linear programming 2015 1 chap 2. the geometry of lp
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Linear Programming 2015 1
Chap 2. The Geometry of LP
In the text, polyhedron is defined as . So some of our earlier re-sults should be taken with modifications.
Thm 2.1:
(a) The intersections of convex sets is convex.
(b) Every polyhedron is a convex set.
(c) Convex combination of a finite number of elements of a con-vex set also belongs to that set.
(recall that S closed for convex combination of 2 points.
S closed for convex combination of a finite number of points)
(d) Convex hull of a finite number of vectors (polytope) is con-vex.
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Pf) (a) Let , convex , since convex
is convex.
(b) Halfspace is convex.
Polyhedron is intersection of halfspaces From (a), P is convex.
( or we may directly show .)
(c) Use induction. True for by definition.
Suppose statement holds for elements. Suppose .
Then
and sum up to 1, hence
(d) Let be the convex hull of vectors and
for some .
and sum up to 1 convex combination of
.
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Extreme points, vertices, and b.f.s’s Def: (a) Extreme point ( as we defined earlier)
(b) is a vertex if such that and . ( is a unique optimal solu-tion of min )
(c) Consider polyhedron and . Then is a basic solution if all equality constraints are active at and linearly independent active constraints among the constraints active at .
( basic feasible solution if is basic solution and )
Note: Earlier, we defined the extreme point same as in the text.
Vertex as 0-dimensional face ( dim() + rank ) which is the same as the basic feasible solution defined in the text.
We defined basic solution (and b.f.s) only for the standard LP. (
Definition (b) is new. It gives an equivalent characterization of ex-treme point. (b) can be extended to characterize a face of .
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Fig. 2.6:
Three constraints active at A, B, C, D. Only two constraints ac-tive at E. Note that D is not a basic solution since it does not satisfy the equality constraint. However, if is given as , D is a basic solution by the definition in the text, i.e. whether a solution is basic depends on the representation of .
𝑥1
𝑥2
𝑥3
P
A
B
C
D
E
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Fig. 2.7: A, B, C, D, E, F are all basic solutions. C, D, E, F are basic feasible solutions.
P
A
BC
D
E
F
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Comparison of definitions in the notes and the text
Notes Text
Extreme point
Geometric definition Geometric definition
Vertex 0-dimensional face Existence of vector which makes as the unique optimal solution for the LP
Basic solu-tion, b.f.s.
Defined for standard form. Set variables at 0 and solve the remaining system. b.f.s. if nonnega-tive.
Defined for general polyhedron. Sat-isfy equality constraints and linearly independent constraints are active. ( 0-dimensional face if feasible)
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Thm 2.3: , then vertex, extreme point, and b.f.s. are equivalent statements.
Pf) We follow the definitions given in the text. We already showed in the notes that extreme point and 0-dimensional face ( , rank , b.f.s. in the text) are equivalent.
To show all are equivalent, take the following steps:
vertex (1) extreme point (2) b.f.s. (3) vertex
(1) vertex extreme point
Suppose is vertex, i.e. such that is unique minimum of
min .
If , then and .
Hence .Hence cannot be expressed as convex combination of two other points in .
extreme point.
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(continued)
(2) extreme point b.f.s.
Suppose is not a b.f.s.. Let .
Since is not a b.f.s., the number of linearly independent vectors in .
Hence nonzero such that .
Consider . But, for sufficiently small positive , and , which implies is not an extreme point.
(3) b.f.s. vertex
Let be a b.f.s. and let .
Let . Then .
, we have , hence optimal.
For uniqueness, equality holds .Since is a b.f.s., it is the unique solution of .
Hence is a vertex.
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Note: Whether is a basic solution depends on the representation of . However, is b.f.s. if and only if extreme point and being ex-treme point is independent of the representation of . Hence the property of being a b.f.s. is also independent of the representation of .
Cor 2.1: For polyhedron , there can be finite number of basic or basic feasible solutions.
Def: Two distinct basic solutions are said to be adjacent if we can find linearly independent constraints that are active at both of them. ( In Fig 2.7, D and E are adjacent to B; A and C are adjacent to D.)
If two adjacent basic solutions are also feasible, then the line segment that joins them is called an edge of the feasible set (one dimensional face).
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2.3 Polyhedra in standard form
Thm 2.4: , , full row rank.
Then is a basic solution satisfies and indices such that are linearly independent and .
Pf) see text.
( To find a basic solution, choose linearly independent columns . Set for all , then solve for . )
Def: basic variable, nonbasic variable, basis, basic columns, basis matrix . (see text)
( )
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Def: For standard form problems, we say that two bases are adja-cent if they share all but one basic column.
Note: A basis uniquely determines a basic solution.
Hence if have two different basic solutions have different bases.
But two different bases may correspond to the same basic solu-tion. (e.g. when )
Similarly, for standard form problems, two adjacent basic solutions two adjacent bases ( nonbasic variables are same.)
Two adjacent bases with different basic solutions two adjacent basic solutions.
However, two adjacent bases only not necessarily imply two adja-cent basic solutions. The two solutions may be the same solution.
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Check that full row rank assumption on results in no loss of generality.
Thm 2.5: , , rank is .
, with linearly independent rows.
Then .
Pf) Suppose first rows of are linearly independent.
is clear. Show .
Every row of can be expressed as .
Hence, for , ,
i.e. is also linear combination of .
Suppose , then ,
Hence,
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2.4 Degeneracy Def 2.10: A basic solution is said to be degenerate if more than
of the constraints are active at .
Def 2.11: , , full row rank.
Then is a degenerate basic solution if more than of the compo-nents of are 0 ( i.e. some basic variables have 0 value)
For standard LP, if we have more than variables at 0 for a basic feasible solution , it means that more than of the nonnegativity constraints are active at in addition to the constraints in .
The solution can be identified by defining nonbasic variables (value ). Hence, depending on the choice of nonbasic variables, we have different bases, but the solution is the same.
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Fig 2.9: A and C are degenerate basic feasible solutions. B and E are nondegenerate. D is a degenerate basic solution.
A
B
C
D
E
P
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Fig 2.11: -dimensional illustration of degeneracy. Here, , . A is nondegenerate and basic variables are . B is degenerate. We can choose as the nonbasic variables. Other possibilities are to choose , or to choose .
A
B
𝑃 𝑥5=0
𝑥4=0𝑥3=0
𝑥2=0 𝑥1=0 𝑥6=0
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Degeneracy is not purely geometric property, it may depend on representation of the polyhedrom
ex) ,
We know that , but representation is different.
Suppose is a nondegenerate basic feasible solution of .
Then exactly of the variables are equal to 0.
For , at the basic feasible solution , we have variables set to 0 and additional constraints are satisfied with equality. Hence, we have active constraints and is degenerate.
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2.5 Existence of extreme points
Def 2.12: Polyhedron contains a line if a vector and a nonzero such that for all .
Note that if is a line in , then for all
Hence is a vector in the lineality space . (in )
Thm 2.6: , then the following are equivalent.
(a) has at least one extreme point.
(b) does not contain a line.
(c) vectors out of , which are linearly independent.
Pf) see proof in the text.
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Note that the conditions given in Thm 2.6 means that the lineal-ity space .
Cor 2.2: Every nonempty bounded polyhedron (polytope) and every nonempty polyhedron in standard form has at least one basic feasible solution (extreme point).
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2.6 Optimality of extreme points
Thm 2.7: Consider the LP of minimizing over a polyhedron . Suppose has at least one extreme point and there exists an op-timal solution.
Then there exists an optimal solution which is an extreme point of .
Pf) see text.
Thm 2.8: Consider the LP of minimizing over a polyhedron . Suppose has at least one extreme point.
Then, either the optimal cost is , or there exists an extreme point which is optimal.
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(continued)
Idea of proof in the text)
Consider any . Let
Then we move to , where and .
Then either the optimal cost is ( if the half line is in and ) or we meet a new inequality which becomes active ( cost does not increase).
By repeating the process, we eventually arrive at an extreme point which has value not inferior to .
Therefore, for any in , there exists an extreme point such that . Then we choose the extreme point which gives the smallest ob-jective value with respect to .
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( alternative proof of Thm 2.8)
. Pointedness of implies .
Hence , where are extreme rays of and are extreme points of and .
Suppose such that , then LP is unbounded.
( For , for . Then as )
Otherwise, for all , take such that .
Then ,
.
Hence LP is either unbounded or an extreme point of which is an optimal solution.
Proof here shows that the existence of an extreme ray of the pointed recession cone ( if have min problem and polyhedron is ) such that is the necessary and sufficient condition for unbound-edness of the LP.
( If has at least one extreme point, then LP is unbounded
an extreme ray in recession cone such that )