Math 164, Lecture 2, Vese
Homework #3, due on Wednesday, October 19, 2005
Remarks:
•each time, solve as many problems as you can from the textbook
•review Sections 3.1 and 4.1 from the textbook.
•T. A. Tristan Roy new office hours: Tuesdays 2-3pm (right after D.S.)
[1] Consider a feasible region Sdefined by a set of linear constraints
S={x:Ax ≤b},
where Ais an m×nmatrix and bis a column vector.
(a) Prove that Sis convex.
(b) Derive the conditions that must be satisfied by a feasible direction p
at a point x∈S.
[2] Consider the problem
minimize f(x)
subject to x1+ 2x2+ 3x3= 6, x1≥0, x2≥0, x3≥0.
Find the sets of all feasible directions at points xa= (0,0,2)T,xb=
(3,0,1)T, and xc= (1,1,1)T.
[3] Solve the following linear program graphically:
minimize
z= 3x1+x2,
subject to
x1−x2≤1,
3x1+ 2x2≤12,
2x1+ 3x2≤3,
−2x1+ 3x2≥9,
x1, x2≥0.
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