Math 171B: Mathematical Programming
Instructor: Jennifer Erway
Spring Quarter 2007
Homework Assignment #5
Due: Friday, May 4
The starred exercises are those that require the use of MATLAB Remember, you MUST do the MATLAB
problems to get credit for the homework.
Exercise 5.1 Let q(x), x ∈Rn, be the quadratic function q(x) = cTx+1
2xTHx, where His symmetric.
(a). Write down an expression for ∇q(x) in terms of c, H and x.
(b). Given an arbitrary point x0and a direction p, write down the Taylor-series expansion of q(x0+p).
(c). For this part, consider q(x) such that His positive definite. If pis a direction such that ∇q(x0)Tp < 0,
show that there exists a positive minimizer α∗of q(x0+αp). Derive a closed-form expression for α∗.
Exercise 5.2 Let f(x) denote a convex continuously differentiable function. Show that if a stationary point
x∗exists then f(x∗) is the unique global minimum of f. (Hint: Use the result that f(x) is convex if and
only if f(y)≥f(x) + f0(x)(y−x) for all xand y).
Exercise 5.3 Write down the function value, gradient vector, and Hessian matrix of the function f(x) =
7x1−x2+ 3x2
1−x2
2at the point (−1,−1)T. If a stationary point exists, find its position and state the nature
of the stationarity. Does f(x) have a bounded minimizer?
Exercise 5.4* Let q(x) be a function of the form q(x) = cTx+1
2xTHx, where cis a vector and His
a constant symmetric matrix. In each of the problems given below, (i) determine if a stationary point x∗
of q(x) exists; (ii) if x∗exists, compute its value and find if it is a minimizer of q; (iii) if x∗is a minimizer
of q(x), find the minimum value of q; (iv) if x∗is a stationary point that is not a minimizer of q, find a
direction psuch that q(x∗+αp)≤q(x∗), for all α≥0.
(a)
c=
1
3
−7
1
, H =
4443
4733
4353
3333
(b)
c=
1
1
1
1
, H =
6420
4733
2353
0333
(c)
c=
1
1
2
, H =
213
132
324
(d)
c=
0
1
5
5
, H =
11 7 3 −2
7 6 5 2
3 5 7 6
−226−7
1