296
D. Irawan and B. Naujoks
provided by the DACE predictor and standard error, respectively. The improvement
I(x) is calculated from Y using the following equation:
I (x) = max(f (x∗) − Y, 0) for minimization problem
I (x) = max(Y − f (x∗), 0) for maximization problem
(8.7)
The expected improvement is then obtained simply by taking the expectation of I(x):
E(I (x)).
EGO uses the point with the largest expected improvement, evaluates the
objective value, and then rebuilds the kriging model with the new design point.
We can opt to use all evaluated designs or take a smaller set, as long as we build a
new kriging model every time a new point is evaluated. The process is repeated in
a loop until a stopping criteria is met (it can be minimum expected improvement,
maximum iteration, etc.).
Now, ParEGO extends this to MOP by using an aggregation method. A set of
weight vectors W is built at the start; at each iteration, one of the weight vectors
is picked randomly and used for aggregation. At each iteration, different weight
vectors will be used but will always be from the set W. The original ParEGO uses the
augmented Tchebycheff aggregation method (see [32] for details). As the problem
is now transformed into single-objective problem, we can follow the original EGO
algorithm for further steps.
To summarize, the ParEGO implementation is nearly the same as the original
EGO, except that at every iteration, the objective function is changed (due to the
changes in aggregation weights). The changing objective function is not a problem
because EGO works on a black-box function, i.e., it does not care what the objective
function is, it only cares for the objective values.
8.5.2 Prescreening Method
Another approach of using surrogate model in (evolutionary) MOP is by applying
the surrogate and exact evaluation tool in cooperation [20]. In ParEGO, the
optimization searches the design space for the best improvement solely based on the
surrogate model; the exact evaluation is only used for enriching the data. ParEGOlike methods use the surrogate model as the optimizer.
In the prescreening method, the exact evaluation is used to improve efficiency
for local search. In essence, it uses a traditional optimizer, guided by the surrogate
model. The role of the surrogate model as a guide is achieved similarly with what
happened in ParEGO-like methods which search for points with “improvement.”
In ParEGO, the expected improvement is used, but other improvement metrics are
actually available as mentioned in [20]. These improvement metrics are then used
to prescreen the offspring, i.e., pick several candidate-offspring with maximum
improvement metrics. This effectively reduces the number of required exact eval-
D. Irawan and B. Naujoks
provided by the DACE predictor and standard error, respectively. The improvement
I(x) is calculated from Y using the following equation:
I (x) = max(f (x∗) − Y, 0) for minimization problem
I (x) = max(Y − f (x∗), 0) for maximization problem
(8.7)
The expected improvement is then obtained simply by taking the expectation of I(x):
E(I (x)).
EGO uses the point with the largest expected improvement, evaluates the
objective value, and then rebuilds the kriging model with the new design point.
We can opt to use all evaluated designs or take a smaller set, as long as we build a
new kriging model every time a new point is evaluated. The process is repeated in
a loop until a stopping criteria is met (it can be minimum expected improvement,
maximum iteration, etc.).
Now, ParEGO extends this to MOP by using an aggregation method. A set of
weight vectors W is built at the start; at each iteration, one of the weight vectors
is picked randomly and used for aggregation. At each iteration, different weight
vectors will be used but will always be from the set W. The original ParEGO uses the
augmented Tchebycheff aggregation method (see [32] for details). As the problem
is now transformed into single-objective problem, we can follow the original EGO
algorithm for further steps.
To summarize, the ParEGO implementation is nearly the same as the original
EGO, except that at every iteration, the objective function is changed (due to the
changes in aggregation weights). The changing objective function is not a problem
because EGO works on a black-box function, i.e., it does not care what the objective
function is, it only cares for the objective values.
8.5.2 Prescreening Method
Another approach of using surrogate model in (evolutionary) MOP is by applying
the surrogate and exact evaluation tool in cooperation [20]. In ParEGO, the
optimization searches the design space for the best improvement solely based on the
surrogate model; the exact evaluation is only used for enriching the data. ParEGOlike methods use the surrogate model as the optimizer.
In the prescreening method, the exact evaluation is used to improve efficiency
for local search. In essence, it uses a traditional optimizer, guided by the surrogate
model. The role of the surrogate model as a guide is achieved similarly with what
happened in ParEGO-like methods which search for points with “improvement.”
In ParEGO, the expected improvement is used, but other improvement metrics are
actually available as mentioned in [20]. These improvement metrics are then used
to prescreen the offspring, i.e., pick several candidate-offspring with maximum
improvement metrics. This effectively reduces the number of required exact eval-