# Working meeting "Dealing with stochastics in optimization problems" - May, 13th 2016

**Organizers: ** Josselin Garnier (Université Paris diderot, laboratoire LPMA & laboratoire JAcques-Louis Lions), Emmanuel Vazquez (Centrale Supélec, département signal & statistiques) and Miguel Munoz Zuniga (IFP Energies Nouvelles, département de mathématiques appliquées)

It will take place on:

May 13, 2016, at Amphithéâtre Hermite, Institut Henri Poincaré, Paris.

**Preliminary Program**

9:15 : Welcome of the participants

9:30 – 9:45 : Introduction – M. Munoz Zuniga (IFPEN) slides .

9:45 – 10:30 :** Maliki Moustapha (ETH Zurich) – Quantile-based optimization using adaptive Kriging models: Application to car body design. slides**

10:30 – 11:15 :** Paul Feliot (Centrale Supélec) – Bayesian constrained multi-objective optimization with application to the design of a commercial aircraft environment control system. slides**

11:15 – 11:45 : Coffee Break

11:45 – 12:30 :** Asma Atamna (INRIA) – Evolution Strategies with Augmented Lagrangian**

Constraint Handling Approach. slides

12:30-14:15: Lunch break

14:15 – 15:00 :** Wim Van Ackooij (EDF) - Differentiability of probability function involving non-linear mappings of Gaussian random vectors. slides **

15:00 – 15:45 :** Nicolas Gayton (IFMA) – Stratégie AK de classification : principe et applications **

15:45 – 16:00 : Conclusion and end of the workshop

**Abstracts**

**Maliki Moustapha – Quantile-based optimization using adaptive Kriging models: Application to car body design**

Uncertainties play an important role in the design of engineering structures.

As they may aect the physical properties and environmental conditions

of a designed structure, they must be fully accounted for throughout the design process. From the point of view of optimization, the most popular

approaches are robust and reliability-based design optimization. In this presentation, we focus on the latter only. More specically, we introduce an

alternative formulation to the state-of-the-art techniques which is based on quantiles as measure of conservatism.

Broadly speaking, optimization under uncertainties require repeated evaluations of the performance function describing the structure behavior. In

practice, the associated models are expensive-to-evaluate. Metamodeling has arisen as a tool to overcome this issue. Here, we consider Gaussian process modeling, also known as Kriging. The remarkable feature of Kriging

is that it also provides a built-in measure of its own accuracy. This feature

is used to propose a methodology for adaptive Kriging optimization based

on quantiles. Such a methodology is aimed at producing a reliable solution

with a reduced computational budget. As an application, a problem related

to the lightweight design of an automotive body shell under frontal impact

is presented.

**Paul Feliot – Bayesian constrained multi-objective optimization with application to the design of a commercial aircraft environment control system**

We present the BMOO algorithm for multi-objective optimization in the presence of non-linear and expensive-to-evaluate constraints and an application to the design of a commercial aircraft environment control system (ECS). The BMOO algorithm implements a Bayesian approach to this optimization problem. The emphasis is on conducting the optimization using a limited number of system simulations and, as a particularity, the algorithm is run on a non-hypercubic design domain and implements hidden constraints handling capabilities. The ECS is composed of two cross-flow heat exchangers, a centrifugal compressor and a radial turbine, the geometries of which are simultaneously optimized to achieve minimal weight and entropy generation of the system as a whole, while respecting strict specifications. While both objectives impact the overall performance of the aircraft, they are shown to be antagonistic and a set of trade-off design solutions is identified.

Registration is free but appreciated.

**Asma Atamna - Evolution Strategies with Augmented Lagrangian Constraint Handling Approach**

Evolution Strategies (ESs) are derivative-free randomized adaptive optimization

algorithms initially designed for unconstrained continuous optimization. In this work,

we consider a simple constrained problem where we have a single linear constraint

and propose a general ES with an augmented Lagrangian approach for constraint handling.

We then discuss how linear convergence can be achieved on our general algorithm

by adopting a Markov chains approach: given some conditions are satisfied by

the transition function of the algorithm, we exhibit a class of functions on which one

can construct a homogeneous Markov chain. We then show how the stability of the

constructed Markov chain leads to linear convergence/divergence of the ES. Stability

being usually difficult to prove “manually”, we will investigate it empirically on convex

quadratic functions, for two particular ESs: the (1+1)-ES with 1/5th success rule

and the (mu,lambda)-ES with median success rule.

**Wim Van Ackooij - Differentiability of probability function involving non-linear mappings of Gaussian random vectors**

In this talk, we will consider probability functions of parameter-

dependent random inequality systems under Gaussian distributions. As a main

result, we provide an upper estimate for the Clarke subdifferential of such

probability functions without imposing compactness conditions. A constraint

qualification ensuring continuous differentiability is formulated. Using these

results, several explicit formulae can be derived from the general result in

case of linear random inequality systems. We will also show how these compare

with other formulae in the same setting. Finally, in the case of a constant

coefficient matrix an upper estimate for even the smaller Mordukhovich

subdifferential is proven. Throughout the talk, we will also discuss several

concrete examples.

**Nicolas Gayton – Stratégie AK de classification : principe et applications**

Les méta-modèles sont couramment utilisés depuis quelques années particulièrement pour l'analyse paramétrique, l'analyse de sensibilité, l'optimisation, l'analyse fiabiliste.

Ces méta-modèles sont en général construits de façon itérative, pour approximer un fonction coûteuse, grâce à un critère d'enrichissement qui permet d'identifier le meilleur point de calcul suivant à réaliser.

L'objectif de la cette intervention est la présentation d'une technique un peu différente de construction d'un méta-modèle de krigeage qui permet de séparer un nuage de points en différentes populations. Cette technique dite de classification est utilisée dans une famille de méthode appelée AK qui sera présentée en lien avec quelques applications à l'ingénierie mécanique.