to work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting

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1 Summary so far z =max{c T x : Ax apple b, x 2 Z n +} I Modeling with IP (and MIP, and BIP) problems I Formulation for a discrete set that is a feasible region of an IP I Alternative formulations for the same set; strengthening a formulation with valid inequalities I Bounds on optimal values I Primal bounds: from feasible solutions I Dual bounds: from relaxations and/or dual problems I Solving IPs I Easy IPs (for which an ideal formulation is known and easy to work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting plane algorithm Most IPs are not easy, and Gomory s algorithm takes a very long time... IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 80 c Marina A. Epelman Divide and conquer approach maximize 9x1 + 5x2 + 6x3 + 4x4 subject to 6x1 + 3x2 + 5x3 + 2x4 10 x3 + x4 1 -x1 + x3 0 -x2 + x4 1 LP-relaxation -- value 16.5 x = (0.8333, 1, 0, 1) x1 = 1, or x1 = 0 xj = {0, 1}, all j x1 = 1 value 16.2 x = (1, 0.8, 0, 0) x1 = 0 value 9.0 x = (0, 1, 0, 1) x1 = 1, x2 = 0 value 13.8 x = (1, 0, 0.8, 0) x1 = 1, x2 = 1 value 16 x = (1, 1, 0, 0.5) x1 = 1, x2 = 0 x3 = 1 value 9 x1 = 1, x2 = 0 x3 = 0 x1 = 1, x2 = 1, x4 =0 value 15.2 x = (1, 1, 0.2, 0) x1 = 1, x2 = 1, x4 = 1 x1 = 1, x2 = 1, x3=0, x4 =0 value 14 x1 = 1, x2 = 1, x3 = 1, x4 = 0 IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 81 c Marina A. Epelman

2 Divide and conquer approach z =max{c T x : x 2 S} Idea: If a problem is hard, partition its feasible region S into subsets, and solve each of the smaller subproblems Proposition 7.1 Let S = S 1 [...[ S K be a decomposition of the set S, andlet z k =max{c T x : x 2 S k } for k =1,...,K. Thenz =max k z k. I We refer to problems max{c T x : x 2 S k } for k =1,...,K as subproblems. I Solution to the original problem can be found by solving subproblems 1,...,K, and taking the best solution found I If any of the subproblems is still too hard to be solved directly, the set S k can be further decomposed, etc., thus forming a tree of subproblems. I Taken to the extreme, this leads to complete enumeration IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 82 c Marina A. Epelman Implicit (and intelligent) enumeration I In principle, this approach can lead to complete enumeration of the feasible solutions! undesirable. I Use information about subproblems to decide which branches of the tree are not worth exploring z =max{c T x : x 2 S} Proposition 7.2 Let S = S 1 [...[ S K be a decomposition of S into smaller sets, and let z k =max{c T x : x 2 S k } for k =1,...,K, U k be an upper bound on z k,andl k be a lower bound on z k. a Then I U =max k U k is an upper bound on z, I L =max k L k is a lower bound on z. a The textbook uses notation z = U and z = L IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 83 c Marina A. Epelman

3 Example: LP-relaxation based B&B (IP) z =max 4x 1 x 2 7x 1 2x 2 apple 14 x 2 apple 3 2x 1 2x 2 apple 3 x 2 Z 2 + Let S be the feasible region of (IP) Begin by solving the LP relaxation of (IP) I If LP is infeasible, then (IP) is infeasible, done I If LP has an integer optimal solution, we have the (IP) optimal solution, done I If optimal solution of LP is not feasible for (IP) I We found a dual bound U z I Do we have a primal bound L apple z? I We need to branch, i.e., create subproblems of (IP), and solve them in the same way (recursively) I If x solves LP and x j 62 Z, decomposes into S \ {x : x j appleb x j c} and S \ {x : x j b x j c +1} IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 84 c Marina A. Epelman Pruning subproblems in a B&B tree Combining bound and feasibility information, there are three potential ways to prune a subproblem S k (i.e., not create any further subproblems from it): 2 I By infeasibility: S k = ; I By optimality: z k =max{c T x : x 2 S k } has been solved (i.e., U k = L k ) I By bound: If a primal bound L is available, U k apple L To be able to prune by bound, keep information on the incumbent: x? 2 S : L = c T x? 2 For briefness, sometimes we refer to S k s as subproblems, although technically they are feasible regions of subproblems! IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 85 c Marina A. Epelman

4 A generic Branch and Bound algorithm 3 Initialization: Add (sub)problem S to the list, incumbent x? void, L = 1 1. Select and remove an active subproblem S k from the list 2. If S k is infeasible, prune it by infeasibility and go to 6.; otherwise, obtain a dual (upper) bound U k 3. If U k apple L, prunes k by bound and go to If solution to S k can be found, prune by optimality; if L < z k, update the incumbent x? to be the optimal solution of S k and L = z k ; go to Decompose S k into further subproblems and add them to the list 6. If the list of active subproblems is empty, stop, report incumbent x? and z = L as the optimal solution/value 3 This algorithm is for a maximization problem; the role of upper and lower bounds is reversed for a minimization problem IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 86 c Marina A. Epelman Designing a Branch and Bound algorithm I How to obtain dual bounds? How to obtain primal bounds? I How to branch? Separate into two parts, or into several parts? I In what order to examine subproblems? In the above example, we... I...used LP relaxations of subproblems to obtain dual bounds; known feasible solutions of subproblems (if known) to obtain primal bounds I...branched by considering fractional optimal values of variables in LP relaxation I... chose active subproblems from the list in arbitrarily order IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 87 c Marina A. Epelman

5 Obtaining bounds Want: good primal and dual bounds that can be obtained quickly I A primal bound becomes available whenever a subproblem is pruned by optimality. I To obtain a dual bound U k, solve a relaxation of the subproblem: I an LP relaxation (see example) I a combinatorial relaxation, e.g., I I Assignment relaxation for TSP I 1-tree relaxation for STSP Lagrangian relaxation I If using LP relaxations: I Better formulations ) better dual bounds ) more pruning by bound! (U k apple L) I Improve each new subproblem s formulation by adding VI s I I VI s remain valid in further branches, so keep them! This is the so-called Branch and Cut algorithm I If a dual problem is available, it can also be used to obtain dual bounds IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 88 c Marina A. Epelman How to partition into subproblems Want: I create a balanced search tree I ideally, create subproblems in which x k solution of the relaxation is not feasible (to improve dual bound) If using LP relaxations: Let x k 62 Z n solve LP relaxation of S k. I Variable dichotomy If xj k 62 Z, create S k,j = S k \ {x : x j applebxj k c} and S k,j+1 = S k \ {x : x j bxj kc +1} I BIP: GUB dichotomy Works with constraints P j2q x j = 1. Choose Q 1 Q: 0< P j2q 1 x j < 1, and create S k,0 = S k [ {x : P j2q 1 x j =0} and S k,1 = S k [ {x : P j62q 1 x j =0}. If using other relaxations: Specific to each problem class; important consideration: should be able to apply the same type of relaxation to the new subproblems IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 89 c Marina A. Epelman

6 Example: LP relaxation-based BIP maximize 9x1 + 5x2 + 6x3 + 4x4 subject to 6x1 + 3x2 + 5x3 + 2x4 10 x3 + x4 1 -x1 + x3 0 -x2 + x4 1 LP-relaxation -- value 16.5 x = (0.8333, 1, 0, 1) L = - ; U = 16.5 xj = {0, 1}, all j Node 1 : x1 = 1 value 16.2 x = (1, 0.8, 0, 0); L = - ; U = 16.5 Node 10 : x1 = 0 value 9.0 x = (0, 1, 0, 1) Pruned by optimality Node 2 : x1 = 1, x2 = 0 value 13.8 x = (1, 0, 0.8, 0); L = - ; U = 16.5 Node 5 : x1 = 1, x2 = 1 value 16 x = (1, 1, 0, 0.5); L = 7 ; U = 16.5 Node 3: x1 = 1, x2 = 0 x3 = 1 value 9 L = 9 ; U = 16.5 Pruned by optimality Node 4: x1 = 1, x2 = 0 x3 = 0 Node 7 : x1 = 1, x2 = 1, x3=0, x4 =0 value 14; L = 14 ; U = 16.5 Pruned by optimality Node 6 : x1 = 1, x2 = 1, x4 =0 value 15.2 x = (1, 1, 0.2, 0); L = 7 ; U = 16.5 Node 9: x1 = 1, x2 = 1, x4 = 1 Node 8: x1 = 1, x2 = 1, x3 = 1, x4 = 0 IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 90 c Marina A. Epelman Example: LP relaxation-based BIP maximize 9x1 + 5x2 + 6x3 + 4x4 subject to 6x1 + 3x2 + 5x3 + 2x4 10 x3 + x4 1 -x1 + x3 0 -x2 + x4 1 LP-relaxation -- value 16.5 x = (0.8333, 1, 0, 1) L = - ; U = 16.5 xj = {0, 1}, all j Node 2 : x1 = 1 value 16.2 x = (1, 0.8, 0, 0); L = 9 ; U = 16.2 Node 7 : x1 = 1, x2 = 0 value 13.8 x = (1, 0, 0.8, 0); L = 14 ; U = 15.2 u 8 = 13.8 < L; Pruned by bound u 2 = 14 U = 14 u 5 = 14 u 3 = 14 Node 6 : x1 = 1, x2 = 1, x3=0, x4 =0 value 14; L = 14 ; U = 16.2 Node 1 : x1 = 0 value 9.0 x = (0, 1, 0, 1); L = 9 ; U = 16.5 Prune by optimality Node 3 : x1 = 1, x2 = 1 value 16 x = (1, 1, 0, 0.5); L = 9 ; U = 16.2 Node 5 : x1 = 1, x2 = 1, x4 =0 value 15.2 x = (1, 1, 0.2, 0); L = 9 ; U = 15.2 Node 4: x1 = 1, x2 = 1, x4 = 1 Node 8: x1 = 1, x2 = 1, x3 = 1, x4 = 0 u 3 = 15.2 IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 91 c Marina A. Epelman

7 How to choose a node (subproblem) Recall: the order in which the nodes are considered a ects which subproblems can be pruned by bound. Some possible choices: I Depth-first I Reach integer solution quickly (improvement of L value) I Fast LP relaxation re-solves I But: if unlucky, will explore a wrong branch for a long tie I Breadth-first I Get quick estimates of the bound early in the process (improvement of U value) I But: too long to find an integer solution I Best-Node First I Choosing the node with the best dual bound might lead to finding a good incumbent quickly, thus eliminating the need to solve the other subproblems. I But: need to solve relaxations at a lot of nodes to get the bound. In practice, a mixture of these strategies is used. IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 92 c Marina A. Epelman A B&B method for TSP min P n i=1 P n j=1 c ijx ij P n i=1 x ij =1, j =1,...,n P n j=1 x ij =1, i =1,...,n No subtours x ij 2 {0, 1} 8i, 8j I Bounding Solve the assignment relaxation of the subproblems I Branching Find a subtour constraint that s violated; suppose the subtour has length k apple n/2. Create k subproblems by adding constraint x ij = 0 prohibiting one of the arcs of one of the subtours. (Note: fixing x ij =0canalsobeinterpreted as changing the cost c ij so that it becomes prohibitively high. Thus, the resulting subproblems are still TSPs, and assignment relaxation can be used for each subproblem.) IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 93 c Marina A. Epelman

8 A combinatorial relaxation of symmetric TSP STSP: atspwithc ij = c ji I Bounding Solve the 1-tree relaxation of the subproblems: ( ) X z STSP =min c e : T is a tour T min T e2t ( ) X c e : T is a 1-tree = z 1-tree e2t I Branching I If a 1-tree is not feasible for STSP, what constraint(s) does it violate? I Can you suggest a branching strategy? I Can you still formulate/solve a 1-tree relaxation of the resulting subproblem? IOE 518: Introduction to IP, Winter 2012 Branch-and-Bound Page 94 c Marina A. Epelman