![](bgc.png)
measurement; this measurement evaluates function:
g(γ,x
o
) = ||γ − h(x
o
)||
2
, (25)
where h(x
o
) =
1
4
∑
4
i=1
M
i
and x
o
= (x
o
1
,y
o
1
,...,x
o
4
,y
o
4
).
Our purpose is then to optimize (a,b) by doing a min-
imum request to the costly evaluation g(γ, x
o
).
We did not have an efficient method for sampling
X
∀ j,g(γ
j
,X) = g
j
at the time of (Dambreville,
2015). Now, we propose to apply algorithm 3 com-
bined with a discretized method (we actually enumer-
ated on 10
6
discretized points of [−5, 5]
2
) for mini-
mizing the expected improvement EI(γ) in order to
process minimizing sequence (γ
k
) . This sequence
converge to isobarycenter of M
1:4
.
Tests and Results. Points M
1
,...,M
4
are (2, −1),
(3,2), (−
3
2
,4), (
1
2
,3). Their barycenter is (1,2).
We used a sampler with M = 5000, N = 10000 and
[] = [−
1
100
,
1
100
]
k
. Variable X is considered uniform
on [−5,5]
8
. Process is stopped at step k
o
, when op-
timum is found; best evaluation and solution are then
respectively 0 and (1,2). The following table sum-
marizes the results of 100 runs. cpu gives the average
computation time:
k
o
3 4 5
% 46 53 1
cpu(s) 1670 2758 2939
Except for outlier, optimum is found almost
equiprobably after 3 or 4 tries. This is similar to the
geometric method
3
. It is noteworthy that our algo-
rithm does not have geometric knowledge of the prob-
lem and deals with eight dimensions model noise.
5 CONCLUSIONS
We proposed an original dichotomous method for
sampling a random vector conditionally to a subva-
riety. This generic approach, inspired from interval
analysis, is accurate and efficient up to a space of di-
mension 11. We have shown how it could be applied
efficiently to the optimization of expensive black-box
function. The work is promizing from an applica-
tive point of view and offers several improvement per-
spectives. We will particularly investigate some relax-
ations techniques applied to the subvariety in order to
enhance the efficiency of the approach in regards to
higher dimensions.
3
Each measure restricts the solution to a circle. After 2 mea-
sures, we usualy have to choose between two points, and
the solution is found equiprobably at step 3 or 4.
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