Lagrangian Relaxation for the Multiple Constrained Robust Shortest Path Problem

Algorithm 1

Subgradient method for updating Lagrangian multipliers.

1. Initialization

  Set initial iteration , initial Lagrangian multipliers , initial

lower bound , initial upper bound , maximum relative gap ,

maximum value of iteration .

2. Update the lower bound

 Solve obtain an optimal solution and optimal value .

 Solve obtain an optimal solution and optimal value .

 Set the lower bound

3. Update the upper bound

 If satisfies all the side constraints (9) then set the upper bound:

 Else if, solve the problem ((8)-(10)) by -shortest path algorithm to obtain a feasible

Solution , and set the upper bound:

 End

4. Update Lagrangian multipliers

  Compute sub-gradient direction

  Updated Lagrangian multipliers

where , ,

the value of is suggested by Fisher [48].

5. Convergence test

  If relative gap or stop; Otherwise , go to

step 2.