数学代写|优化算法作业代写optimisation algorithms代考| Multi-objective Optimization

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|优化算法作业代写optimisation algorithms代考|Multi-objective Optimization

Many optimization problems in the real world have $k$ possibly contradictory objectives $f_{i}$ which must be optimized simultaneously. Furthermore, the solutions must satisfy $m$ inequality constraints $g$ and $p$ equality constraints $h$. A solution candidate $x$ is feasible, if and only if $g_{i}(x) \geq 0 \forall i=1,2, \ldots, m$ and $h_{i}(x)=0 \forall i=1,2, . ., p$ holds. A multi-objective optimization problem (MOP) can then be formally defined as follows:

Definition 7 (MOP). Find a solution candidate $x^{\star}$ in $\mathbb{X}$ which minimizes (or maximizes) the vector function $\mathbf{f}\left(x^{\star}\right)=\left(f_{i}\left(x^{\star}\right), f_{2}\left(x^{\star}\right), \ldots, f_{k}\left(x^{\star}\right)\right)^{T}$ and is feasible, (i.e., satisfies the $m$ inequality constraints $g_{i}\left(x^{\star}\right) \geq 0 \forall i=1,2, . ., m$, the $p$ equality constraints $\left.h_{i}\left(x^{\star}\right)=0 \forall i=1,2, . ., p\right)$.

As in single-objective optimization, nature-inspired algorithms are popular techniques to solve such problems. The fact that there are two or more objective functions implies additional difficulties. Due to the contradictory feature of the functions in a MOP and the fact that there exists no total order in $\mathbb{R}^{n}$ for $n>1$, the notions of “better than” and “optimum” have to be redefined. When comparing any two solutions $x_{1}$ and $x_{2}$, solution $x_{1}$ can have a better value in objective $f_{i}$, i.e., $f_{i}\left(x_{1}\right)<f_{i}\left(x_{2}\right)$, while solution $x_{2}$ can have a better value in objective $f_{j}$. The concepts commonly used here are Pareto dominance and Pareto optimality.

Definition 8 (Pareto Dominance). In the context of multi-objective global optimization, a solution candidate $x_{1}$ is said to dominate another solution candidate $x_{2}$ (denoted by $\left.x_{1} \preccurlyeq x_{2}\right)$ if and only if $\mathbf{f}\left(x_{1}\right)$ is partially less than $\mathbf{f}\left(x_{2}\right)$, i.e., $\forall i \in{1, \ldots, k} f_{i}\left(x_{1}\right) \leq f_{i}\left(x_{2}\right) \wedge \exists j \in{1, \ldots, k}: f_{j}\left(x_{1}\right)<f_{j}\left(x_{2}\right)$.

The dominance notion allows us to assume that if solution $x_{1}$ dominates solution $x_{2}$, then $x_{1}$ is preferable to $x_{2}$. If both solution are non-dominated (such as candidate (1) and (2) in Fig. 12), some additional criteria have to be used to choose one of them.

Definition 9 (Pareto Optimality). A feasible point $x^{\star} \in \mathbb{X}$ is Paretooptimal if and only if there is no feasible $x_{b} \in \mathbb{X}$ with $x_{b} \preccurlyeq x^{*}$.

This definition states that $x^{\star}$ is Pareto-optimal if there is no other feasible solution $x_{b}$ which would improve some criterion without causing a simultaneous worsening in at least one other criterion. The solution to a MOP.

数学代写|优化算法作业代写optimisation algorithms代考|Countermeasures

In order to obtain an accurate approximation to the true Pareto front, many nature-inspired multi-objective algorithms apply a fitness assignment scheme based on the concept of Pareto dominance, as commented before. For example, NSGA-II [61, 62], the most well-known multi-objective technique, assigns to each solution a rank depending on the number of solutions dominating it. Thus, solutions with rank 1 are non-dominated, solutions with rank 2 are dominated by one solution, and so on. Other algorithms, such as SPEA2 $[247,248]$ introduce the concept of strength, which is similar to the ranking but also considers the number of dominated solutions.

While the use of Pareto-based ranking methods allows the techniques to search in the direction of finding approximations with good convergence, additional strategies are needed to promote spread. The most commonly adopted approach is to include a kind of density estimator in order to select those solutions which are in the less crowded regions of the objective space. Thus, NSGA-II employs the crowding distance [61] and SPEA2 the distance to the $\mathrm{k}$-nearest neighbor [62].

数学代写|优化算法作业代写optimisation algorithms代考|Constraint Handling

How the constraints mentioned in Definition 7 are handled is a whole research area in itself with roots in single-objective optimization. Maybe one of the most popular approach for dealing with constraints goes back to Courant [48] who introduced the idea of penalty functions $[73,44,201]$ in 1943: Consider, for instance, the term $f^{\prime}(x)=f(x)+v[h(x)]^{2}$ where $f$ is the original objective

function, $h$ is an equality constraint, and $v>0$. If $f^{\prime}$ is minimized, an infeasible individual will always have a worse fitness than a feasible one with the same objective values.

Besides such static penalty functions, dynamic terms incorporating the generation counter $[111,157]$ or adaptive approaches utilizing additional population statistics $[95,199]$ have been proposed. Rigorous discussions on penalty functions have been contributed by Fiacco and McCormick [73] and Smith and Coit [201].

During the last fifteen years, many approaches have been developed which incorporate constraint handling and multi-objectivity. Instead of using penalty terms, Pareto ranking can also be extended by additionally comparing individuals according to their feasibility, for instance. Examples for this approach are the Method of Inequalities (MOI) of Zakian [245] as used by Pohlheim [164] and the Goal Attainment method defined in [76]. Deb $[56,58]$ even suggested to simply turn constraints into objective functions in his MOEA version of Goal Programming.

有限元方法代写

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MATLAB代写

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