Introduction to Optimization — Serie 4 - IFOR

Institute for Operations Research
ETH Zurich HG G21-23
Dennis Michaels dennis.michaels@ifor.math.ethz.ch
Kathrin Ballerstein kathrin.ballerstein@ifor.math.ethz.ch
Spring term 2011
Introduction to Optimization — Serie 4
http://www.ifor.math.ethz.ch/teaching/lectures/intro_ss11
Submission: Wednesday, Mar 23, 2011 (before the exercise)
HG G 21: Box “Einführung in die Optimierung”
Exercise 11: LP Modeling
Consider the polyhedron P = {x ∈ R n | a ′ i x ≤ b i, i = 1,...,m}. A ball B with center y and radius r
is defined as the set of all points within Euclidean distance r from y, i.e. B = {x | ‖y−x‖ 2 ≤ r}. We
are interested in finding a ball with the largest possible radius, which is entirely contained in P. The
center of such a ball is called the Chebychev center of P.
a)Provide a Linear Program for the problem of finding the Chebyshev center of P.
b)Solve your LP with a Linear Optimization software (e.g. CPLEX) for the polyhedron P = {x ∈
IR 3 | Ax ≤ b}, where ⎛ ⎞ ⎛ ⎞
4 −2 4 4
−2 1 2
16
A =
2 4 −4
⎜−1 0 0
b =
16
⎟ ⎜0
.
⎟
⎝ 0 −1 0 ⎠ ⎝0⎠
0 0 −1 0
(5 points)
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Exercise 12: LP Optimality
Find all necessary conditions on the parameters s,t ∈ R such that the Linear Program given by
maximize x 1 +x 2
subject to sx 1 +tx 2 ≤ 1
x 1 ,x 2 ≥ 0
a) is unbounded, c) has multiple optimal solutions,
b) is infeasible, d) has one unique optimal solution.
In particular, argue for each case why your conditions obtained are the only ones.
(4 points)
Exercise 13: Optimal solutions of LPs in standard form
Consider a polyhedron P = {x ∈ IR n | Ax = b,x ≥ 0} in standard form. Suppose that the matrix
A ∈ IR m×n and that its rows are linearly independent. For each one of the following statements, state
whether it is true or false. If true, provide a proof, else provide a counterexample.
(a)If n = m+1, then P has at most two basic feasible solutions.
(b)The set of all optimal solutions is bounded.
(c)At every optimal solution, no more than m variables can be positive.
(d)If there is more than one optimal solution, then there are uncountably many optimal solutions.
(e)If there are several optimal solutions, then there exist at least two basic feasible solutions that
are optimal.
(f)Consider the problem of minimizingmax{c ′ x,d ′ x} overP. If the problem has an optimal solution,
it must have an optimal solution which is an extreme point of P.
(6 points)
Exercise 14: Extreme points of polyhedra
Let P := {x ∈ R n | Ax ≤ b} be a bounded polyhedron, let c be a vector in R n , and let γ be some
scalar. We define
Q := {x ∈ P | c ′ x = γ}.
Assume that Q is nonempty. Show that every extreme point of Q is either an extreme point of P or
a convex combination of two adjacent extreme points of P.
(5 points)
Certificate conditions: Bachelor/Master students: To obtain the ‘Testat’ at least 50% of both the first seven assignments
and last six assignments have to be solved correctly.
PhD students: To obtain the ‘credit points’ for this course, at least 60% of both the first seven assignments and the last six
assignments have to be solved correctly.
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