3. (Total: 10 points) In this question you deal with the quadratic optimization problem 1 min...

 

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3. (Total: 10 points) In this question you deal with the quadratic optimization problem 1 min xHx, subject to x F XER 2 for different choices of the feasible set F. The matrix H is 1 -1 -1 - H = -1 2 1 1 For each case of F (cases (a), (b), (c) below) you should answer the question: "Is there a vector x* E F such that f(x*) In other words is there a solution to min f(x for all x F?" 1 T -xHx ? 2x If your answer is "yes" then you should also calculate all minimizers x*. You should carefully justify your answers. a) no constraint: F = R [2] b) one constraint: F = {x R | x1 x2 + x3 = 0} [4] c) two constraints: F = {x R | x1 + x2 = 0, and x2 + x3 = 0} [4] 3. (Total: 10 points) In this question you deal with the quadratic optimization problem 1 min xHx, subject to x F XER 2 for different choices of the feasible set F. The matrix H is 1 -1 -1 - H = -1 2 1 1 For each case of F (cases (a), (b), (c) below) you should answer the question: "Is there a vector x* E F such that f(x*) In other words is there a solution to min f(x for all x F?" 1 T -xHx ? 2x If your answer is "yes" then you should also calculate all minimizers x*. You should carefully justify your answers. a) no constraint: F = R [2] b) one constraint: F = {x R | x1 x2 + x3 = 0} [4] c) two constraints: F = {x R | x1 + x2 = 0, and x2 + x3 = 0} [4]