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Vishwani D. Agrawal James J. Danaher Professor ECE Department, Auburn University Auburn, AL 36849

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 vagrawal@eng.auburn.edu http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr07.

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Vishwani D. Agrawal James J. Danaher Professor ECE Department, Auburn University Auburn, AL 36849

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  1. ELEC 7770Advanced VLSI DesignSpring 2007Linear Programming – A Mathematical Optimization Technique Vishwani D. Agrawal James J. Danaher Professor ECE Department, Auburn University Auburn, AL 36849 vagrawal@eng.auburn.edu http://www.eng.auburn.edu/~vagrawal/COURSE/E7770_Spr07 ELEC 7770: Advanced VLSI Design (Agrawal)

  2. 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. ELEC 7770: Advanced VLSI Design (Agrawal)

  3. 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. ELEC 7770: Advanced VLSI Design (Agrawal)

  4. A Single-Variable Problem • Consider variable x • Problem: find the maximum value of x subject to constraint, 0 ≤ x ≤ 15. • Solution: x = 15. ELEC 7770: Advanced VLSI Design (Agrawal)

  5. Single Variable Problem (Cont.) • Consider more complex constraints: • Maximize x, subject to following constraints • x ≥ 0 (1) • 5x ≤ 75 (2) • 6x ≤ 30 (3) • x ≤ 10 (4) 0 5 10 15 x (1) (2) (3) (4) All constraints satisfied Solution, x = 5 ELEC 7770: Advanced VLSI Design (Agrawal)

  6. 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 (1) • 450 man-hours of labor, 10x1 + 15x2 ≤ 450 (2) • x1 ≥ 0 (3) • x2 ≥ 0 (4) ELEC 7770: Advanced VLSI Design (Agrawal)

  7. Solution: Two-Variable Problem 40 30 20 10 0 P = 2200 Best solution: 24 chairs, 14 tables Profit = 45×24 + 80×14 = 2200 dollars (1) Tables, x2 (24, 14) P = 0 0 10 20 30 40 50 60 70 80 90 Chairs, x1 increasing (2) Profit decresing ELEC 7770: Advanced VLSI Design (Agrawal)

  8. 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 (1) • 450 man-hours of labor, 10x1 + 15x2 ≤ 450 (2) • x1 ≥ 0 (3) • x2 ≥ 0 (4) ELEC 7770: Advanced VLSI Design (Agrawal)

  9. Solution: $64 Profit/Chair P = 2880 40 30 20 10 0 Best solution: 45 chairs, 0 tables Profit = 64×45 + 80×0 = 2880 dollars (1) Tables, x2 (24, 14) P = 0 0 10 20 30 40 50 60 70 80 90 Chairs, x1 (2) increasing Profit decresing ELEC 7770: Advanced VLSI Design (Agrawal)

  10. Primal problem Fixed resources Maximize profit Variables: x1 (number of chairs) x2 (number of tables) Maximize profit 45x1 + 80x2 Subject to: 5x1 + 20x2 ≤ 400 10x1 + 15x2 ≤ 450 x1 ≥ 0 x2 ≥ 0 Solution: x1 = 24 chairs, x2 = 14 tables Profit = $2200 Dual Problem Fixed profit Minimize cost Variables: w1 ($ cost/board of wood) w2 ($ cost/man-hour) Minimize cost 400w1 + 450w2 Subject to: 5w1 +10w2 ≥ 45 20w1 + 15w2 ≥ 80 w1 ≥ 0 w2 ≥ 0 Solution: w1 = $1, w2 = $4 Cost = $2200 Primal-Dual Problems ELEC 7770: Advanced VLSI Design (Agrawal)

  11. 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. ELEC 7770: Advanced VLSI Design (Agrawal)

  12. LP for n Variables n minimize Σcj xj Objective function j =1 n subject to Σaij xj ≤ bi,i = 1, 2, . . ., m j =1 n Σcij xj = di,i = 1, 2, . . ., p j =1 Variables: xj Constants: cj, aij, bi, cij, di ELEC 7770: Advanced VLSI Design (Agrawal)

  13. 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 ELEC 7770: Advanced VLSI Design (Agrawal)

  14. Basic Ideas of Solution methods Extreme points Extreme points Objective function Objective function Constraints Constraints Interior-point methods: Successively iterate with interior spaces of analytic convex boundaries. Simplex: search on extreme points. ELEC 7770: Advanced VLSI Design (Agrawal)

  15. 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. ELEC 7770: Advanced VLSI Design (Agrawal)

  16. Solving TSP: Five Cities Distances (dij) in miles (symmetric TSP, general TSP is asymmetric) ELEC 7770: Advanced VLSI Design (Agrawal)

  17. 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 ELEC 7770: Advanced VLSI Design (Agrawal)

  18. A Greedy Heuristic Solution Tour length = 10 + 5 + 12 + 6 + 27 = 60 miles (non-optimal) ELEC 7770: Advanced VLSI Design (Agrawal)

  19. ILP Variables, Constants and Constraints 4 x14 ε [0,1] d14 = 12 5 x15 ε [0,1] 1 d15 = 27 x12 ε [0,1] d12 = 18 x13 ε [0,1] d13 = 10 2 3 x12 + x13 + x14 + x15 = 2 four other similar equations ELEC 7770: Advanced VLSI Design (Agrawal)

  20. Objective Function and ILP Solution 5 i - 1 Minimize ∑ ∑ xij × dij i = 1 j = 1 ∑ xij = 2 for all i j ≠ i ELEC 7770: Advanced VLSI Design (Agrawal)

  21. 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 ELEC 7770: Advanced VLSI Design (Agrawal)

  22. 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 i ε S j ε S ELEC 7770: Advanced VLSI Design (Agrawal)

  23. ILP Solution 4 d54 = 6 d41 = 12 5 1 d25 = 20 d13 = 10 2 3 d32 = 5 Total length = 53 ELEC 7770: Advanced VLSI Design (Agrawal)

  24. 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. ELEC 7770: Advanced VLSI Design (Agrawal)

  25. Test Minimization by ILP n minimize Σti Objective function i=1 n subject to Σfij ti ≥ 1,j = 1, 2, . . ., m i=1 ELEC 7770: Advanced VLSI Design (Agrawal)

  26. 3V3F: A 3-Vector 3-Fault Example Test vector i Variables: t1, t2, t3 ε [0,1] Minimize t1 + t2 + t3 Subject to: t1 + t2 ≥ 1 t2 + t3 ≥ 1 t1 + t3 ≥ 1 Fault j ELEC 7770: Advanced VLSI Design (Agrawal)

  27. 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 ELEC 7770: Advanced VLSI Design (Agrawal)

  28. 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. ELEC 7770: Advanced VLSI Design (Agrawal)

  29. 3V3F: LP Relaxation and Rounding • LP relaxation: t1, t2, t3 ε (0.0, 1.0) • Solution: t1 = t2 = t3 = 0.5 • Recursive rounding: • 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 ILP – Variables: t1, t2, t3 ε [0,1] Minimize t1 + t2 + t3 Subject to: t1 + t2 ≥ 1 t2 + t3 ≥ 1 t1 + t3 ≥ 1 ELEC 7770: Advanced VLSI Design (Agrawal)

  30. Recursive Rounding Algorithm • Obtain a relaxed LP solution. Stop if each variable in the solution is an integer. • Round the variable closest to an integer. • Remove any constraints that are now unconditionally satisfied. • Go to step 1. ELEC 7770: Advanced VLSI Design (Agrawal)

  31. 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. ELEC 7770: Advanced VLSI Design (Agrawal)

  32. Four-Bit ALU Circuit ELEC 7770: Advanced VLSI Design (Agrawal)

  33. CPU Time: ILP vs. Recursive Rounding 100 75 50 25 0 ILP Recursive Rounding CPU s 0 5,000 10,000 15,000 Vectors ELEC 7770: Advanced VLSI Design (Agrawal)

  34. N-Detect Tests (N = 5) ELEC 7770: Advanced VLSI Design (Agrawal)

  35. 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. ELEC 7770: Advanced VLSI Design (Agrawal)

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