local branching and its application of accelerating ...be… · local branching and its application...

Local branching and its application of acceleratingBenders decomposition

Yi Fang

Department of Industrial and Management Systems EngineeringWest Virginia University

Feb 19, 2013

Fang (WVU) Accelerating B.D. by local branching Feb 19, 2013 1 / 24

Outline

1 Local Branching

2 Benders decomposition

3 Accelerating Benders decomposition by local branching


Local Branching

Introduction

I Local branching method was developed by Fischetti and Lodi in 2003;

I The method was developed for general MIPs;

I The method is exact in nature;

I The idea of local branching is to divide solution space into smallersubregions such that generic solver can solve subproblems moreefficiently;

I It aims at finding high-quality solutions at early stages of thecomputation.


Local branching framework

A general MIP:

(P) min cT x (1)

s.t. Ax ≥ b (2)

xj ∈ {0, 1} ∀j ∈ B 6= 0 (3)

xj ≥ 0, integer ∀j ∈ G (4)

xj ≥ 0, ∀j ∈ C (5)



Local branching constraint:Given a feasible solution x̄ of (P) and let S̄ = {j ∈ B : x̄j = 1}, the localbranching constraint is defined as:

∆(x , x̄) =∑j∈S̄

(1− xj) +∑

j∈B\S̄

xj ≤ k (6)

For example: x̄ = {1, 0, 0, 1}, x = {0, 1, 0, 0}, then

∆(x , x̄) = (1− x1) + (1− x4) + x2 + x3 = 1 + 1 + 1 + 0 = 3;

The meaning of ∆(x , x̄) is to count the number of binary variables flippingtheir value ( with respect to x̄) either from 1 to 0 or from 0 to 1.



The local branching constraint can be used as a branching criterion for (P).Given the incumbent solution x̄ , the solution space can be partitioned by:

∆(x , x̄) ≤ k (left branch) or

∆(x , x̄) ≥ k + 1 (right branch)

where k is the neighborhood-size parameter. The basic idea of localbranching is illustrated in Fig. 1


The basic local branching scheme

The triangles marked by the letter ”T” represent the application of theblack-box exact MIP solver (e.g. ILOG-Cplex).


An enhanced heuristic solution scheme

I Imposing a time limit on the left-branch nodes

(a) The incumbent solution has been improved within time limit(this situation is illustrated in Fig. 2)

(b) Time limit is reached with no improved solution (this situation isillustrated in Fig. 3 )



I Diversification

(a) ”Soft” diversification: enlarging the current neighborhood;

(b) ”Strong” diversification: finding a solution (may worse than theincumbent one) that is not ”too far” from current reference solutionx̄ .


Local branching algorithm

Four different cases may arise after each time MIP solver is called:

1 Improved solution has been found and proven to be optimal (i.e., MIPsolver was capable of solving current problem optimally);

2 Improved solution has been found, but cannot prove its optimality(i.e., MIP solver was not capable of proving its optimality due to timelimit);

3 No improved solution has been found and it is proven that improvedsolution doesn’t exist;

4 No improved solution has been found but there is no guarantee thatsuch a solution doesn’t exist (i.e., MIP solver was not capable ofsolving the problem within time limit).



Case 1: Improved solution and optimal

• The last local branching constraint is reversed (i.e., reverse to rightbranch constraint in the form of ∆(x , x̄) ≥ k + 1);

• The improved solution will be the new reference solution x̄ ;

• The incumbent solution and the best upper bound is updated;

• The process is repeated.



Case 2: Improved solution but not optimal

• The last local branching constraint ∆(x , x̄) ≤ k is replaced by ”tabu”constraint ∆(x , x̄) ≥ 1;

• By adding the ”tabu” constraint, we may cut off the optimal solutionof the overall problem (remember we have general integer andcontinuous variables in the general case);

• Using a ”refining” procedure to solve above issue:

(1) Fixing the binary variables by adding ∆(x , x̃) ≤ 0 , where x̃ isthe new improved solution obtained;

(2) Solve the problem by MIP solver and obtain optimal solution;

(3) Replace x̃ by the optimal solution obtained;

(4) This optimal solution will be the new reference solution.



Case 3: Unimproved solution with guarantee

• The last local branching constraint is reversed (i.e., reverse to rightbranch constraint in the form of ∆(x , x̄) ≥ k + 1);

• A soft or strong diversification is performed;

(a) ”Soft” diversification is used if improved solution is obtained inthe previous iteration. In this situation, solution space is enlarged.

(b) ”Strong” diversification is used if no improved solution isobtained in the previous iteration. In this situation, the upper boundis set to +∞ to accept a worse solution, and the exploration isaborted as soon as the first feasible solution is found.



Case 4: Unimproved solution, but there may exist one

• If ”Soft” diversification is used, the last local branching constraint∆(x , x̄) ≤ k is replaced by ∆(x , x̄) ≤ k − dk2 e;• If ”Strong” diversification is used:

(1) the last local branching constraint ∆(x , x̄) ≤ k is replaced∆(x , x̄) ≥ 1 (to escape from x̄ ;

(2) the upper bound is reset to +∞ (to accept worse solution);


Observations

• As to the parameter k , the authors states that values in the range[10, 20] is effective in most cases;

• If there is time left after solving left branches, the remaining time isused to solve right branch to prove optimality (corresponding to node7 in Fig. 1);

• If there is no time limit and the diversification strategies are not usedfor infinity time, then the algorithm acts as exact method; otherwisethe method can be view as a local search heuristic.


Benders decomposition

A general MIP

(O.P.) min cT1 x + cT2 y (1)

s.t. A1x = b1 (2)

A2x + Ey = b2 (3)

x ∈ {0, 1} (4)

y ∈ R (5)


Benders decomposition algorithm


Accelerating Benders decomposition by local branching

Introduction

• By using Benders decomposition, an integer problem needs to besolved for every iteration;

• The integer problem becomes more difficult each time a new cut isadded;

• The lower bound is nondecreasing since a new cut is added to theRMP in every iteration, but there is no guarantee that the upperbound is decreasing;

• Rei et al. (2009) proposes to use local branching to accelerateBenders decomposition;

• The aim is to find better upper bounds as well as multiple optimalitycuts at each iteration of the Benders decomposition algorithm.



Local branching subproblem formulation

(Pt) min cT1 x + cT2 y

s.t. A1x = b1

A2x + Ey = b2

∆(x , x j) ≥ 1, j ∈ Jt

∆(x , x i ) ≥ ki , i ∈ I t

∆(x , x t) ≤ k ,

x ∈ {0, 1}y ≥ 0

(P̄t) min cT1 x + cT2 y

s.t. A1x = b1

A2x + Ey = b2

∆(x , x j) ≥ 1, j ∈ Jt

∆(x , x i ) ≥ ki , i ∈ I t

∆(x , x t) ≥ k + 1,

x ∈ {0, 1}y ≥ 0

where (x t , y t) is the current feasible solution to (1)-(5); x j are previousfeasible solutions; set I t ⊆ Jt



Let (x t+1, y t+1) be the optimal solution to Pt ,

• if (x t+1, y t+1) is better than (x t , y t), a new left branch is created byusing the new solution.

• if (x t+1, y t+1) is not better than (x t , y t) or Pt is infeasible, theneighborhood size k is increased by 1, and a tabu constraint is addedto Pt .



General algorithm structure

• Starting from a feasible solution (x0, y0);

• At iteration ν of the relaxation algorithm, the solution xν to therelaxed master problem is a starting point for local branching(xν doesnot have to be feasible);

• At the end of local branching phase, an upper bound to problem(1)-(5) will be obtained;

• A pool of different feasibility cuts can be generated and added to therelaxed master problem;

• These processes are repeated until predetermined time limit is reachedor an ε̄− opt solution is found.



What if subproblem Pt is infeasible

• Pt is defined by the local branching constraints ∆(x , x j) ≥ 1,∆(x , x i ) ≥ ki , and ∆(x , x t) ≤ k;

• If Pt is infeasible, then a subregion of the neighborhood of xν

contains no feasible solution;

• Adding constraints ∆(x , x i ) ≥ ki and ∆(x , x t) ≥ k + 1 to the relaxedmaster problem;

• These constraints can be used to replace the feasibility cut constraintif local branching phase is used each time he RMP is solved.


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