Math 171A Homework Assignment # 4
Instructor: Jiawang Nie
Assigned Date: February 3, 2010 Due Date: February 10, 2010
1. (4 points) Consider the matrix of active constraints of a certain LP
Aa=
−1−8 4 −7−3
−4 2 0 4 −2
−6 0 −3 3 5
00527
.
Use Matlab to find a direction palong which the residual of the third constraint increases,
but all the other residuals remain the same. If the rows of Aaare labeled 1 through 4, what
are the indices of the active set after a positive step along this direction?
2. (5 points) Let Fbe the feasible set defined as
F={x∈R2:|x1|+ 2|x2| ≤ 2,|x1| ≤ 1,|x2| ≤ 1}.
Is the point ¯x= (1,0) feasible for F? Is this ¯xa vertex? If not, find a vertex from it using
the method described in class. Explain your reasons.
3. (4 points) Assume that pis a vector in the null-space of a matrix A. Show that if ais
any vector such that aTp6= 0, then amust be linearly independent of the rows of A. Does
the converse result hold, i.e., if ais linearly independent of the rows of Aand plies in the
null-space of A, is aTp6= 0?
4. (4 points) Consider the feasible region Fof points in Rnsatisfying the constraints Ax ≥b,
where Ais a nonzero m×nmatrix. Assume that Fcontains at least one point. Show that,
if a nonzero vector pexists such that Ap ≥0, then F is unbounded.
5. (3 points) Show that the feasible region Fdefined by Ax ≥band x≥0 is either empty or
convex.
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