![](bg6.png)
p
opt
− p
relaxCPLEX
p
opt
− p
relaxSDP
>
p
′
opt
− p
relaxCPLEX
p
′
opt
− p
relaxSDP
(19)
On average, the gap improves from 1.80% to
1.71% with original SDP relaxation and 1.56% with
addition of valid equalitites. This latter improve-
ment is promising, even though it comes at high ad-
ditional computational cost, particularly on the larger
instances. This can be ascribed to the fact that SDP
solvers are only in their infancy, especially compared
to a commercial solver like CPLEX.
Finally, the randomized rounding yields satisfying
results: due to the random aspect of the procedure,
there are still some data set where the continuous re-
laxation gives better results than the semidefinite re-
laxation, but on average the loss of optimality reduces
from 7.75% to 6.41% and 5.59%, which is significant
when considering the huge amount at stake.
5 CONCLUSIONS AND
PROSPECTS
We investigated in this paper, semidefinite relaxations
for a MIQP (Mixed-Integer Quadratic Program) ver-
sion of the scheduling of nuclear power plants out-
ages. Comparison of the results obtained on signifi-
cant data sets shows the following main results. First,
our MIQP is extremely hard to solve with CPLEX.
Second, semidefinite relaxations provide a tighter
convex relaxation than the continuous relaxation. In
our experiments the gap between the optimal solu-
tion and the continuous relaxation is on average equal
to 1.80% whereas the semidefinite relaxation yields
an average gap of 1.56%. Third, the computational
time for computing these semidefinite relaxations is
reasonable. Exploiting those results in a randomized
rounding procedure instead of the result of the contin-
uous relaxation leads to a significant improvement of
the feasible solution.
In the view of these preliminary results, additional
investigations will concern i) introduction of more
valid inequalities, ii) evaluation of others SDP resolu-
tion techniques, for instance Conic Bundle for facing
problems of huge size.
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