The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
Chapter 2
Generalized Convex Functions
Convexity is one of the most frequently used hypotheses in optimization theory. It
is usually introduced to give global validity to propositions otherwise only locally
true, for instance, a local minimum is also a gl
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
The Simplex Method
The Simplex Method provides an algorithm for calculating solutions of the canonical linear
programming problem
LP:Min cfw_cTx Ax = b, x 0.
The Simplex Method is actually an application of Gaussian elimination based on the information gi
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 1
(i.) Convert the following problems to linear programs.
z = x + y + z
min
x+y 1
s.t
2x + z = 3
min
s.t
z = maxcfw_cT1 x + d1 , , cTp x + dp
Ax = b
x0
(ii.) Solve the following linear programs graphically and find all the ext
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 5
(i.) To approximate a function g over the interval [0, 1] by a polynomial function p(x) =
an xn + an1 xn1 + + a0 , we minimize the criterion
1
f (a) =
[g(x) p(x)]2 dx.
0
Find the equations satised by the optimal coecients a = (a0
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 3
(i.) Consider the following linear programming problems
max
z = cT x
s.t.
Ax b
min
z = cT x
s.t.
Ax b
(1)
(2)
(a) Write the duals to these problems.
(b) If both (1) and (2) are feasible, prove that if one of these primal problems h
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 6
(i.) Consider a constrained minimization problem
min
f (x) = 3x1 + 4x2
s.t.
(x1 + 1)2 + x22 = 1
(x1 1)2 + x22 = 1.
It is obvious that x = (0, 0) is the only feasible point and thus the optimal solution.
(a) Is x a regular point?
(b
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 7
(i.) Consider a zerosum game played by two players, Peter and Harriet, where Peter has a
set of possible strategies X and Harriet has a set of possible strategies Y . If Peter chooses
a strategy x X and Harriet chooses a strategy
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 2
(i.) Prove that the set of optimal solutions to a linear programming problem is a convex set.
(ii.) Let A Rmn and b Rm . Show that,
(a) if cfw_y Rm : AT y 0, bT y > 0 is nonempty, then cfw_x Rn : Ax = b, x 0 is empty;
(b) if cfw_y
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
41
Algebra and Duality
4. Algebra and Duality
Example: nonconvex polynomial optimization
Weak duality and duality gap
The dual is not intrinsic
The cone of valid inequalities
Algebraic geometry
The cone generated by a set of polynomials
An algebr
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 0
(i.) Suppose f is a twice differentiable function defined on [a, b] with f 0 (a) = f 0 (b) = 0. Show that
there exists an such that
f 00 ()
4
f (b) f (a).
(b a)2
(ii.) Suppose f (n+1) (x) 6= 0 for all x R. A Taylor series can
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 2
(i.) Consider the linear program
min
z = cT x
s.t.
Ax + Is = e
x, s 0
where cT = (0, ., 0, ), eT = (1, ., 1), is a small positive number, say = 250 , and
1
2
A=
1
2
1
.
.
.
.
2
1
is an n n square matrix. So there are 2n variables.
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 3
(i.) Prove the following complementary slackness properties.
(a) If x is optimal for the primal and y is optimal for the dual, then xT (c AT y) = 0
and y T (Ax b) = 0.
(b) If x and y are feasible for their respective problems, and
The Hong Kong University of Science and Technology
Deterministic Models in Operations Research
IELM 5230

Fall 2015
1
Homework Assignment 1
(i.) Convert the following problems to linear programs.
z = x + y + z
min
x+y 1
s.t
2x + z = 3
min
s.t
z = maxcfw_cT1 x + d1 , , cTp x + dp
Ax = b
x0
(ii.) Solve the following linear programs graphically and find all the ext