**Advanced Lectures on Systems Engineering:** **Exam** **questions**

22th Jan. 2020, K. Taji

**1.** Let 0 *< α <* 1 be a parameter. Then verify that the following function defined on
the set*S* =*{x∈ ℜ*^{2} *|*0*≤x*_{1} *≤*1, 0*≤x*_{2} *≤*1*}* is convex or not.

*f** _{α}*(x) =

*−x*1+ 1 if 1 +*α < x*1+*x*2

*−*2x_{1}*−x*_{2}+*α*+ 2 if *x*_{1}+*x*_{2} *≤*1 +*α, α* *≤x*_{2}
2(*−x*_{1}*−x*_{2}+*α*+ 1) if *x*_{2} *≤α*

**2.** Let consider the following quadratic programming problem for given*n*^{2} real numbers
*a** _{ij}*(i, j = 1, . . . , n).

min 1 2

∑*n*
*i=1*

∑*n*
*j=1*

(x*i**−x**j* *−a**ij*)^{2}
s.t.

∑*n*
*i=1*

*x** _{i}* = 0
1. Write the KKT condition of the problem.

2. Show that the optimal solution is unique for any *a** _{ij}*(i, j = 1, . . . , n).

3. What will be the solution set if the constraint

∑*n*
*i=1*

*x** _{i}* = 0 is deleted?

**3.** Consider the following nonlinear optimization problem.

min *−*2x_{1}*x*^{2}_{2}

s.t. ^{1}_{2}*x*^{2}_{1}+*x*^{2}_{2}*−* ^{3}_{2} *≤*0
*x*_{1} *≥*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 setting*X* =

( *x*_{1} *x*_{2}
*x*_{2} *x*_{3}

)

.

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.