Advanced Lectures on Systems Engineering: Exam questions

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22th Jan. 2020, K. Taji

1. Let 0 < α < 1 be a parameter. Then verify that the following function defined on the setS ={x∈ ℜ2 |0≤x1 1, 0≤x2 1} is convex or not.

fα(x) =

−x1+ 1 if 1 +α < x1+x2

2x1−x2+α+ 2 if x1+x2 1 +α, α ≤x2 2(−x1−x2+α+ 1) if x2 ≤α

2. Let consider the following quadratic programming problem for givenn2 real numbers aij(i, j = 1, . . . , n).

min 1 2

n i=1

n j=1

(xi−xj −aij)2 s.t.

n i=1

xi = 0 1. Write the KKT condition of the problem.

2. Show that the optimal solution is unique for any aij(i, j = 1, . . . , n).

3. What will be the solution set if the constraint

n i=1

xi = 0 is deleted?

3. Consider the following nonlinear optimization problem.

min 2x1x22

s.t. 12x21+x22 32 0 x1 0

1. Find the global minimizer with verification.

2. Discuss whether the second order suﬃcient condition at the global minimizer holds or not.

4. Consider the following semidefinite programming problem (SDP).

min

⟨( 0 1/2 1/2 1

)

, X

s.t.

⟨( 1 0 0 1

)

, X

= 2

X is a 2×2 symmetric positive semidefinite matrix.

1. Find the equivalent nonlinear programming problem by settingX =

( x1 x2 x2 x3

)

.

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2. Find an interior feasible point.

3. Find the global optimal minimizer.

4. Find the dual problem as the formulate of SDP (Use S and λ as variables).

5. Find an interior feasible point of the dual problem.

6. Find the optimal solution of the dual problem.

Deadline: 31st Jan. 2020. To report box at Kikai-Koukuu oﬃce room.

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