Pareto optimization

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In mathematics and in operations research , Pareto optimization (according to Vilfredo Pareto ; also multi- criteria optimization , multi-criteria optimization or vector optimization ) describes the solution of an optimization problem with several goals, i.e. a multi- criteria or multi-criteria problem.

In economics one called Pareto optimum one resource allocation with the property that no one can be made better off without another worse off. In the simplest case of an economy with two people and two goods, the Pareto optima can be illustrated using the so-called Edgeworth box . Pareto optimality, or synonymously Pareto efficiency, can thus be viewed as the absence of waste.

Overview with a technical focus

For many optimization tasks, several, fundamentally independent objectives can be defined, for example, in the case of engines, the efficiency , the maximum output and the pollutant emissions. Often there is no solution that is best for all goals at the same time; the goals are often opposing and improvement in one goal deterioration in another. For example, you can find yourself in the situation that you can only increase the maximum output of a motor (an improvement) if at the same time the efficiency drops (a deterioration).

The usual procedure for handling such tasks is to take the goals of interest as sub-goals and to combine them into a common goal function using weighting factors. In this way you get a simple problem - instead of several goals you only have one goal, the “multi-criteria optimization” is thus reduced to one (single) (overall) criterion. This is solved with one of the procedures mentioned under Operations Research and an optimal solution for the common objective function is determined.

In the case of target values ​​that cannot be converted into one another, such as in the example given, the weighting factors to be applied are arbitrary and, within certain limits, subjective. This also results in a corresponding arbitrariness in finding the "best" solution to the optimization problem that is sought. A sensible approach in such cases is the separate optimization for all possible combinations of weighting factors. As a rule, you will not find a single best solution, as the target criteria usually conflict with each other (as above, the maximum output and the efficiency).

Since no unequivocally best solution is defined, a set of solutions to the optimization problem is determined in which an improvement of one objective function value can only be achieved by worsening another, i.e. the set of optimal compromises. This solution set is called the Pareto set or Pareto optimum of the underlying Pareto optimization problem, and its elements are called Pareto optimal. It should be noted that in general the Pareto set cannot be fully determined by varying weighting factors.

Once the Pareto set of the given optimization problem has been found, subjective assessments of the importance of the individual subgoals (different weighting factors) can be given. The Pareto set then contains at least one solution for any relative partial target weightings, which is optimal for this weighting.

Dimension and visualization

In an optimization problem with n goals, the Pareto set will represent an ( n -1) -dimensional hyper interface (in a linear optimization problem this interface is a section of a hyperplane ). The Pareto optimum of a two-criterion problem (e.g. power versus torque of a prime mover) is a strictly monotonically decreasing, not necessarily continuous limit line in a power-efficiency diagram.

Direct visualization of the Pareto set is no longer possible from four dimensions. Instead, the solution space can be captured interactively using tools such as the star diagram .

literature

  • Matthias Ehrgott: Multicriteria Optimization. Lecture Notes in Economic and Mathematical Systems 491, Springer Verlag, 2000.

Individual evidence

  1. ^ Weimann: Economic Policy. 4th edition. Springer 2006, p. 17.