To solve single and multi-objective optimization problems, evolutionary algorithms have been created. We use the non-dominated sorting genetic algorithm (NSGA-II) to find the Pareto front in a two-objective portfolio query, and its extended variant NSGA-III to find the Pareto front in a three-objective portfolio problem, in this article. Furthermore, in both portfolio problems, we quantify the Karush-Kuhn-Tucker Proximity Measure (KKTPM) for each generation to determine how far we are from the effective front and to provide knowledge about the Pareto optimal solution. In the portfolio problem, looking for the optimal set of stock or assets that maximizes the mean return and minimizes the risk factor. In our numerical results, we used the NSGA-II for the portfolio problem with two objective functions and find the Pareto front. After that, we use Karush-Kuhn-Tucker Proximity Measure and find that the minimum KKT error metric goes to zero with the first few generations, which means at least one solution converges to the efficient front within a few generations. The other portfolio problem consists of three objective functions used NSGA-III to find the Pareto front and we use Karush-Kuhn-Tucker Proximity Measure and find that The minimum KKT error metric goes to zero with the first few generations, which means at least one solution converges to the efficient front within a few generations. Also, the maximum KKTPM metric values don’t show any convergence until the last generation. Finally, NSGA-II is effective only for two objective functions, and NSGA-III is effective only for three objective functions.

Genetic algorithms (GA) have been widely employed as optimization and search methods in a variety of problem domains [

Genetic algorithms are the most common algorithms for solving many real-life applications. These algorithms are inspired by natural selection. Genetic algorithms are population search algorithms, which introduced the idea of survival of the best fitness function. Most of these algorithms incorporate the genetic operation to obtain the new chromosome (solution). The basic genetic operations are selection, crossover, and mutation [

In the past, the portfolio optimization problem was designed to find the configuration of assets that generated the maximum expected return which was the main criterion. However, this design changed in 1952, a new variable with the expected return was introduced by Harry Markowitz that called the risk of each portfolio. Thereafter, analysts began to incorporate a risk-return trade-off in their models [

In this article, we solve the portfolio problem [

This study aims to find the Pareto front for portfolio problems with two and three objective functions using the methods NSGA-II and NSGA-III which are simpler and easy to apply. Those methods can address portfolio optimization problems without simplification and with decent results in a fair amount of time, and it has a lot of practical applications. we obtained solutions for the portfolio models using NSGA-II and NSGA-III same as theoretical solutions.

The main contributions of this paper are as follows:

A genetic algorithm can be used to find the Pareto front for portfolio optimization problems which are the same as those found in other approaches.

A genetic algorithm can handle the portfolio optimization problems without simplification and with decent results in a fair amount of time.

The two cases studied were presented to prove the applicability of the genetic algorithm.

The framework of NSGA-II and NSGA-III are elaborated in algorithms 1 and algorithm 2.

In the following part of the article, we first give a literature review of NSGA-II, NSGA-III, and portfolio optimization problems in the second section. After that, we explain the concept of the NSGA-II method. The NSGA-III is discussed in the fourth section. The Karush-Kuhn-Tucker proximity measure (KKTPM) is analyzed in section five for multi-objective optimization problems [

There are many studies in NSGA-II. Deb et al. [

There are some studies in NSGA-III. Deb and Jian [

Markowitz [

The NSGA-II protocols [

It employs three principles: 1. an apparent diversity-preserving mechanism; 2. an elitist principle; and 3. non-dominated alternatives are stressed.

Consider a community of size

Now we demonstrate a crowded tournament collection operator. The crowded comparison operator

Definition: If all of the above assumptions are true, the crowded tournament selection operator [

Crowding gap; To find the estimation density of solutions around a given solution

The lowest and highest objective function values are denoted by

NSGA-III begins [

The following procedures are carried out at descent

For a

The

The authentic analysis generated an output scalarization feature (ASF) for a given repeated (solution)

The reference point

Thereafter, the KKTPM calculation process advanced for single-objective optimization problems to the ASF showed previously. So that the ASF formulation produce the objective function not differentiable, a smooth transformation of the ASF (a performance scalarization function) problem was made firstly by inserting slack variables

At this moment, the KKTPM optimization problem for the previous single-objective problem for

The added term in the objective function permits a penalty correlated with the violation of the complementary slackness condition. The restrictions

The optimal objective value

Exact KKTPM

In this section, we will solve a portfolio problem in special cases using NSGA-II in the first model and NSGA-III in the second model. After that, we will show figures for each model. But, one must know the main portfolio problem. The portfolio is a set of assets or securities

To illustrate the mechanism of the evolutionary algorithms and KKT proximity measure using an evolutionary multi-objective (EMO) algorithm, we thought of three and two-objective Portfolio problems. NSGA-II is used to solve two objective problems, while NSGA-III is used to solve three objective problems. We use the SBX recombination operator [

Consider the three-security problems with expected returns vector and covariance matrix [

Let

The minimum KKT error metric goes to zero with the first few generations, which means at least one solution converges to the efficient front within a few generations. Also, the maximum KKTPM metric values don’t show any convergence until the last generation.

The solutions found in genetic algorithms are the same as those found in other approaches, and they are as effective. The genetic algorithm, on the other hand, is simpler and easier to apply. A genetic algorithm can address portfolio optimization problems without simplification and with decent results in a fair amount of time, and it has a lot of practical applications. NSGA-II and NSGA-III are used to address portfolio problems in models I and II. We measure the smallest, first quartile, median, third quartile, and highest KKTPM values as a function of generation, and the figure shows that KKTPM values decrease with generation. The obtained solutions for the portfolio models using the genetic algorithms same as theoretical solutions. NSGA-II is effective only for two objective functions, and NSGA-III is effective only for three objective functions. NSGA-II can be solving real-life optimization problems with two objective functions, and NSGA-III can be solving real-life optimization problems with three objective functions. In the future direction of this work, we will extend the proposed algorithms with more real-life applications with many objective functions.