Statistics of Pareto Fronts

Abstract

We consider multiobjective optimization problems affected by uncertainty, where the objective functions or the restrictions involve random variables. We are interested in the evaluation of statistics such as medians, quantiles and confidence intervals for the Pareto front. We present a method for the determination of such statistics which is independent of the representation used to describe the Pareto front. In a second step, we start from a sample of Pareto fronts and we use a Generalized Fourier Series approach to generate a larger sample of about 10⁵ Pareto fronts with a reasonable computational cost. These large samples are used to obtain more accurate statistics. Examples show that the method is effective to calculate.

Appendix A: Generalized Fourier Series

In the framework of UQ, we are interested in the representation of random variables: let us consider a couple of random variables $\left( {\varvec{U},\varvec{X}} \right)$, such that $\varvec{X} = \varvec{X}\left( \varvec{U} \right)$, that is $\varvec{X}$ is a function of $\varvec{U}$. If $\varvec{X} \in V$, where $V$ is a separable Hilbert space, associated to the scalar product $\left( { \bullet , \bullet } \right)$, we may consider a convenient Hilbert basis (or total family) $\Phi = \left\{ {\varphi_{i} } \right\}_{{i \in {\mathbb{N}}}}$ and look for a representation $\varvec{X}$ given by  [2]:

$$ \varvec{X} = \varvec{X}\left( \varvec{U} \right) = \sum\nolimits_{{i \in {\mathbb{N}}}} {\varvec{x}_{\varvec{i}} \varphi_{i} } \left( \varvec{U} \right). $$

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If the family is orthonormal, $\left( {\varphi_{i} , \varphi_{j} } \right) = \delta_{ij}$ and the coefficients of the expansion are given by $\varvec{x}_{\varvec{i}} = \left( {\varvec{X},\varphi_{i} \left( \varvec{U} \right)} \right)$. Otherwise, we may consider the approximations of $\varvec{X}$ by finite sums:

$$ \varvec{X} \approx P_{n} \varvec{X} = \mathop \sum \limits_{1 \le i \le n} \varvec{x}_{\varvec{i}} \varphi_{i} \left( \varvec{U} \right). $$

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In this case, the coefficients $\varvec{x}_{\varvec{i}}$ are the solutions of the linear system ${\mathbf{\mathcal{A}}}\varvec{x} = {\mathbf{\mathcal{B}}}$, where ${\mathbf{\mathcal{A}}}_{{\varvec{ij}}} = \left( {\varphi_{i} , \varphi_{j} } \right)$ and ${\mathbf{\mathcal{B}}}_{\varvec{i}} = \left( {\varvec{X}, \varphi_{i} } \right)$. We have:

$$ \mathop {\lim }\limits_{n \to \infty } P_{n} \varvec{X} = \varvec{X}. $$

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In UQ, the Hilbert space $V$ is mainly $L^{2} \left( {\Omega ,P} \right)$, where $\Omega \subset {\mathbb{R}}^{n}$ and $P$ is a probability measure, with $\varvec{ }\left( {\varvec{Y},\varvec{Z}} \right) = {\text{E}}\left( {\varvec{YZ}} \right)$. Classical families $\Phi$ are formed by polynomials, trigonometric functions, Splines or Finite Elements approximations. Examples of approximations may be found in the literature (see, for instance, [2, 3]). When X is a function of a second variable – for instance, t – we denote the function $\varvec{X}\left( {t |\varvec{U}} \right)$ and we have:

$$ \varvec{X}\left( {t |\varvec{U}} \right) = \mathop \sum \limits_{{i \in {\mathbb{N}}}} \varvec{x}_{\varvec{i}} \left( t \right)\varphi_{i} \left( \varvec{U} \right) \approx P_{n} \varvec{X}\left( {t |\varvec{U}} \right) = \mathop \sum \limits_{1 \le i \le n} \varvec{x}_{\varvec{i}} \left( t \right)\varphi_{i} \left( \varvec{U} \right). $$

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The reader may refer to [4] to get more information and MATLAB codes for the evaluation of the coefficients $\varvec{x}_{\varvec{i}}$, namely in multidimensional situations. In practice, we use a sample from $\left( {t |\varvec{U}} \right)\,:\varvec{X}\left( {t |\varvec{U}_{1} } \right), \ldots ,\varvec{X}\left( {t |\varvec{U}_{{\varvec{ns}}} } \right)$ in order to evaluate the means forming ${\mathbf{\mathcal{A}}}$ and ${\mathbf{\mathcal{B}}}$.