (40 marks) Consider the problem of finding lower and upper bounds for the function
f(x) = x1^3 + x2^3 + x3^3 .
(a) Find the gradient vector and Hessian matrix of f, and state the circumstances under which the Hessian is positive definite or negative definite.
(b) Consider the following constrained optimisation problem
min(x1^3 + x2^3 + x3^3)
subject to(x1 + x2 + x3)^2 = 96
x1^2 + x2^2 + x3^2 = 48
i. Explain why finding a solution to this minimization problem can also provide a solution to the corresponding maximisation problem.
ii. Find an expression for the Lagrangian.
iii. State the KKT conditions, and find all solutions for x and the Lagrange multipliers.
iv. Are there any local or global maxima? You may use MATLAB or R to support your answer for this part if you like.
HINT: x1^3 + x2^3 + x3^3 = x1^3 + 3/2(x2 + x3)(x2^2 + x3^2) - 1/2(x2 + x3)^3.
(c) Repeat (b) for the modified problem
min(x1^3 + x2^3 + x3^3)
subject to(x1 + x2 + x3)^2 ? 96
x1^2 + x2^2 + x3^2 ? 48

Added by Carlos P.

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