1 a polynomial relaxation-type algorithm for linear programming sergei chubanov university of...
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A polynomial relaxation-type algorithm for linear programming
Sergei ChubanovUniversity of Siegen, [email protected]
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Relaxation method
x
Project the current point onto the half-spacegenerated by a constraint which is not satisfied:
Agmon, and Motzkin and Schoenberg (1954)
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Relaxation method
is exponential
z
*x
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Relaxation method
*xz
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Outline
A strongly polynomial algorithm
which either finds a solution or
proves that there are no 0,1-solutions
A polynomial algorithm
for linear programming
10 x
bAx
uxu
b Ax -
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dCx
bAx
Linear system
hx
k
1lll
m
1iii
k
1lll
m
1iii dbcah 0l
is induced by the system if and only if
)A,,A(a in1ii )C,,C(c nl1ll
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1 dCx,bAx **
)r,z(Bxhx
:x*
an induced inequality(ii)
(i)
Given , construct one of the two objects:),r,z(
z r
Task
hxsuch that
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bAx
zr
maxc2r
Elementary case
)z(px* 1
dCx
bAx*
*
irc
dzc
i
ii
r)z(pz
zproj)z(p }bAx|x{
maxc Cis a row of of max. length
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bAx
z
T)z(pz:h )z(hp:
)z(p
r
Elementary case
ic:h id:
zr
r)z(pz rc
dzci
i
ii
r,zBxdxc ii
r,zBx
z)z(pzx)z(pz TT
ii dxc
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Divide-and-conquer algorithm
1. If , then the elementary case. maxc2
r
,r
2
1,z2. D&C returns or 11xh *x
3. Calculate :z
r
2
1,zBxh|xr,zB 11
,r
2
1,z D&C returns or 22xh *x
4. Calculate withhx r,zBxhx
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Recursion
rz
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Recursion
Recursive call for the same center and a smaller radius
rz
r2
1
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11xh
Recursion
Recursive call either produces an approximate solution or a valid inequality
rz
r2
1
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11xh
Recursion
Recursive call either produces an approximate solution or a valid inequality
rz z
r2
1
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11xh
Recursion
Recursive call for the same radiusand another center which isthe projection of the currentcenter onto the half-space.
rz z
r2
1
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11xh
22xh
Recursion
The second recursive call either produces an approximate solution or an induced inequality
rz z
r2
1
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11xh
22xh
21 h)1(hh
21 )1(
Recursion
Y
2211 xh,xh|xY
hx
rz z
r2
1
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.0,hh 21
11xh 22xh
Recursion
The algorithm may failto construct an inducedinequality
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max
2
crlog
…
……
maxc2
2
r
r
Depth of recursion
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2
maxcrOAt most recursive calls
Running time
21
2
max2223 crNnmnnmmO
CN nonzero components of
bAx m equations
n variables
Running time
Azb)AA(Az)z(p 1TT
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D&C algorithm
• Not faster than the relaxation method• Can solve the task, but not always
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1t
dtCx
btAx
0
0
dCx
bAx
x
t
1 )t,x( )1,x( x
Parameterized system
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11t
dtCx
btAx
1
0
Strengthened parameterized system
D&C is applied to
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(I) 1t,dtCx,btAx 00
:),h( )r),t,x((B)t,x()t,x(h 00
:)t,x(
is induced by the strengthened parameterized system
)t,x(h
(II)
Task
),r),t,x(( 00 Given
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If D&C finds an approximate solution to the strengthened parameterized system
is an exact solution to the parameterized system
)t,x(
)t,x(
If D&C finds a solution
is a solution to the system in questionxt
1
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)1(ttdxc)t,x(ak
1ii
k
1iiii
bA
The two recursive calls at the iteration where it fails produce the inequalities
where a are linear combinations of the rows ofa
)1(ttdxc)t,x(ak
1ii
k
1iiii
and
0txctd)t,x(a
txctd)t,x(a
k
1iiii
k
1iiii
If D&C fails
= 0
= 021 hh
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Px*
0xcdxcdk
1i
*iii
k
1i
*iii
0 0 contradiction infeasible
0i or i*
i dxc 0i .dCx,bAx,x
dxc ii
or
0 0
)1,x( x
dCx,bAx|xP
If D&C fails
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)1(ttdxc)t,x(ak
1ii
k
1iiii
a is a linear combination of the rows of bA
)1()t,x(hk
1ii
)t,x(h is induced by the original parameterized system
If D&C returns an inequality
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)1()t,x(h:)t,x(Hk
1ii
)0,(0
)t,x(h:)t,x(H
The smaller balldoes not contain any solution of the original parameterized system
If D&C returns an inequality
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H),0,(distH),0,(dist 00
)1,x( **
k
1i
*iii
k
1ii
*
xcd
k
1ii
k
1i
*iii
* xcdh
If D&C returns an inequality
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h
)1(H),0,(dist
k
1ii
0
H),0,(disth
hH),0,(dist
h
hH),0,(dist
hH),0,(dist
k
1ii
00
00
H),0,(distH),0,(dist 00
If D&C returns an inequality
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1rrH),0,(distH),0,(dist ** 00
1r)1,x( ****
** rx
If D&C returns an inequality
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1k1k
xcd
k
1ii
k
1i
*iii
k
1ii
*
i
1r3k2r *
1r1r1kr1rrH),0,(dist **** 0
1
Case 1.
If D&C returns an inequality
0
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1k2
xcd
1k
xcd*
lll
lk
1i
*iii
k
1ii
*
,max iil
1r1r1k2rH),0,(dist ** 0
2
1xcd *
ll ,Px* Case 2.
1r3k2r * 1
If D&C returns an inequality
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i
1rH),0,(dist * 0
,il
2
1xcd,rx,Px *
ll***
or
*** rx,Px
Px1r1,x * Pxrx *
1r3k2r * 1
If D&C returns an inequality
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Pxrx * i
,il *ll rx,Px
2
1xcd
If D&C returns an inequality
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Algorithm
If D&C fails, then either no solutions or
If D&C generates an induced inequality, either no solutions or
*ll rx,Px
2
1xcd
Pxdxc ll
nll ZPxdxc
)1r(3k2r * 1
Repeated application of the following argument:
Pxrx *
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Algorithm
The algorithm either finds a solution or decides that there are no 0,1- solutions in strongly polynomial time
10 x
bAx
2
max2223 crNnnmnnmmO
1)1n(3n4r 2
1
2
1* )1n(r
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Algorithm
If the system is feasible and the bounds are tight, a solution can be found in strongly polynomial time
uxu
b Ax - 10
x
bxA
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n
n
n
K0
uK1x0
uK1x0
uuxx
bbAx
ux
bAx
0 (1) (2)
(1) is feasible if and only if (2) has an integer solution
jiij ubA!nK
General case
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ux
bxA
0
10
z
bzz2a i
n
1jjssjsl
1s
0l
l1ulog
0sjsij
j
(2) (3)
(2) has an integer solution if and only if (3) has an integer solution
122u1s
0l
l1ulog
0sjs
s1ulog
0sjsj
jj
0u j
By solving (3) we also solve (1) a polynomial algorithm for linear programming
Polynomial algorithm