# 数学代写|凸优化作业代写Convex Optimization代考|MATH3204

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## 数学代写|凸优化作业代写Convex Optimization代考|The Generalization to the Multi-Objective Case

Recently several papers have been published which propose multi-objective optimization algorithms that generalize single-objective optimization algorithms based on statistical models of objective functions [53, 101, 105, 106, 142, 224, 252]. The numerical results included there show the relevance of the proposed algorithms to the problems of multi-objective optimization with black-box expensive objectives. We present here a new idea for constructing relevant algorithms.

A multi-objective minimization problem can be stated almost identically to the single-objective problem considered in the previous subsection:
$$\min _{\mathbf{x} \subset \mathbf{A}} \mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_2(\mathbf{x}), \ldots, f_m(\mathbf{x})\right)^T, \mathbf{A} \subset \mathbb{R}^d$$ however, the concept of solution in this case is more complicated. For the definitions of the solution to a multi-objective optimization problem with nonlinear objectives, we refer to Chapter 1 .

In the case of multi-objective optimization, a vector objective function $\mathbf{f}(\mathbf{x})=$ $\left(f_1(\mathbf{x}) \cdot f_2(\mathbf{x}) \ldots \ldots f_m(\mathbf{x})\right)^T$ is considered. The same arguments, as in the case of single-objective optimization, corroborate the applicability of statistical models. The assumptions on black-box information and expense of the objective functions together with the standard assumptions of rational decision making imply the acceptability of a family of random vectors $\Xi(\mathbf{x})=\left(\xi_1(\mathbf{x}), \ldots, \xi_m(\mathbf{x})\right)^T, x \in \mathbf{A}$, as a statistical model of $\mathbf{f}(\mathbf{x})$. Similarly, the location and spread parameters of $\xi_i(\mathbf{x})$, denoted by $m_i(\mathbf{x}), s_i(\mathbf{x}), i=1, \ldots, r$, are essential in the characterization of $\xi_i(\mathbf{x})$. For a more specific characterization of $\Xi(\mathbf{x})$, e.g., by a multidimensional distribution of $\Xi(\mathbf{x})$, the available information usually is insufficient. If the information on, e.g., the correlation between $\xi_i(\mathbf{x})$ and $\xi_j(\mathbf{x})$ were available, the covariance matrix could be included into the statistical model. However, here we assume that the objectives are independent, and the spread parameters are represented by a diagonal matrix $\Sigma(\mathbf{x})$ whose diagonal elements are equal to $s_1, \ldots, s_m$. Similarly to the case of single-objective optimization, we assume that the utility of choice of the point for the current computation of the vector value $\mathbf{f}(\mathbf{x})$ has the following structure
$$v_{n+1}(\mathbf{x})=V_{n+1}\left(\mathbf{m}(\mathbf{x}), \Sigma(\mathbf{x}), \mathbf{y}^n\right),$$
where $\mathbf{m}(\mathbf{x})=\left(m_1(\mathbf{x}), \ldots, m_m(\mathbf{x})\right)^T$, and $\mathbf{y}^n$ denotes a vector desired to improve.

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

Theoretical and algorithmic achievements of multi-objective optimization have implied also the expansion of respective applications. Among applied multiobjective optimization problems, expensive multimodal black-box problems are rather frequent. However, they constitute still a relatively little researched subfield of multi-objective optimization and deserve more attention from researchers. Since the statistical models based single-objective optimization algorithms well correspond to the challenges of single-objective global optimization of expensive black-box functions, they were generalized to the multi-objective case. As shown in the previous sections, the theoretical generalization is rather straightforward. Some experimental investigation was performed to find out how much the generalization corresponds to the expectations of their suitability for multi-objective optimization.
General methodological concepts of testing and comparison of mathematical programming algorithms and software are well developed; see [131]. The methodology, called Competitive Testing in [82], should normally be applied for the comparison of the well-established algorithms. This methodology is also extended for testing and comparison of multi-objective optimization algorithms; see, e.g., [42, $60,135,266,267]$. In the case of the well-researched classes of problems (e.g., convex multi-objective optimization), this methodology is universally applicable, only the selection of test functions should be specially selected taking into account the properties of the considered sub-class of problems, e.g., considered in $[66,129]$. The tests, based on special cases of real world applied problems, can be very useful for evaluating the efficiency of the respective algorithms; see, e.g., [154] where multi-objective portfolio problems are used for testing the algorithms aimed to distribute solutions uniformly in the Pareto optimal set.

However, the standard testing methodology is not well suitable for the algorithms considered in this chapter. The first difficulty is caused by the main feature of the targeted problems: they are supposed to be expensive. Therefore, a solution, found by an optimization algorithm applied, normally is rather rough. An optimization algorithm is as much useful as much its application aids a decision maker in making a final decision in the conditions of uncertainty reduced because of the application of the algorithm. The quantitative assessment of such a criterion of an algorithm is difficult.

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