Transcript Lecture 15: Linear Programming - A Mathematical Optimization Technique

ELEC 7770
Advanced VLSI Design
Spring 2007
Linear Programming – A Mathematical
Optimization Technique
Vishwani D. Agrawal
James J. Danaher Professor
ECE Department, Auburn University
Auburn, AL 36849
[email protected]
http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr07
Spring 07, Mar 13, 15
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What is Linear Programming
 Linear programming (LP) is a mathematical

method for selecting the best solution from the
available solutions of a problem.
Method:
 State the problem and define variables whose
values will be determined.
 Develop a linear programming model:
 Write the problem as an optimization formula (a
linear expression to be minimized or maximized)
 Write a set of linear constraints
 An available LP solver (computer program) gives
the values of variables.
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Types of LPs
 LP – all variables are real.
 ILP – all variables are integers.
 MILP – some variables are integers, others are

real.
A reference:
 S. I. Gass, An Illustrated Guide to Linear
Programming, New York: Dover, 1990.
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A Single-Variable Problem
 Consider variable x
 Problem: find the maximum value of x subject to

constraint, 0 ≤ x ≤ 15.
Solution: x = 15.
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Single Variable Problem (Cont.)
 Consider more complex constraints:
 Maximize x, subject to following constraints




0
x≥0
5x ≤ 75
6x ≤ 30
x ≤ 10
(1)
(2)
(3)
(4)
5
10
(3)
15
x
(2)
(1)
(4)
All constraints
satisfied
Solution, x = 5
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A Two-Variable Problem
 Manufacture of x1 chairs and x2 tables:
 Maximize profit, P = 45x1 + 80x2 dollars
 Subject to resource constraints:
 400 boards of wood,
5x1 + 20x2 ≤ 400
 450 man-hours of labor, 10x1 + 15x2 ≤ 450
 x1 ≥ 0
 x2 ≥ 0
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(1)
(2)
(3)
(4)
6
Solution: Two-Variable Problem
40
Tables, x2
30
Best solution: 24 chairs, 14 tables
Profit = 45×24 + 80×14 = 2200 dollars
(1) 20
(24, 14)
10
0
0
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10
20
30
40
50
60
Chairs, x1
70
(2)
decresing
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80
90
increasing
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Change Chair Profit, $64/Unit
 Manufacture of x1 chairs and x2 tables:
 Maximize profit, P = 64x1 + 80x2 dollars
 Subject to resource constraints:
 400 boards of wood,
5x1 + 20x2 ≤ 400
 450 man-hours of labor, 10x1 + 15x2 ≤ 450
 x1 ≥ 0
 x2 ≥ 0
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(1)
(2)
(3)
(4)
8
Solution: $64 Profit/Chair
40
Tables, x2
30
Best solution: 45 chairs, 0 tables
Profit = 64×45 + 80×0 = 2880 dollars
(1) 20
10
(24, 14)
0
0
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10
20
30
40
50
Chairs, x1
decresing
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(2)
70
80
90
increasing
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Primal-Dual Problems
 Primal problem
 Dual Problem
 Variables:
 Variables:
 Maximize profit 45x1 + 80x2
 Subject to:
 Minimize cost 400w1 + 450w2
 Subject to:
 Fixed resources
 Maximize profit
 Fixed profit
 Minimize cost
 x1 (number of chairs)
 x2 (number of tables)




5x1 + 20x2
10x1 + 15x2
x1
x2
 Solution:
≤ 400
≤ 450
≥0
≥0
 x1 = 24 chairs, x2 = 14 tables
 Profit = $2200
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 w1 ($ cost/board of wood)
 w2 ($ cost/man-hour)




5w1 +10w2
20w1 + 15w2
w1
w2
≥ 45
≥ 80
≥0
≥0
 Solution:
 w1 = $1, w2 = $4
 Cost = $2200
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The Duality Theorem
 If the primal has a finite optimal solution, so
does the dual, and the optimum values of the
objective functions are equal.
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LP for n Variables
n
minimize
Σ
cj xj
Objective function
j =1
n
subject to
Σ aij xj
≤ bi,
i = 1, 2, . . ., m
= di,
i = 1, 2, . . ., p
j =1
n
Σ cij xj
j =1
Variables: xj
Constants: cj, aij, bi, cij, di
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Algorithms for Solving LP
 Simplex method
 G. B. Dantzig, Linear Programming and Extension, Princeton, New
Jersey, Princeton University Press, 1963.
 Ellipsoid method
 L. G. Khachiyan, “A Polynomial Algorithm for Linear Programming,”
Soviet Math. Dokl., vol. 20, pp. 191-194, 1984.
 Interior-point method
 N. K. Karmarkar, “A New Polynomial-Time Algorithm for Linear
Programming,” Combinatorica, vol. 4, pp. 373-395, 1984.
 Course website of Prof. Lieven Vandenberghe (UCLA),
http://www.ee.ucla.edu/ee236a/ee236a.html
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Basic Ideas of Solution methods
Extreme points
Constraints
Extreme points
Objective
function
Simplex: search on extreme points.
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Constraints
Objective
function
Interior-point methods: Successively
iterate with interior spaces of
analytic convex boundaries.
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Integer Linear Programming (ILP)
 Variables are integers.
 Complexity is exponential – higher than LP.
 LP relaxation
 Convert all variables to real, preserve ranges.
 LP solution provides guidance.
 Rounding LP solution can provide a non-optimal
solution.
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Solving TSP: Five Cities
Distances (dij) in miles (symmetric TSP, general TSP is asymmetric)
City
j=1
j=2
j=3
j=4
j=5
i=1
0
18
10
12
27
i=2
18
0
5
12
20
i=3
10
5
0
15
19
i=4
12
12
15
0
6
i=5
27
20
19
6
0
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Search Space: No. of Tours
 Asymmetric TSP tours
 Five-city problem: 4 × 3 × 2 × 1 = 24 tours
 Nine-city problem: 362,880 tours
 14-city problem: 87,178,291,200 tours
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A Greedy Heuristic Solution
Tour length = 10 + 5 + 12 + 6 + 27 = 60 miles (non-optimal)
City
j=1
j=2
j=3
j=4
j=5
i=1
(start)
0
18
10
12
27
i=2
18
0
5
12
20
i=3
10
5
0
15
19
i=4
12
12
15
0
6
i=5
27
20
19
6
0
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ILP Variables, Constants and Constraints
x14 ε [0,1]
d14 = 12
4
5
x15 ε [0,1]
d15 = 27
x12 ε [0,1]
d12 = 18
1
x13 ε [0,1]
d13 = 10
2
3
x12 + x13 + x14 + x15 = 2
four other similar equations
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Objective Function and ILP Solution
5 i-1
Minimize ∑ ∑ xij × dij
i=1 j=1
∑ xij = 2
j≠i
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for all i
xij
j=1
2
3
4
5
i=1
0
0
1
0
0
2
1
0
0
0
0
3
0
1
0
0
0
4
0
0
0
0
1
5
0
0
0
1
0
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ILP Solution
d54 = 6
4
5
d45 = 6
1
d21 = 18
d13 = 10
2
3
d32 = 5
Total length = 45
but not a single tour
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Additional Constraints for a Single Tour
 Following constraints prevent subtours. For any
subset S of cities, the tour must enter and exit
that subset:
∑ xij ≥ 2 for all S, |S| < 5
iεS
jεS
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ILP Solution
d41 = 12
4
d54 = 6
5
1
d25 = 20
d13 = 10
2
3
d32 = 5
Total length = 53
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ILP Example: Test Minimization
 A combinational circuit has n test vectors that detect m

faults. Each test detects a subset of faults. Find the
smallest subset of test vectors that detects all m faults.
ILP model:
 Assign an integer variable ti ε [0,1] to ith test vector such that ti =
1, if we select ti, otherwise ti= 0.
 Define an integer constant fij ε [0,1] such that fij = 1, if ith vector
detects jth fault, otherwise fij = 0. Values of constants fij are
determined by fault simulation.
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Test Minimization by ILP
n
minimize
Σ ti
Objective function
i=1
n
subject to
Σ fij ti
≥ 1,
j = 1, 2, . . ., m
i=1
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3V3F: A 3-Vector 3-Fault Example
Test vector i
fij
i=1
i=2
i=3
Variables: t1, t2, t3 ε [0,1]
Fault j
Minimize t1 + t2 + t3
j=1
1
1
0
j=2
0
1
1
Subject to:
t1 + t2 ≥ 1
t2 + t3 ≥ 1
j=3
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1
0
1
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t1 + t3 ≥ 1
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3V3F: Solution Space
t3
ILP solutions
(optimum)
1
Non-optimum
solution
LP solution
(0.5, 0.5, 0.5)
and recursive
rounding
t2
1
1
t1
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Characteristics of ILP
 Worst-case complexity is exponential in number of


variables.
Linear programming (LP) relaxation, where integer
variables are treated as real, gives a lower bound on the
objective function.
Recursive rounding of relaxed LP solution to nearest
integers gives an approximate solution to the ILP
problem.
 K. R. Kantipudi and V. D. Agrawal, “A Reduced Complexity
Algorithm for Minimizing N-Detect Tests,” Proc. 20th International
Conf. VLSI Design, January 2007, pp. 492-497.
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3V3F: LP Relaxation and Rounding
ILP – Variables: t1, t2, t3 ε [0,1]
LP relaxation: t1, t2, t3 ε (0.0, 1.0)
Minimize t1 + t2 + t3
Solution: t1 = t2 = t3 = 0.5
Subject to:
Recursive rounding:
t1 + t2 ≥ 1
t2 + t3 ≥ 1
t1 + t3 ≥ 1
(1) round one variable, t1 = 1.0
Two-variable LP problem:
Minimize t2 + t3
subject to t2 + t3 ≥ 1.0
LP solution t2 = t3 = 0.5
(2) round a variable, t2 = 1.0
ILP constraints are satisfied
solution is t1 = 1, t2 = 1, t3 = 0
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Recursive Rounding Algorithm
1. Obtain a relaxed LP solution. Stop if each
variable in the solution is an integer.
2. Round the variable closest to an integer.
3. Remove any constraints that are now
unconditionally satisfied.
4. Go to step 1.
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Recursive Rounding
 ILP has exponential complexity.
 Recursive rounding:
 ILP is transformed into k LPs with progressively


reducing number of variables.
Number of LPs, k, is the size of the final solution, i.e.,
the number of non-zero variables in the test
minimization problem.
Recursive rounding complexity is k × O(np), where k ≤
n, n is number of variables.
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Four-Bit ALU Circuit
ILP
Recursive rounding
Initial
vectors
Vectors
CPU s
Vectors
CPU s
285
14
0.65
14
0.42
400
13
1.07
13
1.00
500
12
4.38
13
3.00
1,000
12
4.17
12
3.00
5,000
12
12.95
12
9.00
10,000
12
34.61
12
17.0
16,384
12
87.47
12
37.0
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CPU Time: ILP vs. Recursive Rounding
100
ILP
CPU s
75
Recursive
Rounding
50
25
0
0
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5,000
10,000
ELEC 7770: Advanced VLSI Design (Agrawal)
15,000 Vectors
33
N-Detect Tests (N = 5)
Circuit
Unoptimized
vectors
c432
Relaxed LP/Recur. rounding
ILP (exact)
Lower
bound
Min.
vectors
CPU s
Min.
vectors
CPU s
608
196.38
197
1.0
197
1.0
c499
379
260.00
260
1.2
260
2.3
c880
1,023
125.97
128
14.0
127
881.8
c1355
755
420.00
420
3.2
420
4.4
c1908
1,055
543.00
543
4.6
543
6.9
c2670
959
477.00
477
4.7
477
7.2
c3540
1,971
467.25
477
72.0
471
20008.5
c5315
1,079
374.33
377
18.0
376
40.7
c6288
243
52.52
57
39.0
57
34740.0
c7552
2,165
841.00
841
52.0
841
114.3
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Finding LP/ILP Solvers
 R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A


Modeling Language for Mathematical Programming,
South San Francisco, California: Scientific Press, 1993.
Several of programs described in this book are available
to Auburn users.
MATLAB?
Search the web. Many programs with small number of
variables can be downloaded free.
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