Integer Programming Chapter 9


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1 1 Integer Programming Chapter 9 University of Chicago Booth School of Business Kipp Martin October 30, 2017
2 2 Outline Branch and Bound Theory Branch and Bound Linear Programming Node Selection Strategies Variable Selection Strategies Problem Formulation Quality Solver Options Epsilon Optimality Preprocessing
3 Branch and Bound Theory In American football, on fourth and long, what do you do? We do the same thing in optimization.
4 Branch and Bound Theory We punt and solve an easier problem. We solve a relaxation of the original problem. We may solve many (many, many, many) relaxed problems.
5 Branch and Bound Key Concepts a problem relaxation basic branch and bound an upper bound a lower bound fathoming a node Reading: Chapter 9 of the text.
6 Branch and Bound Theory Problem (PR) min{f (x) x Γ R n } is a relaxation of problem (P) min{g(x) x ˆΓ R n } if and only if ˆΓ Γ and f (x) g(x) for all x ˆΓ.
7 Branch and Bound Theory It follows that if (PR) is a relaxation of (P) the optimal solution value of (PR) is less than or equal to the optimal solution value of (P). If (PR) is relaxation of (P), then (P) is a restriction of (PR).
8 Relaxation Example Let u be an arbitrary set of multipliers for the Ax = b. Then the original linear program min c x LP(x) s.t. Ax = b x 0 is equivalent to min c x + u (b Ax) LP(x, u) s.t. Ax = b x 0
9 Relaxation Example Let s rewrite LP(x, u) slightly as min (c u A)x + u b LP(x, u) s.t. Ax = b x 0 This is going to seem a bit weird, but just go with me on this. Define a problem relaxation of LP(x, u) by min (c u A)x + u b LPR(x, u) x 0 I formed problem LPR(x, u) from problem LP(x, u) by deleting the Ax = b constraints hence the term relaxation.
10 10 Relaxation Example Variation on a Theme: Here is another simple relaxation. Take the integer programming problem: min c x (1) s.t. Ax = b (2) x 0 (3) x j {0, 1}, j B (4) x j Z, j I (5)
11 11 Relaxation Example and replace it with the linear programming relaxation: min c x (6) s.t. Ax = b (7) x 0 (8) x j 1 (9) For now we will work with the linear programming relaxation. We will come back to LPR(x, u) later.
12 Branch and Bound Theory The solution technique for (MIP) used most often in practice is branchandbound. Branchandbound is a philosophy for problem solution, not really a specific algorithm. There are three major factors affecting the efficiency of this method.
13 Branch and Bound Theory 1. Selection of a problem relaxation: Want a relaxation that is tight! Want a relaxation that is easy to solve! want to have our cake and it too! 2. Problem branching/separation: from a current candidate problem create new candidate problems. The new candidate problems are restrictions of the parent candidate problem. 3. Problem selection: select a problem from the candidate list. It is important to make the selection in such a fashion that the gap between the upper and lower bounds closes rapidly and that feasible solutions are found.
14 14 Branch and Bound Linear Programming Step 1: (Initialization) If there is a known feasible solution x to (MIP) set z UB c x, if there is no known feasible solution set z UB. The feasible solution x which gives the smallest possible value for z UB is known as the incumbent and z UB is an upper bound on the optimal solution value of (MIP). Solve (MIP), the linear programming relaxation of (MIP). If (MIP) is infeasible then (MIP) is infeasible. If the (MIP) is integer for all j I, stop with an optimal solution to (MIP). Otherwise, add problem (MIP) to the list of candidate problems and go to Step 2. Step 2: (Problem Selection) Select a candidate problem for separation from the list of candidate problems and go to Step 3. A common rule for candidate problem selection is to select the candidate problem with the smallest (assuming a minimization) linear programming relaxation value.
15 Branch and Bound Linear Programming Step 3: (Branching/Separation) The candidate problem (CP) under consideration has at least one fractional integer variable. 3.a Select a fractional integer variable, x k = n k + f k for branching purposes. Here n k is a nonnegative integer and f k is in the open interval (0, 1). 3.b Create two new mixed integer programs from (CP). Create one new candidate problem by adding the constraint x k n k + 1 to the constraint set of (CP). Create the second new candidate problem from (CP) by adding the constraint x k n k to the constraint set of (CP). The two new candidate problems are restrictions of the parent candidate problem since they are created by adding a constraint to the parent.
16 Branch and Bound Linear Programming 3.c Solve the linear programming relaxation of the two new candidate problems. 3.c.i If the linear programming relaxation is infeasible drop the newly created problem from further consideration. 3.c.ii If the linear programming relaxation is integer for all j I update the incumbent value z UB and make this solution the incumbent if necessary. Drop this problem from further consideration. 3.c.iii If the linear programming relaxation has at least one fractional variable x j for j I and the objective function value is strictly less than z UB, add this problem to the list of candidate problems. Go to Step 4
17 Branch and Bound Linear Programming Step 4: (Optimality Test) Delete from the list of candidate problems any problem with an objective value of its relaxation which is not strictly less than z UB. Stop if this list is emptied. Otherwise, go to Step 2. If the list is emptied and z UB = problem (MIP) is infeasible, otherwise z UB is the optimal solution value of (MIP). In Steps 3 and 4, a candidate problem is deleted when the linear programming relaxation is infeasible, or integral, or has a linear programming relaxation value larger than z UB. This process of deleting candidate problems is known as fathoming.
18 Branch and Bound Linear Programming min x 1 x 2 s.t. x 1 + 2x 2 5 9x 1 + 4x x 1 2x 2 4 x 1, x 2 0 x 1, x 2 Z Solution: x 1 = , x 2 = Objective function value is Set z UB =. Select x 1 as the branching variable.
19 Branch and Bound Linear Programming Add the constraint x 1 0 gives min x 1 x 2 Linear Program 2 : s.t. x 1 + 2x 2 5 9x 1 + 4x x 1 2x 2 4 x 1 0 x 1, x 2 0 The optimal solution is x 1 = 0, x 2 = 2.5 with optimal objective function value 2.5.
20 Branch and Bound Linear Programming Add the constraint x 1 1 gives min x 1 x 2 Linear Program 3 : s.t. x 1 + 2x 2 5 9x 1 + 4x x 1 2x 2 4 x 1 1 x 1, x 2 0 The optimal solution is x 1 = 1.0, x 2 = 2.25 with an optimal objective function value of There are now two candidate problems. Select candidate problem 3 for further branching since it has the smallest linear programming relaxation value.
21 21 Branch and Bound Linear Programming x 1 <= 0 1 x 1 = x 2 = x 1 >= x 1 = 1.0 x 2 = 2.25 x 1 = 0.0 x 2 = 2.5 x 2 <= 2 x 2 >= 3 x 1 = x 2 = 2.0 x 1 <= x 1 >= 2 Infeasible 6 7 Infeasible x 1 = 1.0 x 2 = 2.0
22 22 Branch and Bound Linear Programming Variable x 2 is the only fractional variable for linear program 3 and is used for branching. Create two new linear programs by using the constraints x 2 2 and x 2 3. First branch on x 2 2. min x 1 x 2 Linear Program 4 : s.t. x 1 + 2x 2 5 9x 1 + 4x x 1 2x 2 4 x 1 1 x 2 2 x 1, x 2 0 The optimal solution is x 1 = , x 2 = 2 with an optimal objective function value of
23 Branch and Bound Linear Programming Next branch on x 2 2. min x 1 x 2 Linear Program 5 : s.t. x 1 + 2x 2 5 9x 1 + 4x x 1 2x 2 4 x 1 1 x 2 3 x 1, x 2 0 Linear program 5 is infeasible. There are now two candidate problems. They are candidate problems 2 and 4. Since linear program 4 has the smallest objective function value select it for branching.
24 Branch and Bound Linear Programming Create linear programs 6 and 7 by branching on variable x 1 = First branch on x 1 1. min x 1 x 2 Linear Program 6 : s.t. x 1 + 2x 2 5 9x 1 + 4x x 1 2x 2 4 x 1 1 x 2 2 x 1 1 x 1, x 2 0 The optimal solution is x 1 = 1, x 2 = 2 with an optimal objective function value of 3. Since this solution is integer it is the new incumbent and provides the new upper bound z UB = 3.
25 Branch and Bound Linear Programming Linear program 7 is infeasible. Candidate problem 2 is the only candidate problem remaining in the list. The linear programming relaxation value of this candidate problem is 2.5 which is worse than the incumbent upper bound value of 3.0. Delete candidate problem 2. There are no candidate problems remaining so the optimal solution to the integer program is x 1 = 1, x 2 = 2 with optimal solution value 3.
26 26 Branch and Bound Linear Programming x 1 <= 0 1 x 1 = x 2 = x 1 >= x 1 = 1.0 x 2 = 2.25 x 1 = 0.0 x 2 = 2.5 x 2 <= 2 x 2 >= 3 x 1 = x 2 = 2.0 x 1 <= x 1 >= 2 Infeasible 6 7 Infeasible x 1 = 1.0 x 2 = 2.0
27 Node Selection Strategies It is necessary to pick a node (candidate problem) from a candidate list. Select a node with the best (relaxed) objective function value. This is a breadth first strategy and can leave many dangling nodes which presents a problem of computer storage. Depth first, or lastin, firstout (LIFO) where the node selected is the most recently created node with the smallest linear programming solution value Select a node based on calculations made related to fractional variables.
28 28 Variable Selection Strategies Pick the fractional variable that is the farthest from being integer Assign priorities (either static or dynamic). Branch on the fractional variable with the highest priority. Use pseudocosts P U k := zj+1 z j f k, P D k := zj+2 z j 1 f k Use estimates from dualsimplex pivots or use strong branching.
29 Problem Formulation Quality THE BIG TAKE AWAY: Different polyhedra may contain exactly same set of integer points! THE BIG TAKE AWAY: Different polyhedra may contain exactly same set of integer points! THE BIG TAKE AWAY: Different polyhedra may contain exactly same set of integer points! WHAT IS THE BIG TAKE AWAY?
30 Problem Formulation Quality Absolutely Critical: Understand the differences between 1. Γ this is an MILP 2. Γ this is an LP 3. conv(γ) this is an LPR In most interesting cases conv(γ) Γ. We want to find conv(γ) or a close approximation.
31 31 Reformulation (Lot Sizing) Dynamic Lot Sizing : Variables: x it units of product i produced in period t I it inventory level of product i at the end of the period t y it is 1 if there is nonzero production of product i during period t, 0 otherwise Parameters: d it demand product i in period t f it fixed cost associated with nonzero production of product i in period t c it marginal production cost for product i in period t h it marginal holding cost charged to product i at the end of period t g t production capacity in period t
32 Reformulation (Lot Sizing) Objective: Minimize sum of marginal production cost, holding cost, fixed cost N T (c it x it + h it I it + f it y it ) i=1 t=1 Constraint 1: Do not exceed total capacity in each period N x it g t, i=1 t = 1,..., T
33 Reformulation (Lot Sizing) Constraint 2: Inventory balance equations I i,t 1 + x it I it = d it, i = 1,..., N, t = 1,..., T Constraint 3: Fixed cost forcing constraints x it M it y it 0, i = 1,... N, t = 1,..., T
34 Reformulation (Lot Sizing) Dynamic Lot Sizing : A standard formulation is: min s.t. N T (c it x it + h it I it + f it y it ) i=1 t=1 N x it g t, t = 1,..., T i=1 I i,t 1 + x it I it = d it, i = 1,..., N, t = 1,..., T x it M it y it 0, i = 1,... N, t = 1,..., T x it, I it 0, i = 1,..., N, t = 1,..., T y it {0, 1}, i = 1,..., N, t = 1,..., T.
35 Reformulation (Lot Sizing) Dynamic Lot Sizing : An alternate formulation: z itk is 1, if for product i in period t, the decision is to produce enough items to satisfy demand for periods t through k, 0 otherwise. z ij,t 1 + t 1 j=1 T z i1k = 1 k=1 T z itk = 0, i = 1,..., N, t = 2,..., T k=t T z itk y it, i = 1,..., N, t = 1,..., T k=t
36 Reformulation (Lot Sizing) Dynamic Lot Size (tvw200) : Tight Loose Rows Columns LP Value What happens when you try to solve?
37 Solver Options Most solvers take options. This is particularly important in integer programming. In GAMS you can communicate options to solvers through an option text file. First we tell GAMS which solver we want: OPTION MIP = CoinCbc; Next we tell GAMS that we want the first option file (you can have more than one, 2, 3,...) lot_size.optfile = 1;
38 Solver Options Here is some code we where we put in GAMS options file opt CoinCbc option file /coincbc.opt/; put opt; put optcr 0 / put reslim / put nodelim / put cuts off / put knapsackcuts on /; putclose opt; Make sure to put this code before the solve statement.
39 Solver Options Here is what the options file does: set the tolerance on integer optimality to zero (optcr 0) set a time limit of seconds set a node limit of turned cutting plane generation off (cuts off) turned knapsack cuts on (knapsackcuts on)
40 Epsilon Optimality Key Take Away: Actually proving optimality in branch and bound can be tough. The closer you get, the harder it becomes to resolve that list bit of integrality gap. If, for example, you set optcr =.01, then branch and bound will terminate (assume minimization here) when.99*ub LB
41 41 Epsilon Optimality Experiment: run the tight version of the lot sizing with ratio 0 and ratio Here is what happens for me: UB LB Nodes Seconds optcr = optcr = Is it worth it?
42 Epsilon Optimality Another Take Away: the data may not be that accurate to being with. Standard Oil story.
43 Preprocessing By preprocessing we mean what is done to a formulation to make it more amenable to solution before solving the linear programming relaxation. Objective: make the linear programming relaxation of the mixedinteger program easy and tight. Try the following eliminate redundant constraints fix variables scale coefficients coefficient reduction rounding improve bounds on variables and constraints probing We work with the following canonical form: a j x j + a j x j a j x j a j x j b (10) j I + j C + j I j C
44 44 Preprocessing Rounding: If C + = C =, a j is integer for all j I + I and α = gcd(a j a j I + I ) then the conical form of the constraint is which is equivalent to then a valid rounding is a j x j a j x j b (11) j I + j I (a j /α)x j (a j /α)x j b/α. (12) j I + j I (a j /α)x j (a j /α)x j b/α. (13) j I + j I
45 Preprocessing Rounding Example: Consider the inequality (assume x 1, x 2 are general integer variables. 2x 1 + 2x 2 3 A feasible solution is x 1 = 1.5 and x 2 = 0. Now let s round, So an equivalent inequality is: α = gcd(a 1, a 2 ) = gcd(2, 2) = 2 (1/2)(2x 1 + 2x 2 ) (1/2)3 x 1 + x Rounding up the righthandside gives: x 1 + x 2 2 Is x 1 = 1.5 and x 2 = 0 feasible?
46 46 Preprocessing Rounding Example: Let s look at some geometry. Plot the feasible regions: Γ 1 = {(x 1, x 2 ) 2x 1 + 2x 2 3, x 1, x 2 0} Γ 2 = {(x 1, x 2 ) x 1 + x 2 2, x 1, x 2 0} What is the relationship between? Γ 1 Z 2 and Γ 2 Z 2 Γ 1 and Γ 2
47 47 Preprocessing Coefficient Reduction: if C = I = then it is valid to reduce the coefficients on the integer variables to b. That is, (10) is equivalent to j I + min{a j, b}x j + j C + a j x j b. (14) Additionally, if C or I is not empty then upper bounds on the variables in these sets are used as follows. Define: Then (10) is equivalent to λ := b + a j h j + a j h j. (15) j C j I min{a j, λ}x j + a j x j a j x j a j x j b. (16) j I + j C + j I j C
48 48 Preprocessing Tightening Bounds: The canonical form a j x j + a j x j a j x j a j x j b j I + j C + j I j C implies for each k C + a k x k b j C + j k a j x j a j x j + a j x j + a j x j. j I + j C j I The smallest the right hand side of can be is b j C + j k a j h j a j h j + a j l j + a j l j. j I + j C j I
49 49 Preprocessing Therefore, it is valid to reset l k to (b j C + j k a j h j a j h j + a j l j + a j l j )/a k j I + j C j I Using similar logic the upper bounds of variables indexed by C I are adjusted by ( a j l j + a j l j a j h j a j h j b)/a k j C + j I + j I j C j k
50 Preprocessing This is an iterative process. The upper and lower bounds on variables are adjusted until there is no improvement in an upper or lower bound. Once the lower and upper bounds are calculated one can apply coefficient reduction on the integer variable coefficients.
51 Preprocessing Example: Dynamic Lot Sizing min s.t. N T (c it x it + h it I it + f it y it ) i=1 t=1 N x it g t, t = 1,..., T i=1 I i,t 1 + x it I it = d it, i = 1,..., N, t = 1,..., T x it M it y it 0, i = 1,... N, t = 1,..., T x it, I it 0, i = 1,..., N, t = 1,..., T y it {0, 1}, i = 1,..., N, t = 1,..., T.
52 52 Preprocessing Example: Dynamic Lot Sizing (Continued) Consider the Big M constraints in canonical form x it My it 0 x it + My it 0 Recall λ := b + a j h j + a j h j. j C j I In this case, what is b, C, I, and λ? What do we get after coefficient reduction?
53 Preprocessing Example: Dynamic Lot Sizing (Continued) Let s tighten the bounds on the x it variables. Assume 5 time periods (we drop the product subscript all products are treated identically). In canonical form: I 4 x 5 d 5 I 3 x 4 + I 4 d 4 I 2 x 3 + I 3 d 3 I 1 x 2 + I 2 d 2 x 1 + I 1 d 1 What is a valid upper bound on x 5? What about I 4? Work through to period 1.
54 54 Preprocessing Feasibility: The canonical form is a j x j + a j x j a j x j a j x j b j I + j C + j I j C if a j h j + a j h j a j l j a j l j < b. j I + j C + j I j C the model instance is not feasible.
55 55 Preprocessing Redundancy: The canonical form is if a j x j + a j x j a j x j a j x j b j I + j C + j I j C a j l j + a j l j a j h j a j h j b. j I + j C + j I j C the constraint is redundant and can be deleted. Solve (possibly relaxations) min a j x j + a j x j a j x j a j x j j I + j C + j I j C s.t. Ax b x 0
56 56 Preprocessing Probing and Variable Fixing: In a mixed 0/1 linear program probing refers to fixing a binary variable x k to 0 or 1 and then observing any resulting implications. Assume variable x k is binary. If x k = 1, k I and j I + a j h j + j C + a j h j j I \{k} a j l j j C a j l j < b + a k then the model is infeasible which implies it is valid to fix variable x k to 0. If x k = 0, k I + and j I + \{k} a j h j + a j h j a j l j a j l j < b j C + j I j C then the model is infeasible which implies it is valid to fix variable x k to 1.
57 Preprocessing How to preprocess in CoinOR Cbc. See ( for free optimization solvers. Declare a new solver interface this is what will get preprocessed. OsiSolverInterface *m_osisolverpre = NULL; CglPreProcess process; m_osisolverpre = process.preprocess(*solver>osisolver, false, 10); Build the Cbc model with the preprocessed solver interface and solve CbcModel *model = new CbcModel( *m_osisolverpre); model>branchandbound();
58 58 Preprocessing Unwind to get the original model back process.postprocess( *model>solver() ); Results with p0033.osil Variables Constraints LP Relax Nodes Without With
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