Integer and Combinatorial Optimization: The Cut Generation LP

Transcription

1 Integer and Combinatorial Optimization: The Cut Generation LP John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY USA March 2019 Mitchell The Cut Generation LP 1 / 15

2 General disjunctions General disjunctions Let Q j = {x R n : A j x b j } for j 1,..., q. Note that any bounds on x are included in the constraints A j x b j, for notational purposes. Our feasible region is the union of polyhedra. P := j=1,...,q For example, we could look at a disjunction on a binary variable. More generally, we could look at different Q j corresponding to different allocations to subsets of the variables. Q j, Mitchell The Cut Generation LP 2 / 15

3 General disjunctions Convex hull The convex hull of P can be written in a lifted space as x R n : x q j=1 zj = 0 A j z j b j z j 0 0 for j = 1,..., q conv(p) = z j 0 0 for j = 1,..., q q j=1 zj 0 = 1 for some z j R n, z j 0 R, for j = 1,..., q The z j 0 variables are the weights in the convex combination for each set Q j. The points 1 z j are feasible in each Q j for positive z j z j 0. If zj 0 = 0 and 0 z j 0 then z j is a ray of Q j. This leads to a general version of lift-and-project. Mitchell The Cut Generation LP 3 / 15

4 Finding a disjunctive cut Disjunctive cuts In this handout, we focus on extending the cut generation LP to find violated constraints. Assume we have a point x conv(p). We want to find a valid cutting plane π T x π 0, valid for each x Q j but violated by x. This constraint must satisfy π (A j ) T λ j, π 0 (b j ) T λ j, λ j 0, for j = 1,..., q. We also need a normalization constraint, perhaps q (e j ) T λ j = 1, j=1 where e j is the vector of ones, with dimension equal to the number of rows in A j. Mitchell The Cut Generation LP 4 / 15

5 Cut generation LP Finding a disjunctive cut This leads to the general form of the cut generation LP: max π,π0,λ j πt x π 0 subject to π (A j ) T λ j for j = 1,..., q π 0 (b j ) T λ j for j = 1,..., q λ j 0 for j = 1,..., q q j=1 (ej ) T λ j = 1 for j = 1,..., q Mitchell The Cut Generation LP 5 / 15

6 AMPL code When we have a simple disjunction (g T x h 0 ) (g T x h 1 ) we can set up the cut generation LP as in the following AMPL code. Mitchell The Cut Generation LP 6 / 15

7 Setting up the parameters and variables param n; param m; param A{1..m, 1..n}; # the matrix A has m rows and n columns param b{1..m}; param g{1..n}; # the disjunction is param h{0..1}; # g^tx <= h0 V g^tx >= h1 param xbar{1..n} >= 0; var pi{1..n}; var pi0; # the cut is pi*x >= pi0 var lambda{0..1,1..m} >= 0; # multipliers for Ax <= b var rho{0..1,1..n} >= 0; # multipliers for x >= 0 var mu{0..1} >= 0 ; # multipliers for the disjunction Mitchell The Cut Generation LP 7 / 15

8 Objective function, first constraint maximize objective: sum{j in 1..n} xbar[j] * pi[j] - pi0; subject to inpolar{j in 0..1, i in 1..n}: pi[i] <= sum{k in 1..m} A[k,i] * lambda[j,k] - rho[j,i] + (-1)**j * g[i] * mu[j]; # need to get pi as a linear combination of the # the constraints, for each part of the disjunction Mitchell The Cut Generation LP 8 / 15

9 Remaining constraints subject to findpi0{j in 0..1}: pi0 >= sum{k in 1..m} b[k] * lambda[j,k] + (-1)**j * h[j] * mu[j]; # need to ensure pi0 is small enough so that the constraint is valid subject to normalize: sum {i in 0..1, j in 1..m} lambda[i,j] + sum {i in 0..1, j in 1..n} rho[i,j] + sum {i in 0..1} mu[i] = 1; # need to bound the set of feasible solutions, # to ensure a finite solution Mitchell The Cut Generation LP 9 / 15

10 Data set Taking the following data set gives the problem considered in the previous handout: # the constraints are # 2x1-2x2 <= 1 # x1 <= 1 # x2 <= 1 # # the disjunction is # x1 <= 0 V x1 >= 1 # param n:=2; param m:=3; param A : 1 2 := ; param b := ; param g := ; param h := ; param xbar := ; x Q 1 x 1 2x 2 = 0 (0.5,0) 1 x 1 Mitchell The Cut Generation LP 10 / 15

11 Output ampl: reset; model disjunction.mod; data disjunction.dat; solve; CPLEX : optimal solution; objective dual simplex iterations (4 in phase I) ampl: display pi, pi0; pi [*] := ; pi0 = 0 Mitchell The Cut Generation LP 11 / 15

12 Output ampl: display lambda, mu, rho; lambda := ; mu [*] := ; rho := ; Mitchell The Cut Generation LP 12 / 15

13 Modified problem Adding the extra constraint 19x x 2 10 gives the data file # the constraints are # 2x1-2x2 <= 1 # 19x1 + 10x2 <= 10 # x1 <= 1 # x2 <= 1 # the disjunction is # x1 <= 0 V x1 >= 1 param n:=2; param m:=4; param A : 1 2 := ; param b := ; param g := ; param h := ; param xbar := ; x 2 1 S Q (0.5,0) 1 S 1 1 = x 1 Mitchell The Cut Generation LP 13 / 15

14 Output CPLEX : optimal solution; objective dual simplex iterations (4 in phase I) ampl: display pi, pi0; pi [*] := ; pi0 = 0 Mitchell The Cut Generation LP 14 / 15

15 Check the validity of the constraint The only other nonzero variables are: multipliers for the disjunction (0.45 for x 1 0 and 0.5 for x 1 1) and the multiplier λ = 0.05 for 19x x 2 10 in the x 1 1 side of the disjunction. Check: 0.05(19x x 2 10) + 0.5( x 1 1) = (0.45x x 2 0). Mitchell The Cut Generation LP 15 / 15