2011年下学期最优化理论与方法考试试卷(A)

中南大学考试试卷

2011--2012学年 1 学期 时间100分钟

最优化理论与方法 课程 48 学时 学分 考试形式： 闭 卷

专业年级： 信科08、应数08 总分100分，占总评成绩 70 %

注：此页不作答题纸，请将答案写在答题纸上,可用中英文作答。

1.（15 points ） For an unconstrained optimization problem:

),(min x f

Let )0(x be a given point, )0(d be a descent search direction at )0(x .

(1) With the exact line search, show that there is a steplength 0α satisfying

.0)()0()0(0)0(=+∇d d x f T α

(2)Show that when applied to a quadratic objective function, the Newton method with the exact line search terminates in at most one iteration.

2. （15 points ）For an unconstrained optimization problem:

.2)(min 2

221x x x f +=

(1) Find a descent direction )0(d of f at .)1,1()

0(T x

=

(2) By the Armijo line search, find a steplength 0α along )0(d at .)0(x

3.（15 points ） （1）Let .2113⎪⎪⎭⎫

⎝⎛=A Find two directions 1d and 2d such that 1d

and 2d are conjugate with respect to the matrix A .

（2）Show that when applied to a quadratic objective function, with the exact line

search, the PRP conjugate gradient method is equivalent to the FR conjugate gradient method.

4. （15 points ） Apply the BFGS method associated with the exact line search to solve the following problem:

.2)(min 1212

221x x x x x x f --+=

The initial point .)0,1()0(T x =

5. （10 points ） Apply the barrier penalty function method associated to solve the following problem:

.

01.

.2

1)(min 212

2221≥-+-+=x x t s x x x x f The initial point .)1,1()0(T x =

6. （15 points ） For a constrained optimization problem:

.

0,0,025,07.

.4)(min 2122

21

212

2

21≥≥≥--≤-+-=x x x x x x t s x x x f

(1) Identify the active constraint(s) at .)4,3()0(T x = (2) Find the set of linearized feasible direction at .)0(x

7. （15 points ） (1) Compute the gradient value 0g of the objective function:

32212

3222122)(x x x x x x x x f --++=

at .)1,1,1()0(T x =

(2) Compute the projection of 0g on the feasible region

}.0,02|{31213=-=+∈=Ωx x x x R x