Mathematics for Business Administration: Multivariable Optimization

327 Pages·2014·3.27 MB·English

Mathematics for Business Administration:
Multivariable Optimization
Universidad de Murcia
Mar´ıa Pilar Mart´ınez-Garc´ıa
UniversidaddeMurcia MathematicsforBusinessAdministration:MultivariableOptimization
These slides are intended for students of Business administration
whose mathematical requirements go beyond the calculus for
functions of one variable. The material includes a basic course on
multivariable optimization problems, with and without constraints,
and the tools of linear algebra needed for solving them.
UniversidaddeMurcia MathematicsforBusinessAdministration:MultivariableOptimization
Contents
Chapter One: Convex sets. Convex and concave functions
Chapter Two: Introduction to multivariable optimization
Chapter Three: Classical optimization
Chapter Four: Constrained Optimization
Appendix: Matrix Algebra and Quadratic Forms
Answers to the Problems
Answers to the Multiple choice questions
UniversidaddeMurcia MathematicsforBusinessAdministration:MultivariableOptimization
ChapterOne:ConvexSets.ConvexandConcaveFunctions
Usefullinks
ReviewproblemsforChapter1
Multiplechoicequestions
Chapter One: Convex Sets. Convex and Concave
Functions
ReturntoContents
ChapterOne:ConvexSets.ConvexandConcaveFunctions
ChapterOne:ConvexSets.ConvexandConcaveFunctions
Usefullinks
ReviewproblemsforChapter1
Multiplechoicequestions
Outline
Convex Sets
Convex and Concave Functions
ChapterOne:ConvexSets.ConvexandConcaveFunctions
ChapterOne:ConvexSets.ConvexandConcaveFunctions
Usefullinks ConvexSets
ReviewproblemsforChapter1 Concaveandconvexfunctions
Multiplechoicequestions
Definition 1
Let x = (x ,x ,...,x ) and y = (y ,y ,...,y ) be any two points
1 2 n 1 2 n
in Rn. The closed line segment between x and y is the set
[x,y] = {z / there exists λ ∈ [0,1] such that z = λx+(1−λ)y}
Definition 2
A set S in Rn is called convex if [x,y] ⊆ S for all x, y in S, or
equivalently, if
λx+(1−λ)y ∈ S for all x, y in S and all λ ∈ [0,1]
Note in particular that the empty set and also any set consisting of
one single point are convex.
ChapterOne:ConvexSets.ConvexandConcaveFunctions
ChapterOne:ConvexSets.ConvexandConcaveFunctions
Usefullinks ConvexSets
ReviewproblemsforChapter1 Concaveandconvexfunctions
Multiplechoicequestions
Intuitively speaking, a convex set must be ”connected” without any
”holes” and its boundary must not ”bend inwards” at any point.
Convex Not convex
ChapterOne:ConvexSets.ConvexandConcaveFunctions
ChapterOne:ConvexSets.ConvexandConcaveFunctions
Usefullinks ConvexSets
ReviewproblemsforChapter1 Concaveandconvexfunctions
Multiplechoicequestions
Definition 3
A hyperplane in Rn is the set H of all points x = (x ,x ,...,x ) in
1 2 n
Rn that satisfy
p x +p x +···+p x = m
1 1 2 2 n n
where p = (p ,p ,...,p ) (cid:54)= 0.
1 2 n
Proposition 4 A hyperplane in Rn is a convex set.
ChapterOne:ConvexSets.ConvexandConcaveFunctions
ChapterOne:ConvexSets.ConvexandConcaveFunctions
Usefullinks ConvexSets
ReviewproblemsforChapter1 Concaveandconvexfunctions
Multiplechoicequestions
Definition 5
A hyperplane H devides Rn into two sets,
H = {(x ,x ,...,x ) ∈ Rn/ p x +p x +···+p x ≥ m},
+ 1 2 n 1 1 2 2 n n
H = {(x ,x ,...,x ) ∈ Rn/ p x +p x +···+p x ≤ m},
− 1 2 n 1 1 2 2 n n
which are called half spaces.
Proposition 6 H and H are convex sets.
+ −
ChapterOne:ConvexSets.ConvexandConcaveFunctions
ChapterOne:ConvexSets.ConvexandConcaveFunctions
Usefullinks ConvexSets
ReviewproblemsforChapter1 Concaveandconvexfunctions
Multiplechoicequestions
Proposition 7 If S and T are two convex sets in Rn, then their
intersection S ∩T is also convex.
The union of convex sets is usually not convex.
ChapterOne:ConvexSets.ConvexandConcaveFunctions