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We develop a new evolutionary method of generating epsilon-efficient solutions of a continuous multiobjective programming problem. This is achieved by discretizing the problem and then using a genetic algorithm with some derived probabilistic stopping criteria to obtain all minimal solutions for the discretized problem. We prove that these minimal solutions are the epsilon-optimal solutions to the original problem. We also present some computational examples illustrating the efficiency of our method.

The goal of multiobjective optimization, also called vector optimization, is to find a certain set of optimal (efficient) elements of a nonempty subset of a partially ordered linear space. However, finding an exact description of this set often turns out to be practically impossible or computationally too expensive. Therefore, many researchers have focused their efforts on approximation pro- cedures and approximate solutions (see e.g. [

More than three decades ago, the notion of

In this section we review the results of [

The RHS (Random Heuristic Search) algorithm, described in [

Each population consists of a finite number of individuals which are elements of a given finite set

To simplify the notation, it is convenient to identify

where

We assume that all the populations appearing in sequence (1) have the same size

where

which is a simplex in

We now define the mapping

called heuristic ( [

A transition rule

Of course, a transition rule defined this way is nondeterministic, i.e., by applying it repeatedly to the same vector

In this subsection we consider a genetic algorithm as a particular case of the RHS. We assume that a single iteration of the genetic algorithm produces the next population from the current population as follows:

1) Choose two parents from the current population by using a selection method which can be described by some heuristic (see [

2) Crossover the two parents to obtain a child.

3) Mutate the child.

4) Put the mutated child into the next population.

5) If the next population contains less than

The only difference between the iteration described above and the iteration of the Simple Genetic Algorithm defined in ( [

To derive our stopping criteria, we will use some properties of mutation which is generally understood as changing one element of the search space to another, with a certain probability. The way of implementing selection and crossover is not important for our model, so we omit the discussion of them (we refer the reader to ( [

We assume that mutation consists in replacing a given individual from

matrix

of obtaining a population

where the subscript sc means that we are dealing with the composition of selection and crossover, and the subscript scm indicates the composition of selection, crossover and mutation. To get a whole new population, one should draw an r-element sample from the probability distribution (8). The probability of generating a population

We now consider the following multiobjective optimization problem. Let

The set of all minimal elements of

In particular, if

In this case,

In this section, we assume that the goal of RHS is to find all elements of

where the (possibly unknown) number

Theorem 1 ( [

Let

Let

Corollary 2 ( [

where

The results of Section 2.3 give no practical way of constructing the set

The algorithm presented below is a combination of the RHS and the base VV (van Veldhuizen) algorithm described in ( [

Then we construct a sequence

where

where we have used the notation

It is shown in ( [

Our final result is the following theorem which shows that, with a prescribed probability, the sets

Theorem 3 ( [

Let

To solve this problem means to find all Pareto optimal (efficient) points of

However, in practical situations this can be very difficult or even impossible. Therefore, we shall consider a discretized version of problem (22).

For any given

where

where the same relation (23) is considered, but now in the finite set

It is natural to ask whether an exact solution of problem (25) yields some sort of approximate solution of problem (22). One of the cases where a positive answer can be given is described in the proposition below. Before formulating it, we must define

Let

where the relation “

Proposition 4 Let

and let

Proof. Let

By the second inclusion in (24), there exists

which contradicts the assumption that

Consider the multiobjective optimization problem (22), where the constraint set

where

where

Proposition 5 For every given

Proof. The inclusion

Let us choose

Take any

satisfies

Then the vector

which completes the proof.

In the sequel we consider the following simple evolutionary algorithm which is a special case of the algorithm described in Subsection 2.4. It does not contain selection and crossover. The mutation process is very simple and consists in replacing the current population by another randomly chosen population of the same size. However, the stopping criteria described above still hold for this algorithm because their proofs make use of the properties of the mutation alone.

Algorithm 1 Parameters:

1) Set

2) Choose randomly a population

3) Construct the set

4) Mutate the population

5) Construct the set

6) If

7) Increase

Proposition 6 After stopping Algorithm 1, the equality

Proof. Apply Theorem 3 and Corollary 2 with

Below we report on testing the algorithm described above on some examples taken from literature. To find the set of minimal elements (i.e., nondominated elements) of finite sets in Steps 3 and 5, we have used the simple algorithm for classifying a population according to non-domination, see Section 4.3.1 of [

Example 7 (Problem SCH in

where

As stated in

Therefore, we can take the Lipschitz constants

Let

(30), let

and

the population size is

Remarks:

1) One should remember that the number

2) The cardinality of

3) According to the performance measure Diversity Metric

Example 8 (Problem FON in

where

with variable bounds

is a Pareto optimal solution of this problem. Let

Then

Note that

Therefore, we can take the Lipschitz constants

Let

let

and

the population size is

Remarks:

1) In practical tests, the set

2) The cardinality of

3) According to the performance measure Diversity Metric

Example 9 (Problem POL in

where

POL is a problem with two nonconvex Pareto fronts that are disconnected in both the objective and decision variable spaces, see [

Let

Let

(30), let

satisfied. Suppose that the population size is

Remarks:

1) In practical tests, the set

between iterations 285 and 350, and has not changed in the later iterations.

2) The cardinality of

3) According to the performance measure Diversity Metric

We have presented a new evolutionary method for generating

The authors are grateful to an anonymous referee for his/her comments which have improved the quality of the paper.

Rahmo, E.-D. and Studniarski, M. (2017) Generating Epsilon- Efficient Solutions in Multiobjective Optimization by Genetic Algorithm. Applied Ma- thematics, 8, 395-409. https://doi.org/10.4236/am.2017.83032