February 19, Integer programming. Outline. Problem formulation. Branch-andbound

Transcription

1 Olga Galinina ELT Network Analysis and Dimensioning II Department of Electronics and Communications Engineering Tampere University of Technology, Tampere, Finland February 19, 2014

2 1 2 Branch-and-bound 3

3 1 2 Branch-and-bound 3

4 Optimization

5 Optimization minimize f (x), x R n subject to x Ω.

6 Optimization minimize f (x), x R n subject to x Ω. Pure integer problem An integer problem in which all variables are required to be integer is NP-hard.

7 Optimization minimize f (x), x R n subject to x Ω. Pure integer problem An integer problem in which all variables are required to be integer Mixed integer problem If some variables are restricted to be integer and some are not is NP-hard.

8 Optimization minimize f (x), x R n subject to x Ω. Pure integer problem An integer problem in which all variables are required to be integer Mixed integer problem If some variables are restricted to be integer and some are not Pure (mixed) binary integer problems (0-1 ) The integer variables are restricted to be 0 or 1 is NP-hard.

9 Example: knapsack problem Maximize the sum of the values of the items in the knapsack so that the sum of the weights must be less than the knapsack s capacity. minimize c T x, x j {0, 1,..., n i } subject to w T x W 1 number of items, each with a weight w i and a value c i 2 to maximize the total value of the items in the knapsack 3 knapsack problems are (usually) easy to solve

10 Relationship to Linear Programming minimize c T x, x j {0, 1,..., n i } subject to Ax = b, x Z+ n An associated linear program (the linear relaxation): minimize c T x, x j {0, 1,..., n i } subject to Ax = b, x R+ n 1 The optimal objective value for (LR) is less than or equal to the optimal objective for (IP) 2 If (LR) is infeasible, then so is (IP) 3 If (LR) is optimized by integer variables, then that solution is feasible and optimal for (IP) 4 (LP) gives a bound on the optimal value of (IP) Rounding the solution of LR will not in general give the optimal solution of (IP)

11 Whereas the simplex method is effective for solving linear programs, there is no single technique for solving integer programs. Three approaches: enumeration techniques, including the branch-and-bound procedure cutting-plane techniques group-theoretic techniques

12 Computational Complexity: LP vs. IP Including integer variables increases enormously the modeling power, at the expense of more complexity LP s can be solved in polynomial time with interior-point methods (ellipsoid method, Karmarkar s ) Programming is an NP-complete problem There is no known polynomial-time There are little chances that one will ever be found Even small problems may be hard to solve

13 Example: Traveling salesman problem Starting from his home, a salesman wishes to visit each of (n1) other cities and return home at minimal cost. He must visit each city exactly once and it costs c ij to travel from city i to city j. What route should he select? minimize n i,j=1 c ijx ij subject to n j=1 x ij = 1 n i=1 x ij = 1, x ij {0, 1} The constraints require that the salesman must enter and leave each city exactly once.

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15 General idea of Branch-and-bound strategy of divide and conquer divide the feasible region into more manageable subdivisions there are a number of branch-and-bound s Utilizes: the value of the objective function (LP) is a lower bound on the (IP) any integer feasible point is always an upper bound on the optimal (LP) value

16 Example maximize z = 5x 1 + 8x 2 subject to x 1 + x 2 6 5x 1 + 9x 2 45 x 1, x 2 0, x 1, x 2 Z

17 Example maximize z = 5x 1 + 8x 2 subject to x 1 + x 2 6 5x 1 + 9x 2 45 x 1, x 2 0, x 1, x 2 Z In an example as simple as this, almost any solution procedure will be effective (even exhaustive search)

18 Example Linear- solution has x 1 = and x 2 = 3 3 4, z 41 First subdivision is into the regions where x 2 3 and x 2 4 Subdividing the feasible region:

19 Example Consider L 1 first: (4, 9 5 ), z = 41 Not integer we subdivide L 1 further, into the regions: L 3 with x 1 2 (infeasible) and L 4 with x 1 1 Consider L 4 : (1, 40 9 ), z = 41 Not integer we subdivide L 4 further, into the regions: L 5 with x 2 4 (infeasible) and L 4 with x 2 5 Consider L 5 : (1, 4), z = 37 no integer x in subdivision can give larger value New bound z z 41 Consider L 6 : (0, 5), z = z 41 Consider L 6 : (3, 3), z = 39, no optimal points in L 6

20 Example Subdividing the feasible region:

21 Enumeration tree

22 Heuristics in Branch-and-Bound Possible choices in Branch-and-Bound Choosing a pending problem Depth-first search Breadth-first search Best-first search (select node with best cost value) Choosing a branching variable closest to halfway two integer values with least cost coefficient which is important in the model (0-1 variable) which is biggest in a variable ordering No known strategy is best for all problems!

23 Summary To subdivide the feasible region to develop bounds z 1 < z < z 2 on minimum value. the upper bound z is the highest value of any feasible integer point the lower bound is given by the optimal value of the associated linear program after subdivision, move to another subdivision and analyze subdivision need not be subdivided if the linear program over L j is infeasible the optimal linear- solution over L j is integer the value of the linear- solution z j over L j satisfies z j z (for min) Can the linear programs corresponding to the subdivisions be solved efficiently? yes, use dual simplex (few operations)

24 Mental break If a test for a disease is 99% accurate, and someone s test is positive, what is the probability the person actually has the disease?.

25 Mental break If a test for a disease is 99% accurate, and someone s test is positive, what is the probability the person actually has the disease?.

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27 Cutting plane solves integer programs by modifying LP until the integer solution is obtained works with a single linear program, which it refines by adding new constraints new constraints successively reduce the feasible region until an integer optimal solution is found BB almost always outperform the cutting-plane the first for IP that could be proved to converge in a finite number of steps

28 Cutting plane A cut relative to a current fractional solution satisfies the following criteria: a) No feasible integer solutions are excluded b) Each constraint reduces the feasible solution region c) Each constraint passes through an integer point d) An optimum solution is eventually found

29 Cutting away the linear- solution

30 Summary 1958, Gomory: IP can be solved by some linear program (the associated linear program plus the added constraints) number of cuts to be added, though finite, is usually quite large BB almost always outperform the cutting-plane

31 Mental break There are 10 red balls in a pool of 23 different balls. An experiment is to draw two ball from the pool. What is the probability, that both are red?

32 Mental break There are 10 red balls in a pool of 23 different balls. An experiment is to draw two ball from the pool. What is the probability, that both are red? The number of all possible outcomes and the number of sought outcomes: ( ) ( ) 2 2 N = = 253, N = = Probability p = N M =