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Vishwani D. Agrawal James J. Danaher Professor Dept. of Electrical and Computer Engineering

ELEC 5270/6270 Spring 2013 Low-Power Design of Electronic Circuits Linear Programming – A Mathematical Optimization Technique. Vishwani D. Agrawal James J. Danaher Professor Dept. of Electrical and Computer Engineering Auburn University, Auburn, AL 36849 vagrawal@eng.auburn.edu

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Vishwani D. Agrawal James J. Danaher Professor Dept. of Electrical and Computer Engineering

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  1. ELEC 5270/6270 Spring 2013Low-Power Design of Electronic CircuitsLinear Programming – A Mathematical Optimization Technique Vishwani D. Agrawal James J. Danaher Professor Dept. of Electrical and Computer Engineering Auburn University, Auburn, AL 36849 vagrawal@eng.auburn.edu http://www.eng.auburn.edu/~vagrawal/COURSE/E6270_Spr13/course.html ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  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. ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  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. ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  4. A Single-Variable Problem • Consider variable x • Problem: find the maximum value of x subject to constraint, 0 ≤ x ≤ 15. • Solution: x = 15. Constraint satisfied x 15 0 Solution x = 15 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  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 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  6. A Two-Variable Problem • Manufacture of chairs and tables: • Resources available: • Material: 400 boards of wood • Labor: 450 man-hours • Profit: • Chair: $45 • Table: $80 • Resources needed: • Chair • 5 boards of wood • 10 man-hours • Table • 20 boards of wood • 15 man-hours • Problem: How many chairs and how many tables should be manufactured to maximize the total profit? ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  7. Formulating Two-Variable Problem • Manufacture x1 chairs and x2 tables to maximize profit: P = 45x1 + 80x2 dollars • Subject to given 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) ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  8. Solution: Two-Variable Problem 40 30 20 10 0 P = 2200 Best solution: 24 chairs, 14 tables Profit = 45×24 + 80×14 = 2200 dollars Man-power constraint (1) Tables, x2 (24, 14) Material constraint (3) P = 0 (4) 0 10 20 30 40 50 60 70 80 90 Chairs, x1 increasing (2) Profit decresing ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  9. Change Profit of Chair to $64/Unit • Manufacture x1 chairs and x2 tables to maximize profit: P = 64x1 + 80x2 dollars • Subject to given 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) ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  10. Solution: $64 Profit/Chair P = 2880 40 30 20 10 0 Best solution: 45 chairs, 0 tables Profit = 64×45 + 80×0 = 2880 dollars Man-power constraint (1) Tables, x2 (24, 14) Material constraint (3) P = 0 (4) 0 10 20 30 40 50 60 70 80 90 Chairs, x1 (2) increasing Profit decresing ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  11. A Dual Problem • Explore an alternative. • Questions: • Should we make tables and chairs? • Or, auction off the available resources? • To answer this question we need to know: • What is the minimum price for the resources that will provide us with same amount of revenue from sale as the profits from tables and chairs? • This is the dual of the original problem. ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  12. Formulating the Dual Problem • Revenue received by selling off resources: • For each board, w1 • For each man-hour, w2 • Minimize 400w1 + 450w2 • Subject to constraints: • 5w1 + 10w2 ≥ 45 • 20w1 + 15w2 ≥ 80 • w1 ≥ 0 • w2 ≥ 0 • Resources: • Material: 400 boards • Labor: 450 man-hrs • Profit: • Chair: $45 • Table: $80 • Resources needed: • Chair • 5 boards of wood • 10 man-hours • Table • 20 boards of wood • 15 man-hours ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  13. The Duality Theorem • If the primal has a finite optimum solution, so does the dual, and the optimum values of the objective functions are equal. ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  14. 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 = $2200 Dual Problem Fixed profit Minimize value Variables: w1 ($ value/board of wood) w2 ($ value/man-hour) Minimize value 400w1+450w2 Subject to: 5w1 + 10w2 ≥ 45 20w1 + 15w2 ≥ 80 w1 ≥ 0 w2 ≥ 0 Solution: w1 = $1, w2 = $4 value = $2200 Primal-Dual Problems ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  15. 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 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  16. 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 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  17. Basic Ideas of Solution methods Extreme points Extreme points Objective function Objective function Constraints Constraints Simplex: search on extreme points. Complexity: polynomial in n, number of variables Interior-point methods: Successively iterate with interior spaces of analytic convex boundaries. Complexity: O(n3.5L), L = no. of int. values ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  18. 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. ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  19. Traveling Salesperson Problem (TSP) 4 6 12 5 27 1 12 18 15 20 19 10 2 3 5 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  20. Solving TSP: Five Cities Distances (dij) in miles (symmetric TSP, general TSP is asymmetric) ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  21. Search Space: No. of Tours • Asymmetric TSP tours • Five-city problem: 4 × 3 × 2 × 1 = 24 tours • Ten-city problem: 362,880 tours • 15-city problem: 87,178,291,200 tours • 50-city problem: 49! = 6.08×1062 tours Time for enumerative search assuming 1 μs per tour evaluation = 1.93×1055 years ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  22. A Greedy Heuristic Solution Tour length = 10 + 5 + 12 + 6 + 27 = 60 miles (non-optimal) ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  23. ILP Variables, Constants and Constraints 4 x14 ε [0,1] d14 = 12 5 x15 ε [0,1] 1 d15 = 27 Integer variables: xij = 1, travel i to j xij = 0, do not travel i to j Real constants: dij = distance from i to j x12 ε [0,1] d12 = 18 x13 ε [0,1] d13 = 10 2 3 x12 + x13 + x14 + x15 = 1 four other similar equations ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  24. Objective Function and ILP Solution 5 i - 1 Minimize ∑ ∑ xij × dij i = 1 j = 1 5 ∑ xij = 1, for all i, i.e., every node i has exactly one outgoing edge. j = 1 j ≠ i ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  25. 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 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  26. Additional Constraints for Single Tour • Following constraints prevent split tours. 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 Remaining set At least two arrows must cross this boundary. Any subset ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  27. ILP Solution 4 d54 = 6 d41 = 12 5 1 d25 = 20 d13 = 10 2 3 d32 = 5 Total length = 53 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  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. ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  29. Why ILP Solution is Exponential? LP solution found in polynomial time (bound on ILP solution) Must try all 2n roundoff points Second variable Constraints First variable Objective (maximize) ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  30. 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 tiε[0,1] to ith test vector Ti such that ti= 1 means we select Ti, otherwise ti = 0 means we eliminate Ti • Define an integer constant fijε [0,1] such that fij= 1, if ith vector Ti detects jthfault Fj, otherwise fij= 0, if ith vector Ti does not detect jth fault Fj Values of constants fij are determined by fault simulation ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  31. Test Data Select test Ti if ti = 1 n tests m faults fij = 1; vector Ti detects fault Fj ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  32. Test Minimization by ILP n minimize Σti Objective function i =1 n subject to Σfij ti ≥ 1,j = 1, 2, . . . , m i =1 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  33. Four-Bit ALU Circuit 74181 14 inputs, 8 outputs ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  34. 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. • B. R. Hunt, R. L. Lipsman, J. M. Rosenberg, K. R. Coombes, J. E. Osborn and G. J. Stuck, A Guide to MATLAB for Beginners and Experienced Users, Cambridge University Press, 2006. • Search the web. Many programs with small number of variables can be downloaded free. ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  35. A Circuit Optimization Problem • Given: • Circuit netlist • Cell library with multiple versions for each cell • Select cell versions to optimize a specified characteristic of the circuit. Typical characteristics are: • Area • Power • Delay • Example: Minimize power for given delay. ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  36. Gate Library: NAND(X), X = 0 or 1 • X: an integer variable for each gate. • X = 0, choose gate with small delay • Delay = d × fo, where fo = number of fanouts for gate • Power = 3 × p × fo • d and p are parameters of technology • X = 1, choose gate with low power • Delay = 2 × d × fo • Power = 0.5 × p ×fo • Normalized gate delay = [(1 – X) + 2 X] ×fo • Normalized power = [3(1 – X) + 0.5 X] ×fo • Normalization: d = 1, p = 1 ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  37. Example: One-Bit Full Adder 2 CO SUM A B C 1 3 2 1 1 3 1 1 Number of fanouts, fo ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  38. Define Arrival Time Variables, Tk Tk = Latest signal arrival time at output of gate k T9 2 T1 = T2 = T3 = 0 CO SUM T1 A B C T5 1 T4 T7 T10 3 2 1 T6 T8 T12 T2 1 3 1 T3 1 T11 Number of fanouts, fo ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  39. Constraint: Gate k in the Circuit • Ti = signal arrival time at ith input of gate k • Tk = signal arrival time at gate k output • Tk≥ Ti + (1 – Xk) fo(k) + 2 Xkfo(k), for all i • Where, fo(k) = fanout number of gate k Xk = 0, choose fast cell for k Xk = 1, choose low power cell for k ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  40. Arrival Time Constraints on Gate 7 T9 2 T7 ≥ T5 + (1 – X7) 2 + 2 X7 ✕ 2 T1 = T2 = T3 = 0 T7 ≥ T6 + (1 – X7) 2 + 2 X7 ✕ 2 CO SUM T1 A B C T5 1 T4 T7 T10 3 2 1 T6 T8 T2 T12 1 3 1 T3 1 T11 Number of fanouts, fo ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  41. Clock Constraints • Ti = 0, for all primary inputs i • To ≤ Tc, clock period, for all primary outputs o Combinational Logic Register Register Clock ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  42. Critical Path Constraints T9 ≤ Tc 2 T1 = T2 = T3 = 0 CO SUM T1 A B C T5 1 T4 T7 T10 3 2 1 T12 ≤ Tc T6 T8 T2 1 3 1 T3 1 T11 Number of fanouts, fo ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  43. Optimization Function • Minimize ∑ 3(1 – Xk) fo(k) + 0.5 Xkfo(k) all gates k ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

  44. Typical Result 45 35 25 15 5 (11, 45) Normalized power (22, 7.5) 5 10 15 20 Normalized delay (Tc) ELEC5270-001/6270-001 Spring 2013 Lecture 4 Feb 6 . . .

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