17 Introduction to Evidence-Based Robust Optimisation
555
to ensure that the maximum value of C over U is admissible. The optimisation and
restoration loops then become as follows:
1. [Optimisation] Given a set of maxima A = A u
A c , solve the constrained
minimisation problem:
min d∈D max u∈A f (d, u)
s.t.
max u∈A C(d, u) ≤ 0
(17.40)
where A u is the set of maxima of f and A c is the set of maxima of C.
2. [Restoration] Given the solution of problem (17.40), d ∗ , solve the two maximisation problems:
max u∈U f (d ∗ , u)
s.t.
C(d, u) ≤ 0
(17.41)
max
u∈U
C(d
∗ , u)
(17.42)
The solution of problem (17.41), u a,f , is added to the archive A u and the
solution of problem (17.42), u a,C is added to A c if max u∈U C(d, u) > 0.
Note that problem (17.42) has to be understood as a maximisation for every
constraint function in C and not as a vector optimisation. This approach pushes
the optimiser to find design solutions that are feasible for all values of the
uncertain variables. If a feasible solution cannot be found, the constraints are
relaxed (line 15), by defining the new constraint C ∗ = C + with the minimum
constraint violation over U .
The optimisation and restoration loops are repeated one after the other for a
prescribed number of iterations and all d ∗ and associated maxima A are stored
in a global archive A g . The global archive is then used to perform a cross-check
of the solutions. Given a finite number of iterations, one might obtain a solution d ∗
associated to non-globally optimal value of u e ∈ A. In order to mitigate this problem
one can evaluate f and C taking multiple pairs d ∗ , u e taken from the archive A.
The overall procedure is summarised in Algorithm 1. First the design vector ¯
d and
the archives—A u , A c , A d —are initialised and a constrained maximisation over the
uncertain domain U is run kipping fixed ¯
d (lines 1 and 2). Then the inner and outer
loops are alternated until the maximum number of iteration is reached (lines 3–
22). In particular, the archive of the design vectors d, A d , is updated after each
outer loop (line 6) while the archives of the uncertainty vectors u F —from the
maximisation of the objective function—and u C —from the maximisation of the
constraint violation—are updated after each inner loop (respectively, in lines 10
and 13) if they are not already saved in the archives. During the last loops of the
555
to ensure that the maximum value of C over U is admissible. The optimisation and
restoration loops then become as follows:
1. [Optimisation] Given a set of maxima A = A u
A c , solve the constrained
minimisation problem:
min d∈D max u∈A f (d, u)
s.t.
max u∈A C(d, u) ≤ 0
(17.40)
where A u is the set of maxima of f and A c is the set of maxima of C.
2. [Restoration] Given the solution of problem (17.40), d ∗ , solve the two maximisation problems:
max u∈U f (d ∗ , u)
s.t.
C(d, u) ≤ 0
(17.41)
max
u∈U
C(d
∗ , u)
(17.42)
The solution of problem (17.41), u a,f , is added to the archive A u and the
solution of problem (17.42), u a,C is added to A c if max u∈U C(d, u) > 0.
Note that problem (17.42) has to be understood as a maximisation for every
constraint function in C and not as a vector optimisation. This approach pushes
the optimiser to find design solutions that are feasible for all values of the
uncertain variables. If a feasible solution cannot be found, the constraints are
relaxed (line 15), by defining the new constraint C ∗ = C + with the minimum
constraint violation over U .
The optimisation and restoration loops are repeated one after the other for a
prescribed number of iterations and all d ∗ and associated maxima A are stored
in a global archive A g . The global archive is then used to perform a cross-check
of the solutions. Given a finite number of iterations, one might obtain a solution d ∗
associated to non-globally optimal value of u e ∈ A. In order to mitigate this problem
one can evaluate f and C taking multiple pairs d ∗ , u e taken from the archive A.
The overall procedure is summarised in Algorithm 1. First the design vector ¯
d and
the archives—A u , A c , A d —are initialised and a constrained maximisation over the
uncertain domain U is run kipping fixed ¯
d (lines 1 and 2). Then the inner and outer
loops are alternated until the maximum number of iteration is reached (lines 3–
22). In particular, the archive of the design vectors d, A d , is updated after each
outer loop (line 6) while the archives of the uncertainty vectors u F —from the
maximisation of the objective function—and u C —from the maximisation of the
constraint violation—are updated after each inner loop (respectively, in lines 10
and 13) if they are not already saved in the archives. During the last loops of the