The Lagrange variable dual ILP problem of (4.5h-o) can be implemented into the AD inner decomposition in two different ways. The most powerful application of the Lagrange variable dual ILP problem is to completely replace AD inner decomposition with the Lagrange variable dual problem. Notice that both the Lagrange variable dual ILP problem and regular AD inner decomposition produce an attack (X) to provide as input to the outer decomposition. Regular inner decomposition can produce an optimal attack, but the Lagrange variable problem usually can produce an attack that is fairly close to optimal. There is no guarantee of the quality of the attack found by the Lagrange Variable dual ILP problem. Since optimal attacks are not found, this application is a heuristic approach. But, as we will show, the savings in computation time of not performing inner decomposition are quite dramatic.
A second application of the Lagrange variable dual ILP is as a warm start for the combination of the iterative Lagrangian relaxation operator model with path enumeration. In this second application, we use the fact that each round of inner decomposition begins with a starting point of no fixed attacks. Decomposition works by constructing attacks one at a time until the optimal attack is found. The Lagrange variable dual ILP problem can be used to find an initial attack that is usually fairly close to optimal. This initial attack from the Lagrange variable problem can then be used as a starting fixed attack (X) for Lagrangian relaxation of the operator model. The result is usually fewer inner decomposition iterations because the starting point attack is close to the optimal attack.
The path enumeration algorithm terminates with the constrained shortest path. The associated cost of the constrained shortest path is used to update the inner
decomposition lower bound. The values of the shortest path variables (Y) are fixed and supplied as input to the AD master problem.
The AD inner master problem used in conjunction with the operator model Lagrangian relaxation is unchanged from (3g–k). Each iteration of the AD master problem represents an opportunity to update the inner decomposition upper bound. When the inner decomposition lower bound is updated, the attack associated with that iteration is saved as the best known attack. We refer to this set of best attack variable values as X
star (X*). Also, the value of the Lagrange multiplier from the AD sub problem used in
this iteration is saved as the best known Lagrange multiplier. We refer to this best known value of the Lagrange multiplier as mu star (μ*). Pseudo-code for the implementation of the Lagrangian relaxation and path enumeration as the AD inner decomposition is presented in the appendix.
Either of the two approaches for inner decomposition can serve as a sub problem for the outer decomposition master problem. The outer decomposition master problem of DAD CSP with Lagrangian relaxation features the minimization of both binary defense variables (W) and path variables (Y). We use Lagrangian relaxation again on the outer decomposition master problem by moving the time budget constraint to the objective function. This transformation again restores the special network structure of the problem and allows us to treat the path variables (Y) as continuous variables. The Lagrange multiplier (μ) is necessary to regain the network structure of the outer master problem, and it can be different for each iteration of outer decomposition (k’). However, we do not treat the Lagrange multiplier (μ) as neither a decision variable nor an iterated value in the relaxed outer master problem. Instead, we choose to import the value of the Lagrange multiplier (μ) that was obtained from inner decomposition. We use the hat symbol (^) to denote the Lagrange multiplier is a fixed value in the relaxed DAD outer master problem.
DAD CSP with Lagrangian Relaxation Outer Master Problem: Additional Parameter:
' ˆk
Lagrange multiplier during iteration k’ of outer decomposition. (Note: Obtained from inner decomposition)
Formulation: ' , m in , ijd ijdk D AD W Y Z (4.5p) Subject to:
'
' ' ' ( , ) , ) ˆ ˆ ' ,DAD ijd ijd ijk ijdk k ij ijdk
i j A d D i j A d D Z c q X Y t Y T k K
(4.5q) ' ' :( , ) :( , ) 1 if 0 \ , ' , 1 if i ijdk jidk j i j A d D j j i A d D i s Y Y i N s t k K t
(4.5r) 0 ( , ) : _ , ijd ijd i j A d D d d h W defense budget
(4.5s) ' ( , ) , , ' , ijdk ijd Y W i j A dD k K (4.5t) 1 ( , ) , ijd d D W i j A
(4.5u) ' 0 ( , ) , , ' , ijdk Y i j A dD k K (4.5v)
0,1 ( , ) , . ijd W i j A d (4.5w) DExpressions (4.5p) and (4.5q) work together to form the outer decomposition master problem to minimize the cost of shortest paths with fixed attacks by the choice of defense and path variables. Equation (4.5q) shows the Lagrangian relaxation of the time budget constraint is also present in the outer master formulation, which regains the network structure of the formulation. Equation (4.5r) is similar to (4.2b), except that iteration subscripts “k” have been added to the path variables for each outer decomposition. Equation (4.5s) restates (4.2f). Equation (4.5t) is similar to (4.2c), but the addition of subscript “k” allows for different path variables for each iteration of outer decomposition. Equation (4.5u) restates (4.2f). Equation (4.5v) allows path variables to have continuous values, since the Lagrangian relaxation regains the network structure. Equation (4.5w) is unchanged from (4.2j).
The new feature in the outer decomposition master problem is the Lagrangian relaxation of the time budget constraint into the iteration dependent objective equations (4.5q). Once again, the purpose of the relaxation is to reduce the number of binary variables that must be considered in the outer master problem. However, the effect of the
reduction is much more dramatic in the outer master problem because the path choice variables (Y) can be different in each iteration (k’) of outer decomposition. Lagrangian relaxation has its greatest effect in the outer master problem because the number of path choice variables (Y) is dependent upon the number of iterations (k’).
Determination of the values of the Lagrange multiplier (μk’) in the outer decomposition is not trivial. The outer decomposition features the defense binary decision variables (W) in the constraints alongside the Lagrange multiplier (μ). Furthermore, the Lagrange multiplier (μ) can have a different value for each iteration of outer decomposition, since the path choice variables (Y) can be different in each outer iteration (k’). The outer master formulation means that the value of the Lagrange multiplier (μk’) depends upon both the choice of defense variables (W) as well as relaxed path choice variables(Y), which also vary during each outer decomposition iteration (k’).
We employ a novel feature to solve the issue of determining the Lagrange multiplier (μk’) for each iteration of outer decomposition. We desire to avoid performing another round of Lagrangian relaxation iterations (L) to find the best value of the Lagrange multiplier for each round of DAD outer iterations (K’). We use the best value of the Lagrange multiplier from the inner decomposition (μ*) as the only value of the Lagrange multiplier for the current iteration (k’) in the outer decomposition. The rationale behind this choice is that the Lagrange multiplier (μ) serves as the amount of penalty that the objective function must incur for failing to adhere to the time budget constraint. The inner decomposition relaxation determined the best value for this penalty (μ) when defense variables (W) were fixed and the attack variables (X) were optimized.
The methodology for the updating of the outer decomposition upper and lower bounds is similar to the approach seen in Chapter I with the DAD minimum cost flow problem. Refer to Chapter I for the specifics of updating bounds.