3.3.1. Problems
Because we will use benchmark problems throughout subsequent sections to test and decide on various design choices for MO-GOMEA, we present all problems that we consider in this chapter in this section first. We also describe how the Pareto-optimal front PF of each problem instance can be obtained, which is used to evaluate the performance of MOEAs.
Scalable benchmark problems Zeromax-Onemax
Zeromax-Onemax [11] has two objectives that are defined over a binary string x of l bits as follows:
fOnemax(x) =
l
X
i=1
xi; fZeromax(x) = l − fOnemax(x) (3.1) Objective fOnemax counts the number of bits set to 1 while fZeromax counts the number of bits set to 0, making the objectives conflicting. Every candidate solution is a Pareto-optimal solution; any increase in the number of bits set to 1 will be a decrease in the number of bits set to 0, and vice versa. The Pareto-optimal front PF of Zeromax-Onemax consists of l + 1 points PF = {(i, l − i) | i ∈ {0, 1, . . . l}}, on a straight line. A point on PF can correspond to many different solutions, with the exception of the two extreme points that correspond to exactly a string of all 1s (maximizing Onemax) and a string of all 0s (maximizing Zeromax). It can be seen that the niches of the solutions having objective values in the middle regions of the front PF are exponentially larger than the niches in the extreme regions of PF. Trap-Inverse Trap
Trap-Inverse Trap [11] also has two objectives. A Trap-k is the well-known mutually exclusive, additively decomposable composition of the order-k deceptive subfunc-tions. In this thesis, we consider Trap-5:
fTrap-5(x) =
l/5
X
i=1
fTrap-5sub
5i
X
j=5i−4
xj
(3.2)
where
fTrap-5sub (u) =
5 if u = 5
4 − u if u < 5 (3.3)
3.3. Benchmark problems
Inverse Trap-5 comprises the same partitions as Trap-5 but each partition is evaluated “inversely”:
fInverse Trap-5sub (u) =
5 if u = 0
u − 1 if u > 0 (3.4)
A candidate solution for which each partition has one objective subfunction that evaluates to 5 is Pareto-optimal. The Pareto-optimal front PF consists of l/5 + 1 points PF = {(5i + 4(l/5 − i), 5(l/5 − i) + 4i) | i ∈ {0, 1, . . . l/5}}, on a straight line.
It is well-known known that in order to solve additively decomposable trap func-tions, it is crucial to preserve the linkage relations between problem variables during variation. Therefore, this benchmark is often used to demonstrate the importance of linkage learning.
Leading Ones Trailing Zeros (LOTZ)
The LOTZ problem consists of two objectives: maximizing the number of consecu-tive bits set to 1 at the beginning and maximizing the number of consecuconsecu-tive bits set to zero at the end.
A candidate solution that consists of a substring of all 1s followed by a substring of all 0s, i.e., having the form of 11 . . . 100 . . . 0, is Pareto-optimal. The two extreme solutions are all 1s (maximizing Leading Ones) and all 0s (maximizing Trailing Zeros). The Pareto-optimal front PF of LOTZ consists of l + 1 points PiF = {(i, l − i) | i ∈ {0, 1, . . . l}}, on a straight line. However, different from the Zeromax-Onemax problem, any point on the Pareto-optimal front PF of LOTZ corresponds to exactly one Pareto-optimal solution.
Multi-objective MAXCUT Definition
Weighted MAXCUT is defined over a weighted undirected graph G = (V, E), where V = (v1, v2, . . . , vl) is the set of l vertices, and E is the set of edges (vi, vj) with corresponding weights wij’s. A maximum cut is a partition of l vertices into two disjoint subsets A and B = V \ A such that the combined weight of all edges (vi, vj) having vi∈ A and vj∈ B is maximized. A cut can be encoded as a binary string x of l bits, in which each bit xi corresponds to a vertex, and all 0-valued bits indicate vertices of set A while all 1-valued bits indicate vertices of set B. The objective function of the weighted MAXCUT problem is defined as follows
fMAXCUT(x) = X
(vi,vj)∈E
wij if xi6= xj
0 otherwise (3.6)
We construct a multi-objective version of the weighted MAXCUT problem by optimizing a different MAXCUT instance in each objective. The instances have identical vertices but different edge weights.
Problem Instances
We experiment with four multi-objective MAXCUT problem instances of size l = 12, 25, 50, 100. Each MAXCUT instance is a fully connected graph having 12l(l − 1) edges. The edge weights are set by following the approach described in [26]. For the 12-vertex and 25-vertex problem instances, we can easily obtain the true Pareto-optimal fronts PF by enumeration. For the 50-vertex and 100-vertex problem in-stances, because PF’s are unknown, we can only obtain reference fronts by consult-ing the results of [26] as follows. We use the maximum population sizes of GOMEA, that reliably solve the single-objective MAXCUT, reported in [26] to set as the cluster size for MO-GOMEA. We then run five different instances of MO-GOMEA with different number of clusters k ∈ {1, 3, 5, 7, 10}. Each MO-GOMEA instance is run 100 times, each time with a budget of 20 million function evaluations. We take the non-dominated front of all results in all these runs over all five instances of MO-GOMEA to be PF in this case. A degree of reliability can be taken from the fact that the optimal extreme points were always found to be in the final reference fronts PF so constructed.
Multi-objective Knapsack Problem Definition
The multi-objective knapsack problem involves l items and m knapsacks. Each knapsack k has a specific capacity ck. Each item i has a weight wi,k and a profit pi,k corresponding to each knapsack k. A solution of the knapsack problem can be encoded as a binary string x = (x1, x2, . . . , xl) ∈ {0, 1}l, where each bit xi
corresponds to an item i, and xi = 1 indicates that item i is selected. Selecting an item i in the multi-objective knapsack context means the item i is placed in every knapsack. A feasible solution is a selection of items such that the total weight does not exceed the capacity of any knapsack. The objectives are to maximize the profits of all knapsacks at the same time as follows.
maximize
x (f1(x), f2(x), . . . , fm(x)) where fk(x) =
l
X
i=1
pi,kxi k = 1, . . . , m
subject to
l
X
i=1
wi,kxi≤ ck k = 1, . . . , m
(3.7)
Constraint Handling
If a solution violates the capacity constraint of any knapsack, we use the repair mechanism proposed in [27] to iteratively remove items until all constraints are satisfied. The item removal order follows the principle that the items with the lowest maximum profit/weight ratio should be discarded first.The maximum profit/weight ratio ri of an item i as follows (as in [27]).
ri=maxm
k=1
pi,k
wi,k
(3.8)