1. Linear Programming Piyush Kumar Welcome to CIS5930

2. Optimization

3. For example This is what is known as a standard linear program.

4. Linear Programming Significance  A lot of problems can be converted to LP formulation oPerceptrons (learning), Shortest path, max flow, MST, matching, … Accounts for major proportion of all scientific computations Helps in finding quick and dirty solutions to NP-hard optimization problems  Both optimal (B&B) and approximate (rounding)

5. Graphing 2-Dimensional LPs Example 1: x 3 012 y 0 1 2 4 3 Feasible Region x  0y  0 x + 2 y  2 y  4 x  3 Subject to: Maximize x + y Optimal Solution These LP animations were created by Keely Crowston.

6. Graphing 2-Dimensional LPs Example 2: Feasible Region x  0y  0 -2 x + 2 y  4 x  3 Subject to: Minimize ** x - y Multiple Optimal Solutions! 4 1 x 3 12 y 0 2 0 3 1/3 x + y  4

7. Graphing 2-Dimensional LPs Example 3: Feasible Region x  0y  0 x + y  20 x  5 -2 x + 5 y  150 Subject to: Minimize x + 1/3 y Optimal Solution x 30 1020 y 0 10 20 40 0 30 40

8. y x 0 1 2 3 4 01 2 3 x 30 1020 y 0 10 20 40 0 30 40 Do We Notice Anything From These 3 Examples? x y 0 1 2 3 4 012 3 Extreme point

9. A Fundamental Point If an optimal solution exists, there is always a corner point optimal solution! y x 0 1 2 3 4 01 2 3 x 30 1020 y 0 10 20 40 0 30 40 x y 0 1 2 3 4 012 3

10. Graphing 2-Dimensional LPs Example 1: x 3 012 y 0 1 2 4 3 Feasible Region x  0y  0 x + 2 y  2 y  4 x  3 Subject to: Maximize x + y Optimal Solution Initial Corner pt. Second Corner pt.

11. And We Can Extend this to Higher Dimensions

12. Then How Might We Solve an LP? o The constraints of an LP give rise to a geometrical shape - we call it a polyhedron. o If we can determine all the corner points of the polyhedron, then we can calculate the objective value at these points and take the best one as our optimal solution. o The Simplex Method intelligently moves from corner to corner until it can prove that it has found the optimal solution.

13. But an Integer Program is Different x y 0 1 2 3 4 012 3  Feasible region is a set of discrete points.  Can’t be assured a corner point solution.  There are no “efficient” ways to solve an IP.  Solving it as an LP provides a relaxation and a bound on the solution.

14. Linear Programs in higher dimensions minimize z = 7x 1 + x 2 + 5x 3 subject to x 1 - x 2 + 3x 3 >= 10 5x 1 + 2x 2 - x 3 >= 6 x 1, x 2, x 3  0 What happens at (1,2,3)? What does it tell us about z * = optimal value of z?

15. LP Upper bounds Any feasible solution to LP gives an upper bound on z * So now we know z * <= 30. How do we construct a lower bound?  z * >= 16? [Y/N]?

16. Lower bounding an LP 7x 1 +x 2 +5x 3 >= (x 1 -x 2 +3x 3 ) + (5x 1 +2x 2 -x 3 ) >= 16 Find suitable multipliers ( >0 ?) to construct lower bounds. How do we choose the multipliers?

17. The Dual maximize z’ = 10y 1 + 6y 2 subject to y 1 + 5y 2 <= 7 -y 1 + 2y 2 <= 1 3y 1 – y 2 <= 5 y 1, y 2  0 What is the dual of a dual? Every feasible solution of the dual gives a lower bound on z *

18. The Primal minimize z = 7x 1 + x 2 + 5x 3 subject to x 1 - x 2 + 3x 3 >= 10 5x 1 + 2x 2 - x 3 >= 6 x 1, x 2, x 3  0 Every feasible solution of the primal is an upper bound on the solution to the dual.

19. Primal – Dual picture 0 Z*Z* Dual Solutions Primal Solutions Strong Optimality Primal = Dual at opt

20. Duality A variable in the dual is paired with a constraint in the primal Objective function of the dual is determined by the right hand side of the primal constraints The constraint matrix of the dual is the transpose of the constraint matrix in the primal.

21. Duality Properties Some relationships between the primal and dual problems: 1.If one problem has feasible solutions and a bounded objective function (and so has an optimal solution), then so does the other problem, so both the weak and the strong duality properties are applicable 2.If the optimal value of the primal is unbounded then the dual is infeasible. 3.If the optimal value of the dual is unbounded then the primal is infeasible.

22. In Matrix terms

23. LP Geometry Forms a n dimensional polyhedron Is convex : If z 1 and z 2 are two feasible solutions then λz 1 + (1- λ)z 2 is also feasible. Extreme points can not be written as a convex combination of two feasible points.

24. LP Geometry The normals to the halfspaces defining the polyhedron are formed by the coefficents of the constraints. Rows of A form the normals to the hyperplanes defining the primal LP pointing inside the polyhedron.

25. LP Geometry Extreme point theorem: If there exists an optimal solution to an LP Problem, then there exists one extreme point where the optimum is achieved. Local optimum = Global Optimum

26. LP: Algorithms Simplex. (Dantzig 1947)  Developed shortly after WWII in response to logistical problems: used for 1948 Berlin airlift.  Practical solution method that moves from one extreme point to a neighboring extreme point.  Finite (exponential) complexity, but no polynomial implementation known. Courtesy Kevin Wayne

27. LP: Polynomial Algorithms Ellipsoid. (Khachian 1979, 1980)  Solvable in polynomial time: O(n 4 L) bit operations. on = # variables oL = # bits in input  Theoretical tour de force.  Not remotely practical. Karmarkar's algorithm. (Karmarkar 1984)  O(n 3.5 L).  Polynomial and reasonably efficient implementations possible. Interior point algorithms.  O(n 3 L).  Competitive with simplex! oDominates on simplex for large problems.  Extends to even more general problems.

28. Ellipsoid Method Courtesy S. Boyd

29. Barrier Algorithms Simplex solution path Barrier central path o Predictor o Corrector Optimum Interior Point Methods

30. Back to LP Basics

31. Standard form of LP

32. Standard form of the Dual

33. Weak Duality

34. Complementary solutions For any primal feasible (but suboptimal) x, its complementary solution y is dual infeasible, with cx=yb For any primal optimal x *, its complementary solution y * is dual optimal, with cx * =y * b=z * Duality Gap = cx-yb

35. Complementary slackness x *, y * are simultaneously optimal for (P) and (D) iff  x * is opt for (P) and y * is opt for (D) OR  For I = 1..m if y i * > 0 oThen a i x * = b i  For J = 1..n if x j * > 0 oThen y * A j = c i a i are rows of A and A j are the columns of A

36. Complementary slackness x *, y * are simultaneously optimal for (P) and (D) iff  x * is opt for (P) and y * is opt for (D) OR  y * (Ax * - b) = 0  (y * A – c)x * = 0 Summary: If a variable is positive, its dual constraint is tight Or if a constraint is loose its dual variable is zero.

37. Complementary Slackness Proof? y * (Ax * - b) - (y * A – c)x * = y * Ax * - y * b - y * Ax * + cx * = cx * - y * b = 0 ( But all terms are non-negative ) Hence all must be zero!

38. Algorithm Design Techniques LP Relaxation  Rounding oRound the fractional solution obtained by solving LP-relaxation. oRuns fast  Primal Dual Schema o(iteratively constructs primal n dual solutions)

39. objective Linear Program LP optimum feasible solutions y x

40. objective IP optimum Integer Program rounding down optimum of LP relaxation feasible solutions = y x optimum of LP relaxation

41. Linear Relaxations What happens if the optimal of a LP- Relaxation is Integral? There are a class of IPs for which this is guaranteed to happen  Transportation problems  MaxFlow problems  In general (Unimodularity) … Exact Relaxation

42. Lower Bounds Assume minimization problem Any relaxation of the original IP has a _____________ optimal objective function value than the optimal objective function value of the original IP z* relaxation z* z* relaxation is called a __________________ on z* Difference between these two values is called the relaxation gap

43. Upper Bounds Any feasible solution to the original IP has a _____________ objective function value than the optimal objective function value of the original IP z feasible z* z feasible is called an __________________ on z* Heuristic techniques can be used to find “good” feasible solutions  Efficient, may be beneficial if optimality can be sacrificed  Usually application- or problem-specific

44. Integrality Gap