Cutting Planes in SCIP

Transcription

1 Cutting Planes in SCIP Kati Wolter Zuse-Institute Berlin Department Optimization Berlin, 6th June 2007

2 Outline 1 Cutting Planes in SCIP 2 Cutting Planes for the 0-1 Knapsack Problem 2.1 Cover Cuts 2.2 Lifted Minimal Cover Cuts 3 Computational Results

3 Outline 1 Cutting Planes in SCIP 2 Cutting Planes for the 0-1 Knapsack Problem 2.1 Cover Cuts 2.2 Lifted Minimal Cover Cuts 3 Computational Results

4 SCIP General cutting planes Complemented mixed integer rounding cuts Gomory mixed integer cuts Strong Chvátal-Gomory cuts Implied bound cuts

5 SCIP General cutting planes Complemented mixed integer rounding cuts Gomory mixed integer cuts Strong Chvátal-Gomory cuts Implied bound cuts Cutting planes for special problems 0-1 knapsack problem 0-1 single node flow problem Stable set problem

6 SCIP General cutting planes Complemented mixed integer rounding cuts Gomory mixed integer cuts Strong Chvátal-Gomory cuts Implied bound cuts Cutting planes for special problems 0-1 knapsack problem 0-1 single node flow problem Stable set problem I want to solve general MIPs! Why do I care about cutting planes for special problems?

7 General Cutting Plane Method min{c T x : x X MIP } X MIP := {x Z n R m : Ax b} min{c T x : x X LP } X LP := {x R n R m : Ax b}

8 General Cutting Plane Method min{c T x : x X MIP } X MIP := {x Z n R m : Ax b} min{c T x : x X LP } X LP := {x R n R m : Ax b} Observation If the data are rational, then conv(x MIP ) is a rational polyhedron we can formulate the MIP as min{c T x : x conv(x MIP )} }{{} LP

9 General Cutting Plane Method min{c T x : x X MIP } X MIP := {x Z n R m : Ax b} min{c T x : x X LP } X LP := {x R n R m : Ax b} Observation If the data are rational, then conv(x MIP ) is a rational polyhedron we can formulate the MIP as min{c T x : x conv(x MIP )} }{{} LP Problem (in general) Complete linear description of conv(x MIP )? Number of constraints needed to describe conv(x MIP ) is extremely large

10 General Cutting Plane Method min{c T x : x X MIP } X MIP := {x Z n R m : Ax b} min{c T x : x X LP } X LP := {x R n R m : Ax b} Observation If the data are rational, then conv(x MIP ) is a rational polyhedron we can formulate the MIP as min{c T x : x conv(x MIP )} }{{} LP Idea Construct a polyhedron Q with conv(x MIP ) Q X LP min{c T x : x conv(x MIP )} = min{c T x : x Q} Start with X LP and add inequalities which are valid for X MIP (but violated by the current LP solution) to X LP

11 Valid Inequalities for X MIP Inequalities valid for a relaxation of X MIP are valid for X MIP Generating valid inequalities for a relaxation is often easier The intersection of the relaxations should be a good approximation of X MIP

12 Valid Inequalities for X MIP Inequalities valid for a relaxation of X MIP are valid for X MIP Generating valid inequalities for a relaxation is often easier The intersection of the relaxations should be a good approximation of X MIP Relaxations of X MIP 1. Linear combinations of constraints defining X MIP (row of the simplex tab., single constraint) 2. Other information Logical implications between binary variables (conflict graph) Logical implications between a binary and a real variable

13 SCIP General cutting planes Complemented mixed integer rounding cuts (Linear comb.) Gomory mixed integer cuts (Row of the simplex tab.) Strong Chvátal-Gomory cuts (Row of the simplex tab.) Implied bound cuts (Logical impl.)

14 SCIP General cutting planes Complemented mixed integer rounding cuts (Linear comb.) Gomory mixed integer cuts (Row of the simplex tab.) Strong Chvátal-Gomory cuts (Row of the simplex tab.) Implied bound cuts (Logical impl.) Cutting planes for special problems 0-1 knapsack problem (Single constraint) 0-1 single node flow problem (Linear comb. and bounds) Stable set problem (Conflict graph)

15 Outline 1 Cutting Planes in SCIP 2 Cutting Planes for the 0-1 Knapsack Problem 2.1 Cover Cuts 2.2 Lifted Minimal Cover Cuts 3 Computational Results

16 0-1 Knapsack Polytope conv(x BK ) X BK := {x {0, 1} n : j N a j x j a 0 } N = {1,..., n} a 0 and a j are integers for all j N a j > 0 for all j N (since binary variables can be complemented) a j a 0 for all j N (since a j > a 0 implies x j = 0)

17 Outline 1 Cutting Planes in SCIP 2 Cutting Planes for the 0-1 Knapsack Problem 2.1 Cover Cuts 2.2 Lifted Minimal Cover Cuts 3 Computational Results

18 Class of Cover Inequalities Definition (Cover) C N j C a j > a 0

19 Class of Cover Inequalities Definition (Cover) C N j C a j > a 0 Theorem If C N is a cover for X BK, then the cover inequality x j C 1 is valid for X BK. j C

20 Separation Problem Let x [0, 1] n \{0, 1} n be a fractional vector with j N a jx j a 0. Find C N with j C a j > a 0 such that j C x j > C 1, or show that no inequality in the class of cover inequalities is violated by x.

21 Separation Problem Let x [0, 1] n \{0, 1} n be a fractional vector with j N a jx j a 0. Find C N with j C a j > a 0 such that j C x j > C 1, or show that no inequality in the class of cover inequalities is violated by x. The separation problem can be formulated as a 0-1 KP.

22 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n.

23 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j > a 0 j C

24 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j > a 0 j C a j z j > a 0 j N

25 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j > a 0 j C a j z j > a 0 j N a j z j a j N

26 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j > a 0 j C a j z j > a 0 j N a j z j a j N Violated cover inequality xj > C 1 j C

27 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j > a 0 j C a j z j > a 0 j N a j z j a j N Violated cover inequality xj > C 1 xj z j > z j 1 j N j N j C

28 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j > a 0 j C a j z j > a 0 j N a j z j a j N Violated cover inequality xj > C 1 xj z j > z j 1 j C j N j N (1 xj )z j < 1 j N

29 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j z j a j N Violated cover inequality (1 xj )z j < 1 j N

30 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j z j a j N Violated cover inequality (1 xj )z j < 1 j N x satisfies all cover inequalities z {0, 1} n with j N a j z j a : j N (1 x j )z j 1

31 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j z j a j N Violated cover inequality (1 xj )z j < 1 j N x satisfies all cover inequalities z {0, 1} n with j N a j z j a : j N (1 x j )z j 1 min{ j N (1 x j )z j : j N a j z j a 0 + 1, z {0, 1} n } 1

32 Separation Problem as 0-1 KP For C N, we introduce the characteristic vector z {0, 1} n. Cover a j z j a j N Violated cover inequality (1 xj )z j < 1 j N x satisfies all cover inequalities z {0, 1} n with j N a j z j a : j N (1 x j )z j 1 min{ j N max{ j N (1 xj )z j : a j z j a 0 + 1, j N z {0, 1} n } 1 (1 xj ) z j : a j z j a j (a 0 + 1), j N j N z {0, 1} n } 1 (1 x j )

33 Heuristic for the 0-1 KP Input : c Q n +, a Q n +\{0}, and b Q + Output: Feasible solution of max{c T x : a T x b, x {0, 1} n } 1 Sort the indices such that c 1 a 1... cn a n 2 ā 0 3 for j 1 to n do 4 if ā + a j b then 5 x j 1 6 ā ā + a j 7 else 8 while j n do 9 x j 0 10 j j return x

34 Heuristic for the 0-1 KP Input : c Q n +, a Q n +\{0}, and b Q + Output: Feasible solution of max{c T x : a T x b, x {0, 1} n } 1 Sort the indices such that c 1 a 1... cn a n 2 ā 0 3 for j 1 to n do 4 if ā + a j b then 5 x j 1 6 ā ā + a j 7 else 8 while j n do 9 x j 0 10 j j return x Solves the LP relaxation and rounds down the solution.

35 Heuristic for the 0-1 KP Input : c Q n +, a Q n +\{0}, and b Q + Output: Feasible solution of max{c T x : a T x b, x {0, 1} n } 1 Sort the indices such that c 1 a 1... cn a n 2 ā 0 3 for j 1 to n do 4 if ā + a j b then 5 x j 1 6 ā ā + a j 7 else 8 while j n do 9 x j 0 10 j j return x Time complexity: O(n log n)

36 Exact Algorithm for the 0-1 KP Input : c Q n +, a Z n +\{0}, and b Z + Output: Optimal solution of max{c T x : a T x b, x {0, 1} n } Algorithm uses dynamic programming

37 Exact Algorithm for the 0-1 KP Input : c Q n +, a Z n +\{0}, and b Z + Output: Optimal solution of max{c T x : a T x b, x {0, 1} n } Algorithm uses dynamic programming Time complexity: O(nb)

38 Practice A separator for cover cuts has only a limited effect on the overall performance of SCIP It seems to be important to separate strong cutting planes (facets or at least faces of reasonably high dimension)

39 Practice A separator for cover cuts has only a limited effect on the overall performance of SCIP It seems to be important to separate strong cutting planes (facets or at least faces of reasonably high dimension) Can we strengthen the cover inequalities?

40 Outline 1 Cutting Planes in SCIP 2 Cutting Planes for the 0-1 Knapsack Problem 2.1 Cover Cuts 2.2 Lifted Minimal Cover Cuts 3 Computational Results

41 Definition (Minimal cover) Class of Minimal Cover Inequalities C N j C a j > a 0 j C\{i} a j a 0 for all i C

42 Class of Minimal Cover Inequalities Definition (Minimal cover) C N j C a j > a 0 j C\{i} a j a 0 for all i C Theorem If C N is a minimal cover for X BK, then the minimal cover inequality x j C 1 defines a facet of j C conv( X BK {x {0, 1} n : x j = 0 for all j N\C} ).

43 (j 1,..., j t ) lifting sequence of the variables in N\C X i := X BK {x {0, 1} n : x ji+1 =... = x jt = 0} Sequential Up-Lifting

44 (j 1,..., j t ) lifting sequence of the variables in N\C X i := X BK {x {0, 1} n : x ji+1 =... = x jt = 0} Sequential Up-Lifting x j C 1 valid for X 0 j C x j + α j1 x j1 C 1 valid for X 1 j C. t x j + α jk x jk C 1 valid for X t = X BK j C k=1

45 (j 1,..., j t ) lifting sequence of the variables in N\C X i := X BK {x {0, 1} n : x ji+1 =... = x jt = 0} Sequential Up-Lifting Theorem For each i = 1,..., t, consider the 0-1 knapsack problem z ji = max{ i 1 x j + α jk x jk : i 1 a j x j + a jk x jk a 0 a ji, j C k=1 j C k=1 x {0, 1} C +(i 1) } and let α ji = ( C 1) z ji. Then for each i = 1,..., t, i x j + α jk x jk C 1 j C k=1 defines a facet of conv(x i ).

46 (j 1,..., j t ) lifting sequence of the variables in N\C X i := X BK {x {0, 1} n : x ji+1 =... = x jt = 0} Sequential Up-Lifting Different lifting sequences may lead to different inequalities!

47 Computing the Lifting Coefficients For each i = 1,..., t, solve the 0-1 KP approximately (O(n log n)) exactly (O(nb)) Zemel: Exact algo to calculate all lifting coefficients (O(n 2 )) Uses dynamic programming to solve a reformulation of the 0-1 KPs (role of the objective function and the constraint is reversed)

48 Practice Using sequential up-lifting to strengthen minimal cover cuts improves the performance of SCIP

49 Practice Using sequential up-lifting to strengthen minimal cover cuts improves the performance of SCIP But, a separator which uses up- and down-lifting performs even better!

50 Class of Minimal Cover Inequalities Theorem If C N is a minimal cover for X BK, then the minimal cover inequality x j C 1 defines a facet of j C conv( X BK {x {0, 1} n : x j = 0 for all j N\C} ). Up-lifting: variables in N\C

51 Class of Minimal Cover Inequalities Theorem If C N is a minimal cover for X BK and (C 1, C 2 ) is any partition of C with C 1, then inequality j C 1 x j C 1 1 defines a facet of conv( X BK {x {0, 1} n : x j = 0 for all j N\C, x j = 1 for all j C 2 } ).

52 Class of Minimal Cover Inequalities Theorem If C N is a minimal cover for X BK and (C 1, C 2 ) is any partition of C with C 1, then inequality j C 1 x j C 1 1 defines a facet of conv( X BK {x {0, 1} n : x j = 0 for all j N\C, x j = 1 for all j C 2 } ). Up-lifting: variables in N\C Down-lifting: variables in C 2

53 Sequential Up- and Down-Lifting Similar theorem as for sequential up-lifting Extension of Zemel s up-lifting procedure can be used In SCIP, the separation problem for the class of lifted minimal cover inequalities using sequential up- and down-lifting is solved heuristically

54 Outline of the Separation Algorithm Step 1 (Cover) Determine a cover C for X BK (separation problem for the class of cover inequalities) Step 2 (Minimal cover and partition) Make the cover minimal by removing vars from C Find a partition (C 1, C 2 ) of C with C 1 Step 3 (Lifting) Determine a lifting sequence of the variables in N\C 1 Lift the inequality j C 1 x j C 1 1 using sequential up- and down-lifting

55 Algorithmic Aspects Step 1 (Cover) Which algorithm do we use to find the cover? Fixing of variables Modification of the separation problem (Gu et al. (1998)) Solving the separation problem exactly or approximately Step 2 (Minimal cover and partition) In which order do we remove the variables? Which partition of the minimal cover do we use? Step 3 (Lifting) Which lifting sequence do we use? Which algorithm do we use to solve the knapsack problems that occur in the sequential lifting procedure?

56 Resulting Algorithm Step 1 (Cover) Fix all vars with x j = 0 to zero and all vars with x j Use the modification of the separation problem Solve the modified separation problem approximately Step 2 (Minimal cover and partition) Nondecreasing order of x j C 2 := {j C : x j Step 3 (Lifting) = 1 to one and nondecreasing order of a j = 1} ( C 1 = 1: change the partition) {j N\C : xj > 0}, C 2, and then {j N\C : xj = 0} (nonincreasing order of a j ) Use an extension of Zemel s procedure

57 Important Aspects Gap Closed % Sepa Time sec (Geom. Mean) (Total) Value Value Default algorithm Resulting algorithm

58 Important Aspects Gap Closed % Sepa Time sec (Geom. Mean) (Total) Value Value Default algorithm Cover 1. modification Resulting algorithm Determination of the cover Solve the separation problem approximately

59 Important Aspects Gap Closed % Sepa Time sec (Geom. Mean) (Total) Value Value Default algorithm Cover 1. modification Cover 2. modification Resulting algorithm Determination of the cover Solve the separation problem approximately Modification of the separation problem

60 Outline 1 Cutting Planes in SCIP 2 Cutting Planes for the 0-1 Knapsack Problem 2.1 Cover Cuts 2.2 Lifted Minimal Cover Cuts 3 Computational Results

61 Question Comparison with other MIP solvers Are the cutting plane separators implemented in SCIP competitive to the ones included in other MIP solvers?

62 Computational Study MIP solvers SCIP (CPLEX as LP solver) CPLEX COIN-OR Branch and Cut solver (COIN-OR LP solver)

63 Computational Study MIP solvers SCIP (CPLEX as LP solver) CPLEX COIN-OR Branch and Cut solver (COIN-OR LP solver) Settings One cutting plane separator Isolated application Presolving disabled (used presolved instances obtained by the presolving routines of CPLEX)

64 Computational Study MIP solvers SCIP (CPLEX as LP solver) CPLEX COIN-OR Branch and Cut solver (COIN-OR LP solver) Settings One cutting plane separator Isolated application Presolving disabled (used presolved instances obtained by the presolving routines of CPLEX) Test set 134 instances (MIPLIB 2003, MIPLIB 3.0, and MIP collection of Mittelmann)

65 Computational Results Gap Closed % in Geom. Mean SCIP 0.81 CPLEX Cbc GMI C-MIR Flow Cover Knapsack Performance measure Gap Closed % (100 db z LP z MIP z LP )

66 Computational Results Gap Closed % in Geom. Mean SCIP 0.81 CPLEX Cbc GMI C-MIR Flow Cover Knapsack Knapsack CPLEX: Time in total 9, 400 sec SCIP and CBC: Time in total 600 sec

67 Question Impact on the overall performance of SCIP How strong is the impact of the individual cutting plane separators on the overall performance of SCIP?

68 Two Types of Tests Impact of the individual cutting plane separators when they are used as the only separators in SCIP 1. Started with running SCIP without any separators 2. Compared the performance with the one of SCIP when we enabled one separator in connection with all other separators of SCIP 1. Started with running SCIP with all separators 2. Compared the performance with the one of SCIP when we disabled one separator

69 Performance measures Nodes Time Gap % Enabling Computational Study

70 Enabling Computational Study Performance measures Nodes Time Gap % Improvement factor for each performance measure Value for SCIP run without any separators Value for SCIP run with one separator enabled Factor by which enabling the separator improves the overall performance (Factor > 1?)

71 Enabling Computational Results Improvement Factor in Geom. Mean Nodes Time Gap % C-MIR Flow Cover Knapsack GMI Impl. B. Clique

72 Performance measures Nodes Time Gap % Disabling Computational Study

73 Disabling Computational Study Performance measures Nodes Time Gap % Degradation factor for each performance measure Value for SCIP run with one separator disabled Value for SCIP run with all separators Factor by which disabling the separator degrades the overall performance (Factor > 1?)

74 Disabling Computational Results Degradation Factor in Geom. Mean Nodes Time Gap % C-MIR GMI Knapsack Flow Cover Impl. B. Clique