Introduction Branch and Price Approach May 19, 2015 Introduction Outline: 1 Introduction Concepts of Branch and Price DANTZIG-WOLFE DECOMPOSITION Remarks Introduction Introduction General introduction. Branch-and-price is when branch and bound is generalized to include generation of columns by solving the pricing problems. Another related approach is branch-and-price-and-cut which includes the generation of columns and rows. Branch and price tightens the LP relaxation by generating a subset of profitable columns associated with variables to join the current basis. These columns are generated iteratively by solving subproblems or pricing problems. Introduction Concepts of Branch and Price Concepts of Branch and Price Branch-and-price builds upon the branch and bound framework It applies column generation throughout the branching and bound tree prior to branching Branching occurs when no profitable columns can be found and the LP solution does not satisfy the integrality conditions. The concept of column generation is outlined below 1 The column generation approach is used when the LP relaxation of a given IP formulation contains too many columns (associated with variables) to handle explicitly and simultaneously. Introduction Concepts of Branch and Price . 2 A restricted version of the master problem that contains only a subset of columns (usually associated with basic variables) is maintained and updated, while the remaining huge number of columns (usually associated by nonbasic variables) are left out of the LP relaxation. 3 Because most of these columns will likely have their associated variables equal to zero in an optimal solution, only profitable columns(associated with nonbasic variables) are generated and added to the current restricted master problem to improve its current LP solution 4 The column generation approach to integer programming is closely related to Dantzig-Wolfe decomposition in linear programming. Introduction Concepts of Branch and Price cont. 5 Optimal is found when there is no column that can be generated with profitable reduced cost. It may or may not satisfy integrality conditions 6 If YES a lower bound for IP is found, NOT the noninteger LP optimum can be used as an upper bound and then the branching occurs. 7 A special (problem specific) branching scheme is usually needed because column generation may destroy the original problem structure. Introduction Concepts of Branch and Price . This can be summarized as Introduction DANTZIG-WOLFE DECOMPOSITION DANTZIG-WOLFE DECOMPOSITION Consider LP problem containing two sets Ax ≤ b(general structure) and Gx ≤ d(special structure) Maximize z = cT x Subject to Ax ≤ b Gx ≤ d x ≥ 0 where c is a profit vector in the object function to be maximized. Let S = fx : Gx ≤ d; x ≥ 0g, and assume S is bounded polyhedron. Any point x 2 S can be represented as a convex combination of all (say t) extreme points of S. Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Denoting these extreme points by x1; x2; :::; xt, any x 2 S can be represented as t X j x = λjx j=1 t X λj = 1 j=1 λj ≥ 0 j = 1; 2; :::; t Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Substituting x LP can be transformed into the master problem (PM) in the variables λ1; :::; λt t X T j Maximize z = (c x )λj j=1 t X j Subject to (Ax )λj ≤ b j=1 t X λj = 1 j=1 λj ≥ 0 j = 1; 2; :::; t Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . From the transformed, a revised simplex tableau for the master problem of size (m + 1) × (m + 1) can be constructed as shown in the left box of the figure below. −1 Suppose we have basic feasible solution λ = (λB; λN) and B of size (m + 1) × (m + 1) is known. The primal solution can be obtained by calculating B−1b, the dual T −1 T T −1 solution by ^cBB = (u ; α) and the objective value by ^cBB b T where ^cB is the profit vector of the basic variables with a profit of T j ^cj = ^cBx for each basic variable λj. Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Consider again the figure 13.2 The right box contains the subproblem subject to the constraints of special structure and the left box contains the master problem. Master problem passes the values of the current dual solution, T −1 (u ; α) = ^cBB to the subproblem for constructing its Objective function. A pivot column is formed and passed to the master program Interaction between master and the subproblem are repeated until dual solution is nonnegative. Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Example: Solve the following LP problem by decomposition Maximize − 3x1 + 7x2 + 5x3 + 4x4 (1) Subject to 2x1 − x2 + 2x3 + 2x4 ≤ 19 (2) − 2x1 + 2x2 + x3 − 3x4 ≤ 21 (3) x1 + x2 ≤ 12 (4) 3x1 − x2 ≤ 15 (5) x3 + x4 ≤ 5 (6) − x3 + x4 ≤ 2 (7) x ≥ 0 (8) Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction DANTZIG-WOLFE DECOMPOSITION Cont . Introduction Remarks Remarks.