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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Multi-objective optimization ： ウィキペディア英語版 Multi-objective optimization Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making, that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective optimization has been applied in many fields of science, including engineering, economics and logistics (see the section on applications for detailed examples) where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives. For a nontrivial multi-objective optimization problem, there does not exist a single solution that simultaneously optimizes each objective. In that case, the objective functions are said to be conflicting, and there exists a (possibly infinite) number of Pareto optimal solutions. A solution is called nondominated, Pareto optimal, Pareto efficient or noninferior, if none of the objective functions can be improved in value without degrading some of the other objective values. Without additional subjective preference information, all Pareto optimal solutions are considered equally good (as vectors cannot be ordered completely). Researchers study multi-objective optimization problems from different viewpoints and, thus, there exist different solution philosophies and goals when setting and solving them. The goal may be to find a representative set of Pareto optimal solutions, and/or quantify the trade-offs in satisfying the different objectives, and/or finding a single solution that satisfies the subjective preferences of a human decision maker (DM). == Introduction == A multi-objective optimization problem is an optimization problem that involves multiple objective functions.〔〔 In mathematical terms, a multi-objective optimization problem can be formulated as : $$ \begin \min &\left(f_1(x), f_2(x),\ldots, f_k(x) \right) \\ \text &x\in X, \end where the integer $k\backslash geq\; 2$ is the number of objectives and the set $X$ is the feasible set of decision vectors. The feasible set is typically defined by some constraint functions. In addition, the vector-valued objective function is often defined as :$f:X\backslash to\backslash mathbb\; R^k,\; \backslash \; f(x)=\; (f\_1(x),\backslash ldots,f\_k(x))^T$. If some objective function is to be maximized, it is equivalent to minimize its negative. The image of $X$ is denoted by $Y\; \backslash in\; \backslash mathbb\; R^k$ An element $x^$ *\in X is called a feasible solution or a feasible decision. A vector $z^$ * := f(x^ *)\in \mathbb R^k for a feasible solution $x^$ * is called an objective vector or an outcome. In multi-objective optimization, there does not typically exist a feasible solution that minimizes all objective functions simultaneously. Therefore, attention is paid to Pareto optimal solutions; that is, solutions that cannot be improved in any of the objectives without degrading at least one of the other objectives. In mathematical terms, a feasible solution $x^1\backslash in\; X$ is said to (Pareto) dominate another solution $x^2\backslash in\; X$, if #$f\_i(x^1)\backslash leq\; f\_i(x^2)$ for all indices $i\; \backslash in\; \backslash left\backslash$ and # for at least one index $j\; \backslash in\; \backslash left\backslash$. A solution $x^1\backslash in\; X$ (and the corresponding outcome $f(x^$ *)) is called Pareto optimal, if there does not exist another solution that dominates it. The set of Pareto optimal outcomes is often called the Pareto front or Pareto boundary. The Pareto front of a multi-objective optimization problem is bounded by a so-called nadir objective vector $z^$ and an ideal objective vector $z^$, if these are finite. The nadir objective vector is defined as :$z^\_i=\; \backslash sup\_\; i=1,\backslash ldots,k$ and the ideal objective vector as :$z^\_i=\backslash inf\_\backslash text\; i=1,\backslash ldots,k.$ In other words, the components of a nadir and an ideal objective vector define upper and lower bounds for the objective function values of Pareto optimal solutions, respectively. In practice, the nadir objective vector can only be approximated as, typically, the whole Pareto optimal set is unknown. In addition, a utopian objective vector $z^$ with :$z^\_i\; =\; z^\_-\backslash epsilon\; \backslash text\; i=1,\backslash ldots,k,$ where $\backslash epsilon>0$ is a small constant, is often defined because of numerical reasons. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multi-objective optimization」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.