1 outline relationship among topics secrets lp with upper bounds by simplex method basic feasible...
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Outline
relationship among topics secrets LP with upper bounds
by Simplex method basic feasible solution (BFS)
by Simplex method for bounded variables extended basic feasible solution (EBFS)
optimality conditions for bounded variables ideas of the proof
examples Example 1 for ideas but inexact Example 2 for the exact procedure
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A Depot for Multiple Products
multi-product by a fleet of trucks
depot
Possible Formulation: objective function
common constraints, e.g., trucks, DC capacity, etc.
network constraints for type-1 product
network constraints for type-1 product
network constraints for type-1 product
....
non-negativity constraints
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A General Type of Optimization Problems
structure of many problems: network constraints: easy other constraints: hard
making use of the easy constraints to solve the problems solution methods: large-scale optimization
column generation, Lagrangian relaxation, Dantzig-Wolfe decomposition …
basis: linear programming, network optimization (and also non-linear optimization, integer optimization, combinatorial optimization)
objective function
network constraints
non-negativity constraints
hard constraints
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Relationship of Solution Techniques
two directions of theoretical development for network programming from special structures of networks from linear programming
ideal: understanding development in both directions
linear prog.
network prog.
int. prog.non-linear prog.dynamic prog.
…
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Relationship of Solution Techniques
minimum cost flow column generation, Dantzig-Wolfe decomposition
Lagrangian relaxation
network algorithms
network simplex
shortest-path algorithms
simplex method
revised simplex method
non-linear optimization
linear algebra
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Our Topics
simplex method for bounded variables linkage between LP and network simplex optimality conditions for minimum cost flow networks
minimum cost algorithms standard, and successive shortest path equivalence among network and LP optimality conditions
revised simplex column generation Dantzig-Wolfe decomposition Lagrangian relaxation
It takes more than one semester to cover these
topics in detail! We will only cover the ideas.
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Secrets
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The Most Beautiful …
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linear algebra
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Maybe the Most Beautiful of All…
algebraic properties
geometric properties
matrix properties
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LP with Upper Bounds
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upper bounds: common in network problems, e.g., an arc with finite capacity
quite some theory of network optimization being from LP
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LP with Upper Bounds
Tmax
. .s t
c x
Ax b
0 x u
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incorporate the upper-bound constraints into the set of functional constraints and solve accordingly
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To Solve LP with Upper Bounds
Tmax
. .s t
c x
Ax b
0 x u
Tmax
. .s t
c x
A bx
I u
0 x
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In the simplex method the lower bound constraints 0 x do not appear in A.
Is it possible to work only with A even with upper-bound constraints?
Yes.
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To Solve LP with Upper Bounds
Tmax
. .s t
c x
Ax b
0 x u
Tmax
. .s t
c x
A bx
I u
0 x
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Amn, m n, of rank m
basic feasible solution (BFS) x of LP, i.e., feasible: Ax b, 0 x basic
non-basic variables: (at least) n-m variables = 0 basic variables: m non-negative variables with linearly
independent columns
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BFS for Standard LP Tmax
. .s t
c x
Ax b
0 x
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Amn, m n, of rank m
extended basic feasible solution ( EBFS ) x of LP with bounded variables, i.e., feasible: Ax b, 0 x u basic solution
non-basic variables: (at least) n-m variables = 0, or = their upper bounds
Basic variables: m variables of the form 0 xi ui, with linearly independent columns
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Extended Basic Feasible Solution of LP with Bounded Variables
Tmax
. .s t
c x
Ax b
0 x u
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Maximum Conditions: BFS x is maximal if 0 for all non-basic variable xj = 0
Minimum Conditions: BFS x is minimal if 0 for all non-basic variable xj = 0
intuition : increase of the objective function by unit increase in xj
maximum condition: no good to increase non-basic xj
minimum condition: no good to decrease non-basic xj
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Optimality Conditions of Standard LP
jc
jc
jc
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Maximum Conditions: EBFS x is maximal if 0 for all non-basic variable xj = 0, and
0 for all non-basic variable xj = uj
Minimum Conditions: EBFS x is minimal if 0 for all non-basic variable xj = 0, and
0 for all non-basic variable xj = uj
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Optimality Conditions of LP with Bounded Variables
jc
jc
jc
jc
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How to Prove?
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optimality conditions of the EBFS from duality theory and complementary slackness
conditions
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General Idea
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primal-dual pair
Theorem 1 (Complementary Slackness Conditions) if x primal feasible and y dual feasible then x primal optimal and y dual optimal iff
xj(yTAjcj) = 0 for all j, and yi(biAix) = 0 for all i
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Complementary Slackness Conditions
Tmax
. .s t
c x
Ax b
0 x
T
T T
min
. .s t
b y
y A c
y
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primal-dual pair
Theorem 2 (Necessary and Sufficient Condition) if x primal feasible then x primal optimal iff there exists dual feasible
y such that x and y satisfy the Complementary Slackness Conditions
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Complementary Slackness Conditions
Tmax
. .s t
c x
Ax b
0 x
T
T T
min
. .s t
b y
y A c
y
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by Theorem 2, primal feasible x and dual feasible (yT, T) are optimal iff xj(yTAj + j - cj ) = 0, j
yi(bi - Aix) = 0, i
j(uj - xj ) = 0, j22
Complementary Slackness Conditions for LP with Bounded Variables
Tmax
. .s t
c x
Ax b
x u
0 x
T T
T T T
min
. .s t
b y + u
y A + c
y
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optimality conditions of the EBFS from duality theory and complementary slackness
conditions
ideas of the proof given an EBFS x satisfying the upper-bound optimality
conditions then possible to find dual feasible variables (yT, T)T
such that x and (yT, T)T satisfy the complementary slackness conditions
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General Idea of the Proof
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max 2x + 5y, min 2x 5y, s.t. x + 2y 20, 2x + y 16, 0 x 2, 0 y 8.
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Example 1. Upper-Bound Constraints as Functional Constraints
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Examples of LP with Bounded Variables
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min 2x 5y, s.t. x + 2y 20, 2x + y 16, 0 x 2, 0 y 8.
max. value = 44 x* = 2 and y* = 8
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Example 1. Upper-Bound Constraints as Functional Constraints
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The following procedure is not exactly the Simplex Method for Bounded
Variables. It primarily brings out the ideas of the exact method.
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y as the entering variable 2y + s1 = 20
y + s2 = 16
y 828
Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables
-5
min 2x 5y, s.t. x + 2y 20, 2x + y 16, 0 x 2, 0 y 8.
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mark the non-basic variable y at its upper bound for y = 8
obj. fun.: -2x – 5y – z = 0 -2x - z = 40
eqt. (1): x + 2y + s1 = 20 x + s1 = 4
eqt. (2): 2x + y + s2 = 16 2x + s2 = 8
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Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables
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x as the entering variable x + s1 = 4
2x + s2 = 8
x 230
Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables
min 2x 5y, s.t. x + 2y 20, 2x + y 16, 0 x 2, 0 y 8.
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for x at its upper bound 2, mark x, and obj. fun.: -2x – z = 40 -z = 44
eqt. (1): x + s1 = 4 s1 = 2
eqt. (2): 2x + s2 = 8 s2 = 4
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Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables
min 2x 5y, s.t. x + 2y 20, 2x + y 16, 0 x 2, 0 y 8.
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satisfying the optimality condition for bounded variables 0 for all non-basic variable xj = 0, and
0 for all non-basic variable xj = uj
z* = -44, with x* = 2 and y* = 832
Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables
jc
jc
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in general, variables swapping among all sorts of status non-basic at 0 basic at 0 basic between 0 and upper bound basic at upper bound non-basic at upper bound
Simplex method for bounded variables: a special algorithm to record all possibilities
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Example 1 Being Too Specific
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The following example follows the exact procedure of the Simplex
Method for Bounded Variables.
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max 3x1 + 5x2 + 2x3 min 3x1 5x2 2x3,
s.t.
x1 + x2 + 2x3 7,
2x1 + 4x2 + 3x3 15,
0 x1 4, 0 x2 3, 0 x3 3.
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Example 2
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potential entering variable: x2
bounded by upper bound 3
define = u2-x2 = 3-x2
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Example 2 by Simplex Method for Bounded Variables
2x
min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3.
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Example 2 by Simplex Method for Bounded Variables
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x1 as the (potential) entering variable
s2 as the leaving variable
a pivot operation as in standard Simplex Method
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Example 2 by Simplex Method for Bounded Variables
min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3.
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which can be an entering variable?
can s1 be a leaving variable? Yes
can x1 be a leaving variable? Yes39
Example 2 by Simplex Method for Bounded Variables
2x
min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3.
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when = 1.25, x1 reaches its upper bound 4
replace x1 by and is a basic variable = 0
result
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Example 2 by Simplex Method for Bounded Variables
2x
1,x 1x
1 2 3 2
1 1 2 3 2
1 2 3 2 1
2 1.5 0.5 1.5
( ) 2 1.5 0.5 1.5
2 1.5 0.5 1.5
x x x s
u x x x s
x x x s u
min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3.
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.
a “normal” pivot operation with aij < 0
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Example 2 by Simplex Method for Bounded Variables
2 1 entering and leaving x x
min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3.
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minimum
z* = -20.75, x1* = 4, x2
* = 1.75, x3* = 0
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Example 2 by Simplex Method for Bounded Variables
min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3.