## 数学代写|凸优化作业代写Convex Optimization代考|CPD131

2022年9月27日

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## 数学代写|凸优化作业代写Convex Optimization代考|Statistical Models Based Algorithms

Multi-objective optimization problems with expensive objective functions are typical in engineering design, where time-consuming computations are involved for modeling technological processes. Frequently, the available software implements an algorithm to compute the values of objective functions, but neither details of implementation nor analytical properties of the objective functions are known. Nevertheless, the continuity of the objective functions can be normally assumed. The complexity of the computational model implies not only the expensiveness of the objective function but also the uncertainty in its properties, so that other analytical properties of $\mathbf{f}(\mathbf{x})$, besides the continuity, cannot be substantiated. Such unfavorable, from the optimization point of view, properties of $\mathbf{f}(\mathbf{x})$ as nondifferentiability, non-convexity, and multimodality cannot be excluded. Difficulties of the black-box global optimization of expensive functions are well known from the experience gained in the single-objective case.

The substantiation of an optimization algorithm based on the ideal black-box model would be hardly possible. To construct algorithms rationally, a hypothetic behavior of objective functions in question should be guessed. Therefore, the concept of black-box should be revisited. We will consider mathematical models of some transparency complemented black-box for which the term gray box seemingly would be suitable. IIowever, following widely accepted tradition we will use the term black-box. Actually, we assume that the analytical properties of the the uncertainty of their behavior can be made. In the deterministic approach, typically an assumption is made which enables the construction of underestimates for function values, e.g., the assumption on Lipschitz continuity of the considered objective function enables the construction of Lipschitz underestimates. For a brief review of Lipschitz optimization, we refer to Section $4.2$ and for the details to [87]. Another approach, based on statistical models of objective functions, is validated in [242]; for a brief review of single-objective optimization methods based on statistical models, we refer to Section 4.3. Since the statistical models and the utility theory-based global optimization is a well theoretically justified approach to the single-objective global optimization of expensive functions, it seems worthwhile to generalize that approach also to the multi-objective case.
Let us recall the notation of the considered problem
$$\min _{\mathbf{x} \in \mathbf{A}} \mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_2(x), \ldots, f_m(\mathbf{x})\right)^T, \mathbf{A} \subset \mathbb{R}^d,$$
where the vector objective function $\mathbf{f}(\mathbf{x})$ is defined over a simple feasible region, e.g., for the concreteness it can be assumed that $\mathbf{A}$ is a hyper-rectangle.

## 数学代写|凸优化作业代写Convex Optimization代考|Statistical Model

To validate, from the decision theory perspective, the selection of a site for current computation/observation of the vector of objectives, a model of objective functions is needed. We consider here an approach based on statistical models of the objective functions. For the consideration of functions under uncertainty, the stochastic function models are developed in the probability theory. Assuming such a classical probabilistic model we assume that the considered objective functions are random realizations of the chosen stochastic function. However, the algorithms considered below can also be interpreted in terms of more general statistical models constructed in [242], using the ideas of subjective probabilities that are discussed from all angles in [58] and [182]. To facilitate the implementation of the corresponding algorithms, the Gaussian stochastic functions normally are chosen for statistical models.

The accepted for the statistical models of separate objectives $f_j(\mathbf{x})$ stochastic functions comprise a vector-valued Gaussian random field $\Xi(\mathbf{x})$ which is accepted for the statistical model of $\mathbf{f}(\mathbf{x})$. In many real world applied problems as well as in the test problems, the objectives are not (or weakly) interrelated. Correspondingly, components of $\Xi(\mathbf{x})$ are supposed to be independent. The correlation between the components of $\Xi(\mathbf{x})$ could be included into the model; however, it would imply some numerical and statistical inference problems requiring a further investigation.
It is assumed that a priori information about the expected behavior (a form of variation over A) of objective functions is not available. The heuristic assumption on the lack of a priori information is formalized as an assumption that $\xi_i(\mathbf{x}), i=$ $1, \ldots, m$, are homogeneous isotropic random fields, i.e., that their mean values $\mu_i$ and variances $\sigma_i^2$ are constants, and that the correlation between $\xi_i\left(\mathbf{x}_j\right)$ and $\xi_i\left(\mathbf{x}_k\right)$ depends only on $\left|\mathbf{x}_j-\mathbf{x}_k\right|_D$. The choice of the exponential correlation function $\rho(t)=\exp (-c t), \quad c>0$, is motivated by the fact that, in the one-dimensional case, such a correlation function ensures Markovian property, and by the positive previous experience in use of the stochastic models with the exponential correlation function for the construction of single-objective global optimization; see, e.g., the monograph [139]

The parameters of the statistical model should be estimated using data on $\mathbf{f}(\cdot)$; since the components of $\Xi(\mathbf{x})$ are assumed to be independent, the parameters for each $\xi_j(\mathbf{x})$ can be estimated separately. In the further described implementation, mean values and variances of $\xi_j(\mathbf{x})$ are estimated using their values at random points generated uniformly over $\mathbf{A}$. Arithmetic means are used as the estimates of mean values, and sample variances are used as the estimates of variances. The choice of the value of the correlation function parameter $c$ depends on the scale of variables; in the case of $\mathbf{A}$, scaled to the unit hyper-cube, $c$ is defined by the equation $c \sqrt{d}=7$. Such a choice is motivated by a supposed uncertainty in the behavior of the objective functions: the correlation between random field values at the maximum distant points is assumed vanishing as $10^{-3}$; on the other hand, the random field values at the closely located points are assumed reasonably correlated, i.e., it is supposed that the correlation coefficient is about $0.5$ in the case the mutual distance constitutes $10 \%$ of the maximum distance in $\mathbf{A}$.

## 数学代写|凸优化作业代写凸优化代考|基于统计模型的算法

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$$\min _{\mathbf{x} \in \mathbf{A}} \mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_2(x), \ldots, f_m(\mathbf{x})\right)^T, \mathbf{A} \subset \mathbb{R}^d,$$

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