# Computing with the Mixed Integer Rounding Cut.

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1 Computing with the Mixed Integer Rounding Cut. Marcos Goycoolea. Universidad Adolfo Ibañez Santiago, Chile. (joint work with Sanjeeb Dash, IBM Research, NY)

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4 A quick history... Gomory s Mixed Integer cut (1960). Introduces the GMI procedure. Nemhauser and Wolsey (1990). Introduces the MIR cut as a generalization of the GMI cut to non-tableau rows. Balas, Ceria, Cornuejols, Nataraj (1996). Showed how to use GMI cuts in effective manner. Bixby et al (2000,2004). Report importance of GMI/MIR cuts in modern software (CPLEX).

5 An overview of my research on computing with MIR cuts (the last 5 years). Effectiveness of generalized MIR inequalities. On the strength of GMI and MIR cuts. On numerically accurate GMI cuts. On GMI cuts from non-optimal bases.

6 In this talk: A more detailed look at the MIR cut. On GMI cuts from non-optimal bases.

7 A closer look at the MIR inequality

8 A simple mixed-integer set: a i x i + a j w j = b x i Z i I j C i I x i,w j 0 i, j 1,...,n

9 A simple mixed-integer set: a i x i + a j w j = b x i Z i I j C i I x i,w j 0 i, j 1,...,n Define â = a a S = {i I :â i ˆb} The Mixed-Integer-Rounding Cut (MIR) : (â i + ˆba i )x i + i S i I\S(ˆb + ˆba i )x i + a j >0 a j w j ˆbb

10 A simple mixed-integer set: x + y b y Z,x 0 The Mixed-Integer-Rounding Cut (MIR) : x + ˆby ˆbb

11 y variables (integer values) x + y b x variables (continuous values)

12 A fractional extreme point y variables (integer values) x + y b x variables (continuous values)

13 MIR inequality cuts it off! y variables (integer values) x + y b 3 2 x ˆb(b y) x variables (continuous values)

14 And defines convex hull of feasible region y variables (integer values) x + y b 3 2 x ˆb(b y) x variables (continuous values)

15 y variables (integer values) x + y b 3 2 ˆby + x ˆbb x variables (continuous values)

16 How to derive the general inequality? i I a i x i + j C a j w j = b ˆb a i x i + a j w j b i S\I i S\I i S I i S I a i x i + a j >0 a i x i + a i x i + a j w j b a i x i + a i x i + i S\I i S I â i x i + i S I a j >0 â i x i + a j w j ˆbb i S I a j >0

17 GMI cuts and optimal LP relaxation bases When the base system is defined by an equality, and b is fractional, the MIR is equivalent to: i I,â i <ˆb â i ˆb x i + i I,â i ˆb 1 â i 1 ˆb x i + 1ˆb j C:a j >0 a j w j ˆb j C:a j <0 a j w j 1 This is the GMI cut (Gomory, 1960). Base system optimal tableau row => violated!

18 GMI-based algorithms. 1. Solve LP relaxation L(i) 2. Apply MIR procedure to optimal tableau rows 3. Add GMI cuts to L(i) and obtain L(i+1) 4. Go to step 1 and repeat.

19 GMI-based algorithms. 1. Solve LP relaxation L(i) 2. Apply MIR procedure to optimal tableau rows 3. Add GMI cuts to L(i) and obtain L(i+1) 4. Go to step 1 and repeat. + Very good bound improvements + Relatively cheap (computational time) - Dense high-rank cuts slow down LP solver - Numerical problems lead to invalid cuts!

20 Overcoming these difficulties 1. Carefully control the floating point arithmetic operations to ensure numerical correctness. 2. Stick to generating rank-1 cuts.

21 Generating multiple rounds of rank-1 GMI inequalities

22 GMI-based algorithms. 1. Solve LP relaxation L(i) 2. Apply MIR procedure to optimal tableau rows 3. Add GMI cuts to L(i) and obtain L(i+1) 4. Go to step 1 and repeat.

23 Generating multiple rounds of rank-1 cuts. 1. Solve LP relaxation L(i) 2. Get x*, optimal solution of L(i) 3. Find a set of linear inequalities implied by L 4. Apply MIR procedure to these inequalities, and keep those that are violated by x* 5. Add MIR cuts to L(i) and obtain L(i+1) 6. Go to step 1 and repeat

24 Generating multiple rounds of rank-1 cuts. 1. Solve LP relaxation L(i) 2. Get x*, optimal solution of L(i) 3. Find a set of linear inequalities implied by L 4. Apply MIR procedure to these inequalities, and keep those that are violated by x* 5. Add MIR cuts to L(i) and obtain L(i+1) 6. Go to step 1 and repeat

25 Finding a set of linear inequalities implied by L: First approach: Use IP to compute appropriate linear combination of rows. Dash, Gunluk and Lodi (2010). Balas and Saxeena (2008).

26 Finding a set of linear inequalities implied by L: First approach: Use IP to compute appropriate linear combination of rows. Dash, Gunluk and Lodi (2010). Balas and Saxeena (2008). Method GAP closed 1-GMI 26.09% DGL 62.53% BS 76.52% < 1h (w/ time limit) < 10 days

27 Our approach: Use heuristics to compute appropriate bases of original LP, and apply GMI procedure.

28 We start with a system: L 1 =min{cx : Ax = b, x 0} After several rounds of cuts we obtain: L i =min{ c x : Ā x = b, x 0} =min{cx : Ax = b, Cx d, x 0} If we solve Li, we obtain: B x an optimal basis an optimal LP solution

29 We start with a system: L 1 =min{cx : Ax = b, x 0} can we After several rounds of cuts we obtain: L i =min{ c x : Ā x = b, x 0} =min{cx : Ax = b, Cx d, x 0} If we solve Li, we obtain: B an optimal basis use these to find a good basis for our original system?? x an optimal LP solution

30 B is an optimal basis of L i =min{ c x : Ā x = b, x 0} =min{cx : Ax = b, Cx d, x 0} This basis contains slack variables corresponding to rows defined by C! Does the optimal basis of Li contain an optimal basis of L1? (among the structural variables)

31 L 1 =min{cx : Ax = b, x 0} L i =min{ c x : Ā x = b, x 0} =min{cx : Ax = b, Cx d, x 0} B = A 0 B C I B A B has full rank!

32 Consider B an optimal basis of L i =min{ c x : Ā x = b, x 0} The basic idea =min{cx : Ax = b, Cx d, x 0} Look for a basis of L 1 =min{cx : Ax = b, x 0} Among the columns of I B That is, ignore non-basic variables (fix them at their corresponding bounds) and consider the remaining subproblem.

33 Graphically

34 2. Separate the point with a GMI cut 5. Add this rank-1 cut to separate x* Graphically 3. Re-optimize (solve L2) and get this point 1. Solve L1 and get this solution 4. Find this basic feasible point for L1

35 Also works with infeasible bases

36 3 techniques to find the basis itself 1. FEAS. Solve LP on unfixed variables with original objective. 2. MATROID (greedy and random objectives). Use greedy matroid algorithm to find max set. 3. SPARSE. Refactorize a basis so as to achieve sparsity.

37 Computational results MIPLIB 3.0 MIPLIB GMI DGL B-S L&P FEAS SPARSE GREEDY RANDOM RANDOM ALL Avg. Performance on MIPLIB Problems

38 We never assumed Cx >= 0 defined valid inequalities! L 1 =min{cx : Ax = b, x 0} L i =min{ c x : Ā x = b, x 0} =min{cx : Ax = b, Cx d, x 0} B = A 0 B C I B A B has full rank!

39 Branch and Gather 1. Input: x* 2. Start branch-and-bound algorithm. 3. At each node of the tree, let C represent branching constraints and locally valid cuts. 4. At each node of tree, apply cut procedure. 5. Collect cuts for some number of nodes.

40 Branch and Gather

41 Computational results MIPLIB 3.0 MIPLIB GMI DGL B-S L&P ALL ALL+BG ALL+BG Avg. Performance on MIPLIB Problems. We use Branch-and-Gather algorithm on root node of each instance.

42

43 Branch-and-Cut If we add cuts at every node of the branch-and-bound tree we dont always do better.

44 Branch-and-Cut Solved in 26 minutes, using 7500 nodes. CPLEX takes 1 hour, using 2,245,200 nodes.

45 Branch-and-Cut Solved in 92 hours, using 265,900 nodes. First solved in 114 days of computing time, and 17 million nodes (problem specific cuts + grid-computing), by Buscieck et al Later solved in 22 hours of computing time by a problem-specific branch-and-cut method (see MIPLIB 2003 webpage).

46 We don t even need a system Cx >= d. Just a point x*.

47 We don t even need a system Cx >= d. Just a point x*. We can use this for non-linear MIPs as well!

48 We don t even need a system Cx >= d. Just a point x*. We can use this for non-linear MIPs as well! ibienst1 -- MINLP from Mittleman s test set. quadratic objective, linear constraints Given x*, linear relaxation optimum. Use FEAS to try to separate. Guaranteed to find a feasible solution -- not necessarily basic.

49 We don t even need a system Cx >= d. Just a point x*. We can use this for non-linear MIPs as well! ibienst1 -- MINLP from Mittleman s test set. quadratic objective, linear constraints Given x*, linear relaxation optimum. Use FEAS to try to separate. Guaranteed to find a feasible solution -- not necessarily basic. In 3 seconds, using our heuristics, we obtain the same bound CPLEX 11,2 obtains in 30 seconds after using 100+ nodes of branching.

50 CONCLUSIONS Given a nonbasic solution x* of an m-row system Ax = b, x >= 0 with m+t non-zeros: 1. One can find a basis such that the GMI cuts from this basis are violated by x* if A is sparse and t is small. 3. The time to find these cuts is comparable to generating a round of GMI cuts. 4. Such cuts help to improve the bound. 5. This procedure can be used to obtain GMI cuts from nodes of a branch-and-bound tree or in nonlinear optimization problems.

51 CONCLUSIONS Given a nonbasic solution x* of an m-row system Ax = b, x >= 0 with m+t non-zeros: 1. One can find a basis such that the GMI cuts from this basis are violated by x* if A is sparse and t is small. 3. The time to find these cuts is comparable to generating a round of GMI cuts. 4. Such cuts help to improve the bound. 5. This procedure can be used to obtain GMI cuts from nodes of a branch-and-bound tree or in nonlinear optimization problems. Questions?

52 Numerically accurate GMI inequalities

53 Floating point arithmetic: basic properties.

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