Quadratic approximations

In this example, we illustrate the use of the QuadApprox plot on the Sobieski's SSBJ problem.

The QuadApprox post-processing performs a quadratic approximation of a given function from an optimization history and provide two plots.

The first plot shows an approximation of the Hessian matrix \(\dfrac{\partial^2 f}{\partial x_i \partial x_j}\) based on the Symmetric Rank 1 method (SR1) [NW06]. The color map uses a symmetric logarithmic (symlog) scale. This plots the cross influence of the design variables on the objective function or constraints. For instance, on the last figure, the maximal second-order sensitivity is \(\dfrac{\partial^2 -y_4}{\partial^2 x_0} = 2.10^5\), which means that the \(x_0\) is the most influential variable. Then, the cross derivative \(\dfrac{\partial^2 -y_4}{\partial x_0 \partial x_2} = 5.10^4\) is positive and relatively high compared to the previous one but the combined effects of \(x_0\) and \(x_2\) are non-negligible in comparison.

The second plot represents the quadratic approximation of the objective around the optimal solution : \(a_{i}(t)=0.5 (t-x^*_i)^2 \dfrac{\partial^2 f}{\partial x_i^2} + (t-x^*_i) \dfrac{\partial f}{\partial x_i} + f(x^*)\), where \(x^*\) is the optimal solution. This approximation highlights the sensitivity of the objective function with respect to the design variables: we notice that the design variables \(x\_1, x\_5, x\_6\) have little influence , whereas \(x\_0, x\_2, x\_9\) have a huge influence on the objective. This trend is also noted in the diagonal terms of the Hessian matrix \(\dfrac{\partial^2 f}{\partial x_i^2}\).

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<gemseo.post.quad_approx.QuadApprox object at 0x76a67c102e70>

from __future__ import annotations

from gemseo import execute_post
from gemseo.settings.post import QuadApprox_Settings

execute_post(
    "sobieski_mdf_scenario.h5",
    settings_model=QuadApprox_Settings(function="-y_4", save=False, show=True),
)