Multi-objective optimization

In mathematics and operations research is called with Pareto optimization (after Vilfredo Pareto, and multi-objective optimization, multi-criteria optimization or vector optimization ) solving an optimization problem with multiple objectives, ie, a multi-criteria or multi-objective problem.

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

Overview with a technical focus

In many optimization tasks can be several, mutually independent in principle, define objectives, for example in engine efficiency, maximum performance and pollutant emissions. Here it is often not possible to optimize all objectives together, one can be found for example in the situation that you can only have maximum performance increase (an improvement ) if at the same time decreases the efficiency ( deterioration ).

The usual procedure for dealing with such tasks is to interpret the interest objectives as part of goals and combine them to a common objective function using weighting factors. There is obtained a simple problem in this way. This can be redeemed with any of those referred Operations Research methods and determines an optimal solution for the joint objective function.

When not mesh umrechenbaren targets, such as in the example given, the weighting factors to be applied are arbitrary and subjective in certain frames. This also results in a corresponding arbitrariness in the identification of the relevant "best" solution of the optimization problem. A useful procedure in such cases, the separate optimization for all possible combinations of the weighting factors. Here you will not find a single best solution is usually because the target criteria are usually in conflict with each other ( as above, the maximum power and efficiency ).

Since there is no clear best solution is defined, it determines a set of solutions of the optimization problem, in which an improvement of an objective function value can be achieved only by deterioration of another, ie, the set of optimal compromises. This solution set is called the Pareto set or Pareto optimum of the underlying Paretooptimierungsproblems whose elements as Pareto- optimal. It should be noted that the Pareto amount generally can not be completely determined due to the variation of the weighting factors.

If the Pareto quantity of the given optimization problem once found, subjective assessments of the importance of the individual sub-goals (different weighting factors) can be specified. The Pareto set contains then for arbitrary relative partial target weightings at least a solution that is optimal at this weight.

Dimension and visualization

Wherein an optimization problem with the targets n Pareto amount will be of an (n- 1) - dimensional hyper - interface (For a nonlinear optimization problem, this interface is a portion of a hyper-plane ). The Pareto optimality of a two -criteria problem ( eg power versus torque of an engine ) is a strictly decreasing, not necessarily continuous boundary line in a power efficiency chart.

At the latest on four-dimensional problems listening to any direct visualization ability. Instead, the solution space must be palpated by interactive tools such as the star chart.