IE521 Master Homework List Version 02
Professor Terry L. Friesz
Spring 2020
1. Answer this true-false question: every linear function is convex.
2. Answer this true-false question: every linear function is concave.
3. Prove the intersection of convex sets is a convex set.
4. Solve the nonlinear program
max f (x1 ; x2 )
x1 + x2
x1
x2
2
=
2
(x1 ) + (x2 )
6
0
0
using the Kuhn-Tucker conditions.
5. Under what conditions does the directional derivative of f (x) obey
iT
f (x + d) h
= rf (x) d
lim
!0
6. What type of convexity, if any, does this function possess
5
X
( i xi
2
2)
i=1
when
i
i
= ( 2)
7. Prove Farkas’lemma using the simplex algorithm.
8. Answer this true-false question: every linear function is convex.
9. Answer this true-false question: every linear function is concave.
10. Solve this linear program using the Kuhn-Tucker conditions
min Z (x1 ; x2 )
0
1
1
=
3x1 + 2x2
x1 2
x2 2
Compare the Kuhn-Tucker multipliers to the dual variables found using
the Simplex algorithm, and comment on your …ndings.
11. Extend the proof of the Fritz John conditions given herein to include dual
variables for nonbinding constraints.
12. Give an example of a merely quasiconvex function.
13. Give an example of a merely pseudoconvex function.
14. Derive the necessary as well as the su¢ cient conditions for an unconstrained NLP.
15. Using the Kuhn-Tucker conditions, develop a simple method of solving
min f (x)
such that x 2 ;
when f (x) : <1 ! <1 is convex on
= fx : a
x
bg
with a 2 <1+ and b 2 <1+ ; moreover
a<b
Your method should employ the elementary projection operator
8
v>b
< b if
b
v if a v b
[v]a =
:
a if
v<a
16. Give precise statements of these theorems and provide proofs:
(a) the intersect of convex sets is itself a convex set
(b) a linear function is both a convex and a concave function
(c) the level sets of a convex function are convex
17. Supply all missing steps and verify all aspects of the derivation of the
KKT conditions from the Lagrange multiplier rule found on pages 33 to
43 of the "NLP Overview" slides.
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