DAD CSP Phase 1 Bounding Heuristic Algorithm 166

The phase 1 bounding heuristic algorithm incorporates all of the mathematical models shown in the previous sections. The Lagrange Variable dual ILP problem is used to produce an initial attack (X) that serves as a smart starting point for the inner decomposition AD sub problem. Multi-cuts are used from path enumeration to give multiple feasible attack cuts to the AD master problem. There are two nested loops in the

algorithm which represent the outer and inner decompositions of the DAD CSP problem instance.

Since our heuristic algorithm uses Lagrangian relaxation, there is no guarantee of convergence on an optimal solution. We introduce a stopping criterion for our heuristic algorithm to terminate whenever it repeats a defense (W) that it had found in a previous iteration. In our tests, we limit the number of outer iterations of the phase 1 algorithm at 199. We observe the algorithm always terminates because the defense plan is repeated twice before reaching 199 iterations.

DAD CSP Phase 1 Bounding Heuristic Algorithm:

Inputs:

Network data with defense, attack, and time budgets

Iteration limits for DAD (outer) and AD (inner) decomposition Optimality tolerances for all models (usually zero)

Outputs:

DAD Upper & Lower Bounds, best known defense (W*) and attack (X2*).

Algorithm:

While DAD Upper bound – DAD Lower Bound ≥ tolerance & DAD iteration ≤ limit: Solve AD CSP Lagrange Variable dual ILP

Obtain initial attack (X) and Lagrange multiplier (μ)

While AD Upper Bound – AD Lower Bound ≥ tolerance & AD iteration ≤ limit: Solve AD CSP with Lagrangian relaxation inner sub problem

Obtain Lagrangian upper and lower bounds and best multiplier (μ)

If Lagrangian relaxation bounds match:

Obtain shortest path (Y) & associated cost

Else:

Perform Recursive Path Enumeration Algorithm Obtain all feasible paths (Y) & shortest path cost

If shortest path cost > AD (inner) Lower bound:

Update AD (inner) lower bound

Save associated attack (X) as best known attack (X*)

Save associated Lagrange multiplier (μ) as best multiplier (μ*) Solve CSP Inner Decomposition AD Master Problem

Obtain attack plan (X) and associated cost

Update AD (inner) upper bound if associated cost decreases Increment inner decomposition iteration counter

Update DAD (outer) Upper Bound with AD (inner) Upper Bound, if it is smaller

Solve DAD CSP Lagrangian Relaxation outer master

If outer objective function value > DAD master lower bound:

Update DAD master lower bound

Save attack associated with this defense as (X2*) Increment DAD iteration counter

If current defense (W) matches a defense from a previous outer iteration: Exit outer while loop

In our testing of the phase 1 bounding algorithm, we limit the maximum value of inner decomposition iterations to one. We observe that limiting the inner decomposition to one iteration step has a positive benefit at the expense of a small degradation in precision. We can simplify the phase 1 bounding algorithm to obtain upper and lower bounds more quickly by limiting the AD inner decomposition to just one iteration. But, the bounds obtained from just one inner decomposition iteration may be less precise than if more iterations were performed. The one round of AD inner decomposition allows for all of the steps of the algorithm to be completed. The shortcoming of using only one iteration of inner decomposition is that the optimal attack (X) may not be found. However, we believe that the tradeoff of weaker bounds in this algorithm is justified by the increase in solution speed, since only one iteration of inner decomposition is performed for each iteration of outer decomposition. The phase 1 bound algorithm is a heuristic, so performing more than one iteration of inner decomposition may not have much benefit for the extra effort required in additional inner decomposition iterations. A secondary benefit of only having one iteration of inner decomposition is that we do not need to further complicate our algorithm by checking for repeated paths.

Proposition 4–2. The phase 1 bounding heuristic algorithm produces valid upper and lower bounds for a DAD CSP problem instance.

Proof of Proposition 4–2. The heuristic approach contains all of the constraint equations that appear in the original DAD CSP problem, except for the time constraint, which has been relaxed to the objective function. The time constraint only effects the choice of path variables (Y), as seen in Equation (4.1g). Each set of attack variable values (X) obtained in the heuristic are feasible attacks because they must obey the attack constraints of the original DAD CSP problem. Each set of defense variable values (W) obtained in the heuristic are feasible defenses because they must obey the defense constraints of the original DAD problem. Our heuristic approach is not concerned with

the values of the path variables (Y), so we need not worry if the values obtained for them via relaxation are feasible. Ahuja et al. prove via weak duality that for any value of the Lagrange multiplier (μ), the value of the Lagrangian function “is a lower bound on the optimal objective function value of the original optimization problem” (1993, p. 605). Our use of the same Lagrange multiplier value from the inner decomposition to the outer master problem produces the same effect of creating a valid lower bound on the overall DAD CSP problem. The phase 1 boundary algorithm also produces a valid upper bound. Since the phase 1 bound algorithm always includes at least one iteration of Lagrangian relaxation and path enumeration of the inner sub problem, we are guaranteed that the output of the inner sub problem is a feasible path (Y) for the original problem instance. The inner upper bound is computed by taking the best feasible path (Y) and the optimized attack (X) for a fixed defense (W). The inner upper bound is simply transferred to the outer decomposition as its upper bound if it is an improvement. Thus, the upper bound calculation is feasible solution to the original problem instance and therefore it is valid. Therefore, our use of Lagrangian relaxation in the phase 1 bound algorithm is guaranteed to produce valid upper and lower bounds on the original DAD CSP problem instance.