Integer Programming as Projection

Transcription

1 Integer Programming as Projection H. P. Williams London School of Economics John Hooker Carnegie Mellon University INFORMS 2015, Philadelphia USA

2 A Different Perspective on IP Projection of an IP onto a subset of variables is not an IP. It is an IP plus congruence relations. 2

3 A Different Perspective on IP Projection of an IP onto a subset of variables is not an IP. It is an IP plus congruence relations. So let s view an IP as inequalities + congruence relations. Then projection of an IP is an IP. 3

4 A Different Perspective on IP This perspective leads to an alternative theory of IP that parallels the traditional theory. Cutting planes. Branching. Duality. 4

5 A Different Perspective on IP Traditional Theory Theory Based on Projection Chvatál-Gomory cuts Linear combination + rounding Unbounded rank Congruence cuts Linear combination + congruences Rank bounded by number of variables 5

6 A Different Perspective on IP Traditional Theory Theory Based on Projection Chvatál-Gomory cuts Linear combination + rounding Unbounded rank Branching on integer variables Unbounded depth Congruence cuts Linear combination + congruences Rank bounded by number of variables Branching on auxiliary variables Depth bounded by number of variables 6

7 A Different Perspective on IP Traditional Theory Theory Based on Projection Chvatál-Gomory cuts Linear combination + rounding Unbounded rank Branching on integer variables Unbounded depth IP dual based on a Gomory function Nested rounding to integer Unbounded nesting depth Congruence cuts Linear combination + congruences Rank bounded by number of variables Branching on auxiliary variables Depth bounded by number of variables IP dual based on min over Gomory functions Nested rounding to multiple of integer Nesting depth bounded by number of variables. 7

8 IP Example Optimal integer solution 8

9 IP Example To project out x 1, pair inequalities as in Fourier-Motzkin elimination. For example, pair C1 and C2: 9

10 IP Example To project out x 1, pair inequalities as in Fourier-Motzkin elimination. For example, pair C1 and C2: Add auxiliary variable u 1 to increase LHS to multiple of 2: 10

11 IP Example To project out x 1, pair inequalities as in Fourier-Motzkin elimination. For example, pair C1 and C2: Add auxiliary variable u 1 to increase LHS to multiple of 2: A congruence relation forces LHS to a multiple of 2: 11

12 IP Example We now have the system which simplifies to 12

13 IP Example After pairing C1 and C3 as well, we get the projected system: This is an IP in the more general sense. 13

14 The projection defines disjunction of 2 inequality systems on 2 integer sublattices (i.e., disjunction of scenarios) u 1 = 0 solution is (9,9) optimal u 1 = 1 solution is (10,10) 14

15 IP Example To project out x 2, we again pair inequalities. For example, if we pair C0 and C13, we get with new auxiliary variable u

16 IP Example To project out x 2, we again pair inequalities. For example, if we pair C0 and C13, we get with new auxiliary variable u 13. Since x 2 1 (mod 2), we have 3x 2 3 (mod 6) and can write 16

17 IP Example To project out x 2, we again pair inequalities. For example, if we pair C0 and C13, we get with new auxiliary variable u 13. Since x 2 1 (mod 2), we have 3x 2 3 (mod 6) and can write If x 2 occurs in more than 1 congruence, we use a generalized Chinese remainder theorem to combine congruences. 17

18 IP Example After pairing C0 and C12, we get the projected system We now have only z and 2 auxiliary variables with a finite domain. 18

19 IP Example After pairing C0 and C12, we get the projected system We now have only z and 2 auxiliary variables with a finite domain. The solutions (u 1,u 2 ) = (0,4), (1,0) of the congruence define 2 scenarios. The optimization problem is trivial in each scenario: z = 9 and z = 10. The better solution z = 9 is the optimal value. Recovery of the optimal solution (x 1,x 2 ) = (2,9) is straightforward. 19

20 Congruence Cuts Recursive generation of Chvátal-Gomory cuts suffices to solve any IP. 20

21 Congruence Cuts Recursive generation of Chvátal-Gomory cuts suffices to solve any IP. To obtain a C-G cut, take a linear combination of constraints and strengthen the result by rounding. 21

22 Congruence Cuts Recursive generation of Chvátal-Gomory cuts suffices to solve any IP. To obtain a C-G cut, take a linear combination of constraints and strengthen the result by rounding. However, there is no bound on the Chvátal rank as a function of the number of variables. 22

23 Congruence Cuts Recursive generation of Chvátal-Gomory cuts suffices to solve any IP. Recursive generation of congruence cuts suffices to solve for the optimal value. To obtain a C-G cut, take a linear combination of constraints and strengthen the result by rounding. However, there is no bound on the Chvátal rank as a function of the number of variables. 23

24 Congruence Cuts Recursive generation of Chvátal-Gomory cuts suffices to solve any IP. To obtain a C-G cut, take a linear combination of constraints and strengthen the result by rounding. However, there is no bound on the Chvátal rank as a function of the number of variables. Recursive generation of congruence cuts suffices to solve for the optimal value. To obtain a congruence cut, take a linear combination of constraints and strengthen the result with a congruence relation. 24

25 Congruence Cuts Recursive generation of Chvátal-Gomory cuts suffices to solve any IP. To obtain a C-G cut, take a linear combination of constraints and strengthen the result by rounding. However, there is no bound on the Chvátal rank as a function of the number of variables. Recursive generation of congruence cuts suffices to solve for the optimal value. To obtain a congruence cut, take a linear combination of constraints and strengthen the result with a congruence relation. The rank is bounded by the number of variables. 25

26 Example of Congruence Cut Take linear combination of C1 and C3 using (say) multipliers 1 and 1 and the congruence x 1 0 (mod 1). First add an auxiliary variable to C1: The linear combination is 26

27 Example of Congruence Cut Take linear combination of C1 and C3 using (say) multipliers 1 and 1 and the congruence x 1 0 (mod 1). First add an auxiliary variable to C1: The linear combination is Now strengthen it with the congruence relation where the mod 2 is based on the coefficient of x 1. 27

28 Congruence Cuts In general, congruence cuts are defined: Given a set S of inequalities and set C of congruence relations, a rank 1 congruence cut is the result of: Selecting any inequality in S and congruence relation in C Taking any nonnegative linear combination of and another inequality in S. The resulting cut is associated with congruence relation 28

29 Congruence Cuts How to find the cuts that solve a problem? The necessary C-G cuts are obtained by taking multipliers from Gomory s cutting plane method. The necessary congruence cuts are obtained by taking multipliers from projection operations that eliminate all the variables. 29

30 Example of Congruence Cuts Multipliers from the projection operations result in congruence cuts: 30

31 Example of Congruence Cuts Multipliers from the projection operations result in congruence cuts: The congruences define 2 integer sublattices: (u 1, u 13 ) = (1,0), (0,4). 31

32 Example of Congruence Cuts Multipliers from the projection operations result in congruence cuts: The congruences define 2 integer sublattices: (u 1, u 13 ) = (1,0), (0,4). Solving this LP in each sublattice yields the optimal IP solution in that sublattice. The inequalities cut off suboptimal values of z. 32

33 Example of Congruence Cuts Multipliers from the projection operations result in congruence cuts: However, these inequalities may not cut off all fractional solutions (x 1,x 2 ) if the LP is dual degenerate. 33

34 Example of Congruence Cuts Multipliers from the projection operations result in congruence cuts: For example, the LP in sublattice (u 1,u 13 ) = (0,4) has optimal solutions (x 1,x 2,z) = (2, 9, 9), (2.4, 9, 9). The latter cannot be cut off by congruence cuts alone. 34

35 Example of Congruence Cuts Multipliers from the projection operations result in congruence cuts: However, the integer solution (x 1,x 2,z) = (2,9,9) can be recovered by retracing the projection steps. 35

36 A New Branching Strategy At each node of the branching tree, solve LP relaxation. Project out a variable x j that violates its congruence relation (even if it is integer). Branch on values of new auxiliary variables that solve the congruence relations. The problems at nodes of the branching tree therefore contain no auxiliary variables. 36

37 Branching Example 37

38 Branching Example Project out x 1 u1 = 0 u! = 1 38

39 Branching Example Project out x 1 u1 = 0 u! = 1 Project out x 2 Project out x 2 (u 12,u 13 ) = (0,4) (u 12,u 13 ) = (0,0) Solution z = 9 Solution z = 10 39

40 Branching The depth of the tree is bounded by the number of variables. Can also use LP values for branch-and-bound. 40

41 IP Duality We can solve an IP dual by constructing a value function that yields the optimal value for all perturbations in RHS. 41

42 IP Duality Traditional value function is a Gomory function. The function is obtained from Gomory s cutting plane method. It contains nested roundings to integers. There is no bound on the nesting depth. 42

43 IP Duality Traditional value function is a Gomory function. The function is obtained from Gomory s cutting plane method. It contains nested roundings to integers. There is no bound on the nesting depth. Proposed value function is a minimum over Gomory functions. The function is obtained from projection operations in integer sublattices. It contains nested roundings to multiple of an integer. Nesting depth is bounded by number of variables. 43

44 Example of Value Function First pair perturbed constraints C1 and C2: Add this to LHS rather than simply the auxiliary variable u 1, where is rounded up to nearest multiple of m. 44

45 Example of Value Function First pair perturbed constraints C1 and C2: Add this to LHS rather than simply the auxiliary variable u 1, where is rounded up to nearest multiple of m. This yields 45

46 Example of Value Function After projecting out x 1 and x 2, we get Note that the congruence relation doesn t depend on perturbations! 46

47 Example of Value Function After projecting out x 1 and x 2, we get Note that the congruence relation doesn t depend on perturbations! Now we have the value function where (u 1,u 13 ) ranges over the solutions of the congruence relation, namely (1,0) and (0,4) 47

48 Example of Value Function These are Chvátal functions (with special kind of rounding). The depth of nested rounding is bounded by number of variables. 48

49 Example of Value Function This is a Gomory function (max over Chvátal functions). The traditional value function is a Gomory function, but with unbounded nesting depth. 49

50 Example of Value Function This is the value function based on projection. A min over Gomory functions with bounded nesting depth. 50

51 Example of Value Function The value functions v( i ) are shift periodic for sufficiently large i All the Gomory functions in the min have the same periodicity. So it is unnecessary to solve the congruence relations to determine the periodicity. 51

52 Example of Value Function Value function v( 1 ) 52

53 Example of Value Function Value function v( 2 ) 53

54 Example of Value Function Value function v( 3 ) 54