Many methods for performing multi-objective optimisation of computationally
expensive problems have been proposed recently. Typically, a probabilistic
surrogate for each objective is constructed from an initial dataset. The
surrogates can then be used to produce predictive densities in the objective
space for any solution. Using the predictive densities, we can compute the
expected hypervolume improvement (EHVI) due to a solution. Maximising the EHVI,
we can locate the most promising solution that may be expensively evaluated
next. There are closed-form expressions for computing the EHVI, integrating
over the multivariate predictive densities. However, they require partitioning
the objective space, which can be prohibitively expensive for more than three
objectives. Furthermore, there are no closed-form expressions for a problem
where the predictive densities are dependent, capturing the correlations
between objectives. Monte Carlo approximation is used instead in such cases,
which is not cheap. Hence, the need to develop new accurate but cheaper
approximation methods remains. Here we investigate an alternative approach
toward approximating the EHVI using Gauss-Hermite quadrature. We show that it
can be an accurate alternative to Monte Carlo for both independent and
correlated predictive densities with statistically significant rank
correlations for a range of popular test problems.

使用 Gauss-Hermite 积分作为新的求解多目标优化问题的方法，比起传统的蒙特卡罗模拟方法，准确性更高，且可以处理两个或者多个相关联的目标变量。

高斯 - 埃尔米特积分的期望超体积改进的高效近似

Efficient Approximation of Expected Hypervolume Improvement using Gauss-Hermite Quadrature

A fast and accurate algorithm for the computation of Gauss-Hermite and
generalized Gauss-Hermite quadrature nodes and weights is presented. The
algorithm is based on Newton's method with carefully selected initial guesses
for the nodes and a fast evaluation scheme for the associated orthogonal
polynomial. In the Gauss-Hermite case the initial guesses and evaluation scheme
rely on explicit asymptotic formulas. For generalized Gauss-Hermite, the
initial guesses are furnished by sampling a certain equilibrium measure and the
associated polynomial evaluated via a Riemann-Hilbert reformulation. In both
cases the $n$-point quadrature rule is computed in $\mathcal{O}(n)$ operations
to an accuracy that is close to machine precision. For sufficiently large $n$,
some of the quadrature weights have a value less than the smallest positive
normalized floating-point number in double precision and we exploit this fact
to achieve a complexity as low as $\mathcal{O}(\sqrt{n})$.

该研究提出了一种基于牛顿方法的快速、精确算法，用于计算 Gauss-Hermite 和广义 Gauss-Hermite 积分的节点和权重，并根据初始值和相应的正交多项式确定初始猜测。