Important terms and ideas for discussing feasible regions in LP problems
[Chapter 2 of "Strategic Mathematics"]
The shape of the feasible region in LP programs is an important feature that makes possible the techniques
used to solve LP problems fairly routinely. We will develop the theory that justifies these methods.
Here is a list of many of the critical terms, ideas, and theorems on which the theory is based [details in
Strategic Mathematics, Chapters 2 & 3]
Definition 2.1.1: For two points X and Y (which may also be considered to be vectors) in Rn, the line
determined by X and Y is {tX + (1-t)Y| t  R}
The segment joining X and Y is the set {tX + (1-t)Y| t  [0,1]} (the subset of the line formed by the
points X & Y & the points between them)
Definition 2.1.6: A set S  Rn is convex if, for every pair of points X and Y in S, the segment joining
X and Y is also in S.
[We will be working with linear equations and functions, and matrix representations; notice this requirement
has strong connections to the idea of linear operations. In particular, the feasible set for an LP problem will
always be a convex set]
Theorem 2.1.8 The intersection of a finite collection of convex sets is convex.
Theorem 2.1.9 For an mxn matrix A and a vector b in Rn, the solution set for the system of linear
equations AX = b is a convex set
Definition 2.1.10 A hyperplane in Rn, is a set H(C,k) = {X  Rn | CTX = k} for some non-zero vector C
in Rn, and constant k. [Note: This says a hyperplane is the solution set for one linear equation in n
unknowns – an n-1 – dimensional subset. In R2 it is a line, in R3 a plane If k = 0, this is an n-1
dimensional subspace of RN]
Theorem 2.1.11 A subset H of Rn is a hyperplane if and only if there is a subspace S of Rn for which H
can be written as H = S + X [= { P + X | P  S} – the algebraic sum] for every X  H.
[Note that this exactly matches what happens with solutions of a system of linear equations – a particular
solution plus any element of a subspace.]
Definition2.1.13, 2.1.14: For any n—dimensional vector (coefficient vector) C and any number k, we have
h+(C,k) = {X  Rn| CTX > k} – the open upper half-space
h-(C,k) = {X  Rn| CTX < k} – the open lower half-space
H+(C,k) = {X  Rn| CTX ≥ k} – the closed upper half-space
H-(C,k) = {X  Rn| CTX ≤ k} – the closed lower half-space
Note that H(C,k) = H+(C,k) H-(C,k) and that Rn,= h+(C,k) H(C,k) h-(C,k) [A hyperplane and its
halspaces completely cover Rn]
Half-spaces are solution sets for linear inequalities and hyperplanes are solution sets for linear equations –
the basic building blocks of the feasible regions for linear programming problems – it is significant that they
Math 438 Geometry of Rn p.2
are convex (and so intersections – which are solutions of systems of inequalities and equations - are also
convex sets.)
Theorem 2.1.16 Half-spaces are convex sets
Corollary 2.1.17 Hyperplanes are convex sets
Corollary 2.1.19 The solution set for m linear inequalities in n unknowns is a convex set in Rn.
Definition 2.1.21 A set S is a convex polyhedron if it is the intersection of half spaces.
Corollary 2.1.22 The fesible set for a linear programming problem is a convex polyhedron
[These are the sets that can be feasible regions for LP problems, remember – and are convex sets]
There are several “hulls” that may be formed using a set of points/vectors in Rn. All are convex sets which
include the original points [Thus “hull” – as on a nut or a ship]. The conical hull and the convex hull are of
particular importance for proving that solutions for linear programming problems can be found only in
certain places – reducing the search to a finite collection of options. [Compare max/min problems on closed
intervals in Calc I – if the function is continuous, we need only check values at critical points and endpoints
– a finite set of points]
Definition 2.2.1 (summarized)
A linear combination of vectors from some space Rn is called
an affine combination if the coefficients total 1
a conical combination if all the coefficients are nonnegative
a convex combination if both conditions are met (coefficients all nonnegative and add up to 1)
Definition 2.2.2:
For a set S of vectors in some Rn,
the linear hull L(S) is the set of all linear combinations of vectors in S (the span of S)
the affine hull Aff(S) is the set of all affine combinations of members of S
the conical hull Coni(S) is the set of all conical combinations of members of S
the convex hull Conv(S) is the set of all convex combinations of members of S
We will be using the convex hull of a set of points and the conical hull of a related set of points to describe
feasible regions of LP problems
Theorem 2.2.7 Conv(S) is a convex set.
Definition 2.2.8 The convex hull of a finite set of points in Rn is called a convex polytope
Definition 2.2.9 A point X in a convex set S is an extreme point of S if it cannot be written as a convex
combination of two distinct points of S
[That is, X is not on the segment determined by two other points of S - whenever we write X = tA + (1t)B with 0 < t < 1(so neither A nor B is equal to X) then at least one of A , B is not in S]
Definition 2.2.11 A bounding ray of a convex polyhedron S is a ray R for which
a.) The vertex of R is an extreme point of S
Math 438 Geometry of Rn p.3
b.) A point X’ of R can be written as a convex combination of distinct points X, Y of S only if X, Y are
on R.
A direction vector of a convex polyhedron is a direction vector of one of its bounding rays.
Definition 2.2.14 A set S in Rn, is a convex cone with vertex at the origin if, for every pair of non-negative
real numbers a and b and every X, Y  S we have aX + bY  S [that is , if every conical
combination of vectors in S [including the origin] is also in S]
[This describes a set which is an infinite cone – or multi-sided [infinite] pyramid – with the peak at the
origin]
Theorem2.2.15 The conical hull Coni(S) of any set S in Rn is a convex cone with vertex at the origin.
Theorem 2.2.16 The conical hull of any set is a convex set.
Theorem2.2.14 [Finite Basis Theorem] For any convex polyhedron F in Rn, If the extreme points are X1, .
. . Xd and the direction vectors are Y1, . . . Yt, then F = Conv({X1, . . . Xd }) + Coni ({Y1, . . . Yt })
This says that a convex polyhedron [any feasible region for any LP problem] can be completely described in
terms of
1. the convex hull of its extreme points and
2. the conical hull of its direction vectors .
Thus we are looking at linear combinations (of special types) of finitely many objects (points and vectors) –
this gives a handle on what points we must test as solutions, and some hope for developing a testing method.
– This will allow us to prove (in Chapter 3)
Theorem 3.1.1 The Extreme Point Theorem If the linear function f(X) = CTX is bounded below (has a
lower bound) on the convex polyhedron F = {X | AX ≤ b, X ≥ 0} then f(X) attains a minimum and this
occurs at one of the extreme points of F.
[Once we prove this, we can focus on setting up search methods – “plug all the extreme points into the
function” is not, typically, a reasonable approach – unlike the Calc I max/min problems, we are likely to
have many extreme points to look at and many variables to keep track of ]