## 数学代写|凸优化作业代写Convex Optimization代考|ELEC4631

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|Multi-Objective P-Algorithm

In this section, a single-objective global optimization algorithm, based on a statistical model of multimodal functions, namely the P-algorithm, is generalized for the case of black-box multi-objective optimization. For the axiomatic definition of statistical models of multimodal black-box functions and the single-objective Palgorithm, we refer to [242,243].

The minimization is considered at the $n+1$-st minimization step. The points where the objective functions were computed are denoted by $\mathbf{x}i, i=1, \ldots, n$, and the corresponding objective vectors are denoted by $\mathbf{y}_i=\mathbf{f}\left(\mathbf{x}_i\right) ; \mathbf{y}_i=\left(y{i 1}, \ldots, y_{i m}\right)$. Based on the discussion in the previous section, the vector-valued Gaussian random field
$$\Xi(\mathbf{x}) \in \mathbb{R}^m, \mathbf{x} \in \mathbf{A} \subset \mathbb{R}^d$$ is accepted as a model of the vector objective function. In the frame of that model, an unknown vector of objectives $\mathbf{f}(\mathbf{x}), \mathbf{x} \neq \mathbf{x}i, i=1, \ldots, n$, is interpreted as a random vector whose distribution is defined by the conditional distribution function of $\Xi(\mathbf{x})$ $$\Pi_x^n(\mathbf{t})=P\left{\Xi(\mathbf{x}) \leq \mathbf{t} \mid \Xi\left(\mathbf{x}_i\right)=\mathbf{y}_i, i=1, \ldots, n\right} .$$ Ihe choice of the current observation point, 1.e., of the point where to compute the vector of objectives at the current minimization step, is a decision under uncertainty. The statistical model (7.2) represents the uncertainty in a result of the decision. With respect to the statistical model (7.2), the choice of the current observation point $\mathbf{x}{n+1} \in \mathbf{A}$ means a choice of a distribution function from the set of distribution functions $\Pi_x^n(\cdot), \quad \mathbf{x} \in \mathbf{A}$. To justify the choice, the methodology of rational decision making under uncertainty can be applied. If the preference of choice satisfies the rationality principles, then there exists a unique (to within a linear transformation) utility function $U(\cdot)$ compatible with the preference of choice [58]:
$$\Pi_x^n(\cdot) \succ \Pi_z^n(\cdot) \text { iff } \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} U(\mathbf{t}) \cdot d \Pi_x^n(\mathbf{t}) \geq \int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty} U(\mathbf{t}) \cdot d \Pi_z^n(\mathbf{t}) .$$
The problem of selection of a utility function for the case of single-objective minimization $\left(\min _{\mathbf{x} \in \mathbf{A}} f(\mathbf{x})\right)$ was considered in [243]. The classical axioms of rational decision making (see, e.g., $[58,182]$ ) are reformulated in terms of the rationality of search for the global minimum in [243], where it has also been shown that the following utility function
$$u(t)=1 \text {, for } t \leq y^n, \text { and } u(t)=0, \text { for } t>y^n$$
is compatible with these axioms; $y^n$ denotes an improvement threshold: a new function value at the current step is considered an improvement if it is smaller than $y^n ; y^n<y_i, i=1, \ldots, n ; y_i=f\left(\mathbf{x}_i\right)$ are the function values computed at previous minimization steps. The average utility defined using the utility function (7.5) means the improvement probability.

## 数学代写|凸优化作业代写Convex Optimization代考|A New Approach to Single-Objective Optimization

Let us consider the current minimization step, where $n$ function values have been computed at the previous steps: $y_i=f\left(\mathbf{x}_i\right), i=1, \ldots, n$. A rational choice of a point for the next computation of the objective function value cannot be performed without the assessment of the uncertainty in the result of the computation. The only objective information on $f(\cdot)$ is $\mathbf{x}_i, y_i, i=1, \ldots, n$. Besides that objective information, normally some subjective information is available, e.g., the experience of solution of similar problems in the past. As shown in [242], very general assumptions on the rational perception of uncertainty imply a random variable model for the objective function value to be computed, i.e. those assumptions imply that a random variable $\xi_x, \mathbf{x} \in \mathbf{A}, \mathbf{x} \neq \mathbf{x}_i, i=1, \ldots, n$, is acceptable as a statistical model of the unknown value of the objective function $f(\mathbf{x})$. We refer to $[216,239]$ for the axiomatic construction of a computationally simple statistical model of objective functions. In case a stochastic function is chosen as a statistical model of $f(\cdot)$, the corresponding random variable is defined by the conditional distribution of this stochastic function.

Let us consider the choice of a point for the current computation of the objective function value. Such a choice in the black-box situation is a decision under uncertainty, and the rational decision theory [58] can be applied to make the choice rationally. The theory suggests to make a decision by maximizing the average utility. To compute the average utility, a utility function is needed besides of a statistical model of uncertainty. A utility function corresponding to the conception of global optimization is proposed in [243]. However, a natural extension of the axioms proposed in [243] to the multi-objective case is difficult.

Any characterization of a random variahle normally includes a location parameter (e.g., the mean value) and a spread parameter (e.g., the standard deviation); we use a minimal description of $\xi_x$ by these two parameters denoted by $m(\mathbf{x})$ and $s(\mathbf{x})$. The dependence of both parameters on the information available at the current optimization step $\left(\mathbf{x}i, y_i, i=1, \ldots, n\right)$ will be included into the notation where needed. Assume that the average utility of computation of the current objective function value at the point $\mathbf{x}$ at the $n$ step $V{n+1}(\cdot)$ depends on $\mathbf{x}$ only via $m(\mathbf{x})$ and $s(\mathbf{x})$ :
$$V_{n+1}\left(m(\mathbf{x}), s(\mathbf{x}), y^n\right)=\int_{-\infty}^{\infty} u(t) d \Pi_x^1(t),$$
where the threshold for the improvement is also shown as a variable influencing the average utility. The subscript can be omitted in the cases where general properties of the average utility are considered independently of the number of steps of the search. The shorthand $v_{n+1}(\mathbf{x})$ is usable where only the dependence of the average utility on $\mathbf{x}$ is of interest.

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## MATLAB代写

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