Problem 8(A) [10 marks]: This problem is concerned with the relations among Hessian matrix, positive curvature (convexity), negative curvature (concavity). Consider the following quadratic form Q(x, y, z) = ax^2 + 4ay^2 + 4az^2 + 4xy + 2axz + 4yz with a ? ?^1 as its parameter. Write down its Hessian matrix; specify the values of a such that Q is positive definite, negative definite; and specify in which case quadratic form has positive curvature (convexity), negative curvature (concavity). You may apply the so-called principal minors methods (cf. any linear algebra textbook for such method) to determine a given symmetric matrix is positive/negative definite.
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Problem 1 [10 points] What are the values of parameter p such that the function of two real variables f(x, y) = x^2 / y^p is convex in the domain y > 0? Answer: Hint: You should know that a symmetric matrix is positive semidefinite if and only if all its principal minors are nonnegative. In particular, 2 x 2 symmetric matrix [ a b ; b c ] is positive semidefinite if and only if a >= 0, c >= 0, and ac - b^2 >= 0.
Suman K.
Sri K.
Consider the problem of finding lower and upper bounds for the function f(x) = x1^3 + x2^3 + x3^3. Find the gradient vector and Hessian matrix of f, and state the circumstances under which the Hessian is positive definite or negative definite. 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. Explain why finding a solution to this minimization problem can also provide a solution to the corresponding maximisation problem. Find an expression for the Lagrangian. State the KKT conditions and find all solutions for x and the Lagrange multipliers. 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. 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.
Adi S.
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