Multicriteria optimization in details
Multicriteria optimization can be done mathematically correctly only when some optimality principle
is used. We use Pareto optimality principle, the essence of which is following.
The multiobjective optimization problem solution is considered to be Pareto-optimal
if there are no other solutions that are better in satisfying all of the
objectives simultaneously. That is, there can be other solutions that
are better in satisfying one or several objectives, but they must be worse
than the Pareto-optimal solution in satisfying the remaining objectives.
In this case multiobjective optimization problem results in finding a
full set of Pareto optimal solutions.
As a rule, it is impossible to find the full infinite set of Pareto optimal
solutions for the particular real-life problems. For this reason the engineering
multiobjective problem statement seeks to determine the finite subset
of criteria-distinguishable Pareto optimal solutions.
It is vital to note, that at the initial stage of optimization
process the accuracy of the response surfaces may be not so good due to
the small number of points in the plan of experiment and the relatively
large size of the current search area. However, during the optimization
process, the number of points in the vicinity of the Pareto optimal points,
is increased. At the same time, the size of the current search area is
decreased. These trends are leading to a more accurate approximation of
the objective functions and, hence, to the optimization process efficiency
increase. Actually, during the optimization process, the information about
the objective functions in the vicinity of the Pareto optimal points is
permanently accumulated.
A number of heuristic procedures are developed for
proposed optimization method efficiency increase as the information
accumulates. These procedures are directed towards the adaptive change
in the numbers of points in the plan of experiment and in the current
search area. The adaptations affect the value of the normal distribution
parameter and the rational selection in the basic optimization algorithm.
The main advantages of the proposed indirect multicriteria optimization
method over traditional mathematical programming strategies lie in the
possibility of the problem solving in case of nonconvex, discontinuous
and stochastic goal functions and constraints; the unnecessity of considerable
adaptation of the mathematical model for the optimization problem solving;
the possibility to obtain a set of EP-optimal decisions under relatively
small number of turns to the mathematical model; a significantly high
probability of locating the global optimum in a multimodal design space.
These advantages are the basis for the wide use of the proposed method
in the real-life problems.
The main advantages of the proposed indirect multiobjective optimization
method over traditional mathematical programming strategies is the following.
- The ability of solving the problems with nonconvex, discontinuous
and stochastic objective functions and constraints;
- It is not necessary to considerable adapt the mathematical model for
the optimization problem solving;
- The possibility to obtain a set of EP-optimal solutions using relatively
small number of evaluations of the mathematical model;
- A significantly high probability of locating the global optimum in
a multimodal design space.
These advantages are the basis for widely using of the proposed method
in the real-life problems.
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