# Introduction to Mathematical Programming IE406. Lecture 20. Dr. Ted Ralphs

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1 Introduction to Mathematical Programming IE406 Lecture 20 Dr. Ted Ralphs

2 IE406 Lecture 20 1 Reading for This Lecture Bertsimas Sections 10.1, 11.4

3 IE406 Lecture 20 2 Integer Linear Programming An integer linear program (ILP) is the same as a linear program except that the variables can take on only integer values. If only some of the variables are constrained to take on integer values, then we call the program a mixed integer linear program (MILP). The general form of a MILP is min s.t. c x + d y Ax + By = b x, y 0 x integer We have already seen a number of examples of integer programs. Product mix problem Cutting stock problem Integer knapsack problem Assignment problem Minimum spanning tree problem

4 IE406 Lecture 20 3 How Hard is Integer Programming? Solving general integer programs can be much more difficult than solving linear programs. There in no known polynomial-time algorithm for solving general MILPs. Solving the associated linear programming relaxation results in a lower bound on the optimal solution to the MILP. In general, an optimal solution to the LP relaxation does not tell us anything about an optimal solution to the MILP. Rounding to a feasible integer solution may be difficult. The optimal solution to the LP relaxation can be arbitrarily far away from the optimal solution to the MILP. Rounding may result in a solution far from optimal. We can bound the difference between the optimal solution to the LP and the optimal solution to the MILP (how?).

5 IE406 Lecture 20 4 Duality in Integer Programming Let s consider again an integer linear program min s.t. c x Ax = b x 0 x integer As in linear programming, there is a duality theory for integer programs. We can dualize some of the constraints by allowing them to be violated and then penalizing their violation in the objective function. We relax some of the constraints by defining, for given Lagrange multipliers p, the Lagrangean relaxation Z(p) = min x X {c x + p (A x b)} where X = {x Z n A x = b, x 0} and A = [(A ), (A ) ].

6 IE406 Lecture 20 5 More Integer Programming Duality Z(p) is a lower bound on the optimal solution to the original ILP, so we consider the Lagrangean dual max Z(p). As long as we can optimize over the set X, we can solve the Lagrangean dual efficiently. As before, the optimal solution to the Lagrangean dual yields a lower bound on the optimal value of the original ILP (weak duality). However, for integer programming, strong duality does not hold. The difference between the optimal solution to the ILP and the optimal solution to the dual is called the duality gap. This is another indication of why integer programming is difficult.

7 It is easy to see why solving the LP relaxation does not necessarily yield a good solution. IE406 Lecture 20 6 The Geometry of Integer Programming Let s consider again an integer linear program min s.t. c x Ax = b x 0 x integer The feasible region is the integer points inside a polyhedron.

8 IE406 Lecture 20 7 Easy Integer Programs As we have already seen, certain integer programs are easy. What makes an integer program easy? All of the extreme points of the LP relaxation are integral. Every square submatrix of A has determinant +1, -1, or 0. We know a complete description of the convex hull of feasible solutions. We have an efficient algorithm for finding an optimal integer solution (other than linear programming). There is no duality gap. Examples of easy integer programs. Minimum cost network flow problems. Assignment problem. Minimum cost spanning tree problem.

9 IE406 Lecture 20 8 Modeling with Integer Variables Why do we need integer variables? We have already seen some examples. If the variable is associated with a physical entity that is indivisible, then it must be integer. Product mix problem. Cutting stock problem. We can use 0-1 (binary) variables for a variety of purposes. Modeling yes/no decisions. Enforcing disjunctions. Enforcing logical conditions. Modeling fixed costs. Modeling piecewise linear functions.

10 IE406 Lecture 20 9 Modeling Binary Choice We use binary variables to model yes/no decisions. Example: Integer knapsack problem We are given a set of items with associated values and weights. We wish to select a subset of maximum value such that the total weight is less than a constant K. We associate a 0-1 variable with each item indicating whether it is selected or not. m max c j x j s.t. j=1 m w j x j K j=1 x 0 x integer

11 IE406 Lecture Modeling Dependent Decisions We can also use binary variables to enforce the condition that a certain action can only be taken if some other action is also taken. Suppose x and y are variables representing whether or not to take certain actions. The constraint x y says only take action x if action y is also taken.

12 IE406 Lecture Example: Facility Location Problem We are given n potential facility locations and m customers that must be serviced from those locations. There is a fixed cost c j of opening facility j. There is a cost d ij associated with serving customer i from facility j. We have two sets of binary variables. y j is 1 if facility j is opened, 0 otherwise. x ij is 1 if customer i is served by facility j, 0 otherwise. min s.t. n c j y j + j=1 n x ij = 1 j=1 x ij y j m i=1 x ij, y j {0, 1} n d ij x ij j=1 i i, j i, j

13 IE406 Lecture Selecting from a Set We can use constraints of the form j T x j 1 to represent that at least one item should be chosen from a set T. Similarly, we can also model that at most one or exactly one item should be chosen. Example: Set covering problem A set covering problem is any problem of the form min c x s.t. Ax 1 x j {0, 1} j where A is a 0-1 matrix. Each row of A represents an item from a set S. Each column A j represents a subset S j of the items. Each variable x j represents selecting subset S j. The constraints say that {j xj =1}S j = S. In other words, each item must appear in at least one selected subset.

14 IE406 Lecture Example: Combinatorial Auctions The winner determination problem for a combinatorial auction is a set covering problem. The rows represent items or services that a buyer is trying to acquire. The columns represent subsets of the items that a particular supplier can provide for a specified cost. The object is to select a subset of the bidders such that cost is minimized, and every item is provided by at least one bidder. This is a set covering problem. Similarly, we can also consider set packing and set partitioning problems.

15 IE406 Lecture Modeling Disjunctive Constraints We are given two constraints a x b and c x d with nonnegative coefficients. Instead of insisting both constraints be satisfied, we want at least one of the two constraints to be satisfied. To model this, we define a binary variable y and impose a x yb, c x (1 y)d, y {0, 1}. More generally, we can impose that exactly k out of m constraints be satisfied with (a i ) x b i y i, i [1..m] m y i k, i=1 y i {0, 1}

16 IE406 Lecture Modeling a Restricted Set of Values We may want variable x to only take on values in the set {a 1,..., a m }. We introduce m binary variables y j, j = 1,..., m and the constraints m x = a j y j, j=1 m y j = 1, j=1 y j {0, 1}

17 IE406 Lecture Piecewise Linear Cost Functions We can use binary variables to model arbitrary piecewise linear cost functions. The function is specified by ordered pairs (a i, f(a i )) and we wish to evaluate it at a point x. We have a binary variable y i, which indicates whether a i x a i+1. To evaluate the function, we will take linear combinations k i=1 λ if(a i ) of the given functions values. This only works if the only two nonzero λ i s are the ones corresponding to the endpoints of the interval in which x lies.

18 IE406 Lecture Minimizing Piecewise Linear Cost Functions The following formulation minimizes the function. k min λ i f(a i ) s.t. i=1 k i=1 = 1, λ 1 y 1, λ i y i 1 + y i, i [2..k 1], λ k y k 1, k 1 i=1 λ i 0, y i = 1, y i {0, 1}. The key is that if y j = 1, then λ i = 0, i j, j + 1.

19 IE406 Lecture Fixed-charge Problems In many instances, there is a fixed cost and a variable cost associated with a particular decision. Example: Fixed-charge Network Flow Problem We are given a directed graph G = (N, A). There is a fixed cost c ij associated with opening arc (i, j) (think of this as the cost to build the link). There is also a variable cost d ij associated with each unit of flow along arc (i, j). Minimizing the fixed cost by itself is a minimum spanning tree problem (easy). Minimizing the variable cost by itself is a minimum cost network flow problem (easy). We want to minimize the sum of these two costs (difficult).

20 IE406 Lecture Modeling the Fixed-charge Network Flow Problem To model the FCNFP, we associate two variables with each arc. x ij (fixed-charge variable) indicates whether arc (i, j) is open. f ij (flow variable) represents the flow on arc (i, j). Note that we have to ensure that f ij > 0 x ij = 1. Min s.t. (i,j) A j O(i) c ij x ij + d ij f ij f ij j I(i) f ji = b i f ij Cx ij f ij 0 i N (i, j) A (i, j) A x ij {0, 1} (i, j) A

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