Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute 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 is a type of vector optimization that has been applied in many fields of science, including engineering, economics and logistics 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 multi-objective optimization problem, it is not guaranteed that a single solution simultaneously optimizes each objective. The objective functions are said to be conflicting. 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, there may exist a (possibly infinite) number of Pareto optimal solutions, all of which are considered equally good. 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). Bicriteria optimization denotes the special case in which there are two objective functions. There is a direct relationship between multitask optimization and multi-objective optimization.
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
min x ∈ X ( f 1 ( x ) , f 2 ( x ) , … , f k ( x ) ) {\displaystyle \min _{x\in X}(f_{1}(x),f_{2}(x),\ldots ,f_{k}(x))}
where the integer k ≥ 2 {\displaystyle k\geq 2} is the number of objectives and the set X {\displaystyle X} is the feasible set of decision vectors, which is typically X ⊆ R n {\displaystyle X\subseteq \mathbb {R} ^{n}} but it depends on the n {\displaystyle n} -dimensional application domain. The feasible set is typically defined by some constraint functions. In addition, the vector-valued objective function is often defined as
f : X → R k x ↦ ( f 1 ( x ) ⋮ f k ( x ) ) {\displaystyle {\begin{aligned}f:X&\to \mathbb {R} ^{k}\\x&\mapsto {\begin{pmatrix}f_{1}(x)\\\vdots \\f_{k}(x)\end{pmatrix}}\end{aligned}}}
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