Ellipsoidal Search

As the slope scaling procedure is only applied indirectly, sometimes ISND solutions suggested by the R1-ISND leave a considerable optimality gap. Given a series of fea-sible solutions produced by sliding problems, we intent to incorporate the promising attributes shown from the solutions, and generate better solutions. A new population-based algorithm is proposed.

The algorithm derives from thepath relinkingscheme: during the slope scaling pro-cedure, each time a sliding problem is solved, the obtained feasible solution is added into areference solution set if applicable; then two reference solutions in the reference set are selected, one as theinitial solutionand the other as theguiding solution. Instead of finding a path between two solutions in a neighborhood as the classical path relinking,

we shape an ellipsoid space, and apply a MIP solver to explore the ellipsoid space. The initial solution is then removed from the reference set. If the MIP solver finds a different solution than the guiding solution, the reference set is updated by the new solution, and the algorithm restarts, otherwise stop. We name this new methodellipsoidal search, and a pseudo code is given in Figure 6.10. In the following, we discuss the major components in the ellipsoidal search, including reference solution selection and ellipsoid shaping.

1. Given areference solution set.

2. SelectguiSolandiniSol.

3. Shape an ellipsoid space.

4. Generate an intermediate feasible solution in the ellipsoid with MIP solver.

5. RemoveiniSolfrom the reference set.

6. If the intermediate feasible solution updates the reference set, go to step 2.

7. Stop.

Figure 6.10: Ellipsoidal Search Procedure

In our algorithm, feasible reference ISND solutions are provided during the slope scaling by solving sliding problems, and the quality and diversity of reference solutions are supported by factor perturbations, which is conducted by long-term memories.

Identification of the initial and the guiding solution is critical as “seeds” are im-portant to an evolutionary algorithm. Several policies for choosing initial and guiding solutions are investigated by Ghamlouche et al. [58] regarding the FMCND problem.

Their conclusion indicates that two policies are of the most interesting and with compa-rable performance. One is to define guiding solution as the best solution in the reference set, while initial solution as the solution with the maximum hamming distance from the guiding solution; the other is to define guiding solution and initial solution as the most distant solutions in the reference set. The success of these policies emphasized the im-portance of diversity of the reference solutions. The first policy is more theoretically sound, as on one hand, it preserves the overall best solution, and on the other hand it provides the maximal variation. We choose to work with this referent selection policy.

Shaping an ellipsoid also has great impacts to the quality and efficiency of the proce-dure. An ellipsoid solution space is built with thelocal branchingidea. Equipped with the initial solution and the guiding solution, we first calculate the hamming distance be-tween the two solutions, denoted it as k. Anelite problem is defined as an ISND with k-distance cut, and fed with theinitial solutionas the starting solution. The elite prob-lem therefore explores thek-opt neighborhood of the initial solution, which is defined

as all solutions with maximalkdesign variable status changed from the initial solution.

Apparently, the k-opt neighborhood of the initial solution covers the guiding solution.

Thek-distance cut in the elite problem dramatically reduces the solution space, however with a large number of design variables, thek-opt neighborhood could still be very big.

Thek-opt neighborhood is further trimmed, and an ellipsoid (illustrated in Figure 6.11) is shaped.

ISND solution space k−opt neighborhood

initial guiding

ellipsoid

Figure 6.11: Illustration of Ellipsoid withink-opt Neighborhood

The process of shaping an ellipsoid space in the elite problem includes variable se-lection and variable fixing. Variable sese-lection chooses only a set of decision variables to be considered in the elite problem in order to further reduce the searching space, and variable fixing preserves some variables with values according to the guiding solution, to ensure the proper searching direction. According to the nature of ISND, we have choices to select or fix binary variables basing on either block or service decisions as blocks and services are interrelated. Compared with block decisions, service decisions generally have more impacts as each service is shared by many blocks. Missing one key service variable in the elite problem will severely deteriorate the solution quality. As a result, in the following we shape ellipsoids based on block decisions, i.e. select and fix block decision variables, and include all service variables accordingly.

Variable selection elaborately builds a block variable set. DenoteBinithe set of open blocks in the initial solution and B_gui the set of open blocks in the guiding solution.

An intuitive proposal is to define the variable set as the union of open blocks in both solutions,B_ini∪ B_gui. However, one may want to further improve the variable set. There are trade-offs: on one hand, one would limit the variable set to improve the efficiency because removing one block from the variable set leads to the deduction of a number

of flow variables; on the other hand, without missing the optimum, one would like to include all blocks with promising characteristics.

To enrich the variable set, we explore the long-term memories. That is, a block not in B_ini∪ B_guialso has its chance to be considered in the elite problem, if it shows steadily in the previous slope scaling iterations, which is regulated by long-term memories, x¯_b (average flow on blockb) andxˆ_b(maximal flow onb),

¯

x_b/ˆx_b > λ (6.11)

whereλis a parameter. Only blocks satisfying inequality (6.11) are included in the elite problem.

An alternative for variable selection is to regulate the size of the block variable set.

Other than the block variables inB_int∪ B_gui, vacancies in the variable set are filled with blocks following the sequence indicated byx¯_b/ˆx_b. Blocks bearing steady flows are filled first, until the variable set reaches the predefined cardinality. The variable set size is defined with parameterϕ,

|B_gui| ×ϕ (6.12)

The two variable selection variants will be compared in Section 7.2. As a result, in the elite problem, only a rather small portion of block variables picked by one of the regulations above is considered.

Variable fixing ensures the searching direction from the initial towards the guiding solution, and accelerates the solving procedure of elite problems. Among the selected block variables, blocks open in both initial and guiding solutions (in setB_int∩ B_gui) are fixed to open (= 1) in the elite problem. Furthermore, the common flows from both reference solutions are also fixed.

Because the slope scaling process actually addresses the R1-ISND problem, and the ellipsoid search intensifies the ISND problem directly, the trajectory included by ellip-soid search is not a superposition with slope scaling.