## Section: New Results

### Platform for prioritized multi-objective optimization

Participant : Jean-Antoine Désidéri.

A multi-objective differentiable optimization
algorithm had been proposed to solve problems presenting a hierarchy in the cost functions,
$\left\{{f}_{j}\left(\mathbf{x}\right)\right\}$ ($j=1,\cdots ,M\ge 2;\phantom{\rule{3.33333pt}{0ex}}\mathbf{x}\in {\Omega}_{a}\subseteq {\U0001d5b1}^{n}$).
The first cost functions for which $j\in \{1,\cdots ,m\}$ ($1\le m<M$) are considered
to be of preponderant importance;
they are referred to as the “primary cost functions” and are subject to a
“prioritized” treatment, in contrast with the tail ones, for which
$j\in \{m+1,\cdots ,M\}$, referred to as the “secondary cost functions”.
The problem is subject to the nonlinear constraints, ${c}_{k}\left(\mathbf{x}\right)=0\phantom{\rule{3.33333pt}{0ex}}(k=1,\cdots ,K)$.
The cost functions $\left\{{f}_{j}\left(\mathbf{x}\right)\right\}$ and the constraint functions $\left\{{c}_{k}\left(\mathbf{x}\right)\right\}$
are all smooth, say ${C}^{2}\left({\Omega}_{a}\right)$.
The algorithm was first introduced in the case of two disciplines ($m=1$, $M=2$),
and successfully applied to optimum shape design
optimization in compressible aerodynamics concurrently with a secondary discipline
[101] [105].
An initial admissible point ${\mathbf{x}}_{A}^{\u2606}$ that is Pareto-optimal with respect to the sole primary cost
functions (subject to the constraints) is assumed to be known. Subsequently, a small
parameter $\epsilon \in [0,1]$ is introduced, and it is established that
a continuum of Nash equilibria $\left\{{\overline{\mathbf{x}}}_{\epsilon}\right\}$ exists for all small enough $\epsilon $.
The continuum originates from ${\mathbf{x}}_{A}^{\u2606}$ (${\overline{\mathbf{x}}}_{0}={\mathbf{x}}_{A}^{\u2606}$).
Along the continuum:
(*i*) the Pareto-stationarity condition exactly satisfied by the primary cost functions
at ${\mathbf{x}}_{A}^{\u2606}$ is degraded by a term $O\left({\epsilon}^{2}\right)$ only, whereas
(*ii*) the secondary cost functions initially decrease, at least linearly with $\epsilon $
with a negative derivative provided by the theory.
Thus, the secondary cost functions are reduced while
the primary cost functions are maintained to quasi Pareto-optimality. In this report, we firstly
recall the definition of the different steps in the computational Nash-game
algorithm assuming the functions all have known first and second derivatives
(here without proofs). Then we show how, in the absence of explicitly known derivatives, the
continuum of Nash equilibria can be calculated approximately via the construction of quadratic
surrogate functions. Numerical examples are provided and commented
[41].