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# Multiobjective Optimization - Science topic

Explore the latest questions and answers in Multiobjective Optimization, and find Multiobjective Optimization experts.

Questions related to Multiobjective Optimization

The steepest descent method proposed by Fliege et al. motivates the research on descent methods for multiobjective optimization, which has received increasing attention in recent years. In the context of convex cases, all the Pareto critical point can be obtained by weighted scalarization method with cheap iterates. Therefore, is it necessary to develop descent methods for convex multiobjective optimization problems?

In his name is the judge

Hi

There is a fuzzy logic control system in python. The system contain 2 inouts and 18 outputs.

inference of system is mamdani and shape function used to be guassian.

Then in term of refine performance of the controller I need to optimize specifications belong to shape functions of both input and output. In order to that I need to use multi objective optimization.

We have 2 input and 1 output in case of this problem. I have developed 3 shape functions for each entrance and 3 for output and the shape function is gaussian so we have 18 parameters totally.

I defined my problem as a function in python. But notice this I there is not any clear relationship between input and output of function. It’s just a function which is so complicated with 2 inputs and 18 outputs.

I made my decision to use NSGAII algorithm and I really don't want to change the algorithm.

So I try every way to optimize my function but I didn’t find any success. By searching about python library which can do multiobjective optimization I find Pymoo as the best solution but I really failed to optimize my function which is complicated custom function, with it.

So It’s really my pleasure if you can introduce me a library in python or suggest me a way that I can use Pymoo in order to this aim.

wish you best

Take refuge in the right.

May I have the MATLAB code of some well-known multi-objective benchmark functions like Schaffer, Fonseca, ZDT1, ZDT6, Srinivas, DTLZ5, DTLZ6, LZ09_F1, LZ09_F2, LZ09_F6, LZ09_F7, LZ09_F9, WFG4, CF1, CF2, CF4, CF5, CF6, CF9, and CF10?

Please answer the question or recommend some articles.

In his name is the judge

Hi

I want to learn multi-objective optimization with NSGAII in python for my research.

Please recommend a good source for learning NSGAII in python.

wish you best

Take refuge in the right.

To solve multi-objective problems using the lexicographic method.

Can we use for each step different algorithms? For instance, when we have two objectives to be minimized, can we use firstly the Genetic Algorithm to optimize the first objective, then use Particle swarm optimization Algorithm to solve the second objective while taking the result of the Genetic Algorithm as a constraint?

Dear Members

I had recently gone through a sample program code (in MATLAB) corresponding to

**Pareto GA**in Appendix II (Program 4, Page: 219) of the book**. However, I failed to understand the***PRACTICAL GENETIC ALGORITHMS*by Haupt**Niching strategy**employed in the program code. Also, kindly explain the**Pareto dominance tournaments**that have been employed or used for mate selection in the example provided in the book. The flow diagram for Pareto GA delineated in Figure 5.2 of the book, mentions an increase in the cost of closely located chromosomes. But the code apparently contains no such steps. Kindly explain the anomaly.We have proposed an algorithm for multiobjective multimodal optimization problems and tested it on CEC 2019 benchmark dataset. We need to show the results on some real problem also. Kindly help.

I am using an evolutionary-based GA optimisation tool that divides the space of possibilities into subsets to perform hypercube sampling within each subset, and thereafter generates multiple generation of results, the best of which will form the Pareto front and through each iteration move closer to the global optimum. However is it possible that through the process of hypercube sampling (and hence disregarding some options) , this tool might miss a true global optimum?

I am doing NSGA ii optimization. For this I have developed some equation through rsm in minitab. Now the problem I am facing is to set the constraints. 4 variable 3 objective. I am getting optimized values which are beyond the experimental response domain. I think a proper constŕaint may solve this.

I am coding a multi-objective genetic algorithm, it can predict the pareto fronts accurately for convex pareto front of multi-objective functions. But, for non-convex pareto fronts, it is not accurate and the predicted pareto points are clustered on the ends of the pareto front obtained from MATLAB genetic algorithm. can anybody provide some techniques to solve this problem. Thanks in advance.

The attached pdf file shows the results from different problems

Hello RG Community,

As we have ZDT, DTLZ, WFG test functions for continuous multiobjective optimization problems, are there any test functions for mixed integer multiobjective optimization problems?

Thanks!

When we do multi-objective optimization using MOGA algorithm in Ansys Workbench, the algorithm stop automatically after few generation saying that the algorithm is converged while the objective function still not converged and shows random behavior? The maximum allowable pareto percentage is 70%. Is the true convergence achieved?

I have been trying to find a way to fit two functions

**simultaneously**using nonlinear least squares (I have to find the optimum 3 variables, common for both models, that fits best both of them). I typically use Python's scipy.optimize.least_squares module for NLLS work, which uses the Levenberg–Marquardt algorithm.I tried some specialised multi-objective optimization packages (like pymoo), but they don't seem suitable for my problem as they rely on evolutionary algorithms that output a set of solutions (I only need one optimum solution per variable) and they are made to work for conflicting objectives.

I also tried to take the sum of the norms of the residuals of the two functions (making it into a single objective problem) and to minimize that by various gradient and non-gradient based algorithms from

**Python's scipy.minimize****package**, but I found this norm becomes so huge (even with parameter bounds!) that I get oveflow error (34, results too large), crashing the programme sooner or later. It didn't crash using Truncated Newton's Method, but the results produced were rubbish (and from running an optimization on this same data on a simpler model, I know they shouldn't be!)I have to perform this fit for a few thousand data sets per experiment, so it has to be quite robust.

Surprisingly, I can not find a way to do multiobjective NLLS (only for linear regression). I have found some papers on this, but I'm not a mathematician so it's quite out of my depth to understand them and apply them in Python..

Has anyone had a similar problem to solve?

Many thanks!

What is the best way to solve multi-objective optimization problems with

**distance-based approaches**when there are no clear**referent points**(ideal and anti-ideal points) to compare with the current solution of the applied model (metaheuristic)?I excluded the possibility of using

**lexicography**and**ε-constraint**based methods because they seem to neglect the multi-criterial nature of the problem.Any advice is welcome. Thanks in advance.

Hi,

I just want to make sure that I understand the mechanics of the NSGA-II (the non-dominating sorting genetic algorithm) for a multiobjective optimization problem, since the resources I have (I am not satisfied with, I would be grateful if anyone can recommend me a good paper or source to read more about the NSGA-II)

Here is what I got so far:

1- We start with a random population lets call P_0 of N individuals

2- Generate an off-spring population Q_0 of size N from P_0 by using binary tournament selection, crossover and mutation)

3- Let R_0=P_0 U Q_0

While (itr<= maxitr) do,

5- Identify the non-dominating fronts in R_0, (F_1,..F_j)

6- create P_1 (of size N) as follows:

for i=1:j

if |P_1| + |F_i| <= N

set P_1=P_1 U F_i

else,

add the least crowded N - |P_1| solutions from F_i to P_1

end

end

7- set P_1=P_0;

8- generate an off-spring Q_0 from P_0 and set R_0=Q_0 U P_0

9- itr=itr+1;

end(do)

**My question**(assuming the previous algorithm is correct, how do I generate Q_0 from P_0 in step 8?

Do I choose randomly any 2 solutions from P_0 and mate them or do I choose according to or is it better to select the parents according to some condition like those who have the highest rank should mate?

Also, if you can leave me some well-written papers on NSGA-II I would be grateful.

Thanks

Any recent literature on epsilon constraint handling method in multiobjective optimization. state the advantage of using this method.

In most MOEA publications, researchers will use IGD or hyper-volume as their evaluation metrics. However, in real-world problems, both nadir points and ideal points are unknown. What's worse, in many cases, the domain knowledge is unavailable. Thus, in this case, how do I compare the performance between different algorithms? For example, how do I compare the performance of different MOEAs on a real-world TSP problem without any domain knowledge?

Hi,

I have a bi-objective minimization problem. The first objective is in seconds and the second one does not have a unit (it counts the number of some product). How can I unscale this objective function properly?

Can I use the monetary value of both of them? If yes, what is the monetary value of time?

Hello,

What is the interpretation of the pareto front graph when using three-objectives such that, we would like to maximize the first objective while minimizing the second and the third objectives.

attached, an example of 3D Pareto-Front that I would like to interpret.

VM:Max

resource wastage:Min

PMs:Min

Thanks

R2 in fact is hood option for evaluations populations of solutions in Many-objective optimization since Hypervolume (at least WFG implementation) is really heavy to compute. I have tried a Quick Hypervolume implementation. However I found out R2 aa a better option when you don't have prior information about the Pareto Front.

Do you know something better than R2 for this purpose?

I have seen many researchers using these three algorithms for solving several different benchmarks. Sometimes I also see SPEA2-SDE as an option. These four algorithms are available in jMetal, which makes comparison really easy to perform.

Do you know some better? Are they available in Java? The implementation uses jMetal?

For the multi-objective optimization problem is it possible to apply the concept of SN ratio to individual outputs obtained through RSM or full factorial design of experiment. Also is it possible that the design of experiment developed by full factorial can match with Taguchi orthogonal array more specifically 2 factors 3 level design problem?? Where for full factorial it is coming 9 experiments.

Thank you....

Hello,

I have some more multi-objective optimization questions using the exhaustive search method(brute-force method) that help would be much appreciated. An answer to all or one of the questions are all very very welcome.

I have 3 different multi-objective optimization problems with 3 discrete variables.

Optimization Problem 1) 2 objective functions

Optimization Problem 2) 2 objective functions + 1 constraint

Optimization Problem 3) 3 objective functions

The ranges for the variables are 15~286, 2~15, 2~6, respectively.

I have been told that the search space is small enough that exhaustive search space is my best bet and that it is easy to implement so I wanted to try it.

My questions are

1) Is it possible to apply the exhaustive search method to all three optimization problems?

2) How would I go about doing this using a computer software?

I was thinking that for

Optimization Problem 1 and 3

I would find all the function values first and then move on to finding and plotting the pareto fronts -> Is this the right approach?

Also, is there any example code I could follow(especially for optimization with three objectives)

For Optimization Problem 2 with a constraint

How would I incorporate this constraint?

Just make the program give me the function solution values that do not violate the constraint (meaning ignoring solutions that do) and than use them in plotting the pareto front?

Is there example codes/programs for doing this?

I would in particular find any Matlab and R codes helpful.

If there is a specific program/software or what not that does this, I would be very grateful as well.

Thank you to everyone in advance!

Which are the PIs those are recommended for optimization model with many (=>4) conflicting objectives ?

How to use them if true pareto fronts are not practically known ?

Is there any methodology for generating the TRUE PF ?

I am researching on the optimization of road alignment between two given points using NSGA-II with two objectives: 1.cost, 2.safety

I implemented the optimization on a case study and got reasonable results. However i couldn't find any similar research with the same criteria to compare and validate my results. Actually it is normal to have different results since specified criteria are different.

Is it sufficient to have reasonable results or do i have to validate them anyway? If so, do you know any method or technic?

Thanks in advance

Dear altruists,

I used GA algorithm with an objective function F_obj= w1* f1 + w2*f2. where w1= 0 to 1 and w2=1-w1.

I am changing w1 value from 0 to 1 and get corresponding Fobj.

For plotting optimal pareto front, I put values w1*f1 in X axis, w2*f2 in Y axis, and corresponding Fobj.

Is it a correct way ? Becuase I am not getting the exact figure. I also attached the result.

Can I request to help me for plotting pareto front ?

I have rewritten an MPC problem to a QP formulation.

The QP solvers quadprog and Gurobi, which uses the interior-point algorithm, give me the same objective function value and optimized values x, but GPAD, a first-order solver, gives me the same optimized values x, but an objective function value, which is a factor 10 bigger compared to quadprog and Gurobi. Do you anyone know a possible explanation?

I am trying to solve a QP problem.

Does anybody know the differences between the interior-point-convex algorithm of quadprog and the barrier method of Gurobi in terms which kind of matrices can the solvers handle the best? Sparse matrices or dense matrices? And what kind of problems: large problems or small problems? Furthermore, which solver needs more iterations (with less expensive cost per iteration) and which solver needs fewer iterations, but expensive cost/time per iteration.

From results, I think Gurobi gives more iterations and quadprog less, but I do not know why?

And what are the differences with GPAD by means of what is described above?

Based on your expertise, what is the better optimizer tool between GAMS ( https://www.gams.com/ ) and Gurobi ( https://www.gurobi.com/ )?

Please also let me know your field of research.

Does anybody has experience on visualizing the Pareto-optimal front of a bi-objective optimization with any R package such as ggplot2, smoof or sth else.

Could anyone suggest me how do I get ideal pareto front distribution of DTLZ3 test problem. I went through some research papers but still it is not clear to me.

I would like to know if any of you know whether there is a direct link between the computational effort of a multi-objective problem depending on whether the objective functions are heterogeneous or not. For example, 1) one objective is cost and a second objective is some kind of security index with a range [0, 1], 2) one objective is cost and we convert the second objective also to some kind of a security related cost (or we can try to convert the first objective to some normalized cost in a way that both objective functions have some homogeniety). Is there any research evidence whether such strategies can improve the computational time?

Sub question: Is there a difference when it comes to heuristic and non-heuristic algorithms in this regards?

Thank you!

Dear community,

I would like to request some references related to results on parameter-dependent Pareto Front, if there are any. I am interested in studying the behavior of the Pareto Front with respect to an external parameter, for multi-objective problems.

Thanks for any recommendation!

Best,

Nathalie

I would like to test the performance of a modified algorithm developed to solve a real-world problem that has these characteristics : (1)Discrete (2)Multi-Objective (3) Black-box (4)Large-scale.

How we can do this? and if there are no such test problems, is it sufficient to show its performance on the real-world problem only? (where the true Pareto Front is unknown)

Best regards,

The attached picture is the result of a bi-objective optimization problem. Genetic algorithm was used. The "missed-out" band (i.e. approximately 41 to 43.5 in the vertical axis) is within the range of the respective objective function (in other words, there are values of design variables which result in values between 41 and 43.5 of the objective function).

The question is, is there any explanation for this discontinuity (physical or mathematical)? Or should I see this as a fault in my solution procedure?

It is worthy to add that I've carried out the procedure several times and with different optimization parameters (population, mutation fractions, etc), but the discontinuity seems to be always there...

how can we validate our mathematical model(multi objective model)?Are there any techniques?

For instance, if i formulate mathematical equations to solve multi-objective optimization problem (1 value should decrease while other value should increase at the same time) then what algorithm can be the best algorithm to dealt with.

Hello.

Is there a fully functional NSGA-III implementation?

There is an implementation in Java and C++, jMetal bases their implementation on : http://web.ntnu.edu.tw/~tcchiang/publications/nsga3cpp/nsga3cpp-validation.htm, the author of which says the algorithm does not scale well beyond 8 objectives.

I'm looking to use NSGA-III on many-objective problems (which it is designed to handle) i.e. 15 objectives.

Thank you

Evolutionary Algorithms: https://www.youtube.com/watch?v=L--IxUH4fac

Multi-Objective Problems: https://www.youtube.com/watch?v=56JOMkPvoKs

For a multi-objective problem with uncertainty in demand, consider the scenario tree (attached herewith) for a finite planning horizon consisting of three time periods. It's a two objective minimization problem in which the augmented e-constraint method is utilized to obtain Pareto optimal solutions (POS).

In the time period T1, only the mean demand is considered. Then in T2, demand follows a certain growth rate for a scenario with expected probability of growth for each scenario. Similar trend is outlined for T3.

The deterministic counterpart envisaged for the problem is a set of time periods with specific pattern of growth rate for mean demand - say 15% in T1, 10% in T2 and 10% in T3.

I want to draw out a comparison of the POS obtained from the stochastic and deterministic analysis. What is the best way to proceed in order to give the decision maker a whole picture of the POS with the scenario and time period considered in both type of analyses?

Do I obtain POS sets for all the 13 scenarios from T1 to T3, or just the 9 scenarios in T3? It'd mean 13 or 9 Pareto fronts for the stochastic analysis alone. In other words - a Pareto front with POS for each time period and scenario! How do I compare whatever I obtained from the stochastic analysis with the deterministic one?

Once again, the aim is to analyze the stochastic analysis and draw out a comparison of the POS obtained from the stochastic and deterministic analysis for the time periods and scenarios considered.

*Comments on the aforementioned approach and recommendations for alternatives are appreciated.*

Hi all,

I would like to know what is the difference between MOGA and NSGA-II, in some literature it classifies as two separate methods but in other it classify NSGA as a MOGA method.

Thank you.

Best Regards,

Rashiga

I have a multi-objective optimization problem whose analysis is sought using weighted sum approach. The objectives are: minimizing transportation costs and minimizing transportation risks.

Consider i sources which supply A

_{i}and j destinations which demand B_{j}quantities of a product. C_{ij}and R_{ij}represent the costs and risks of transportation. Q_{ij}is the unit of products to be shipped from i to j. I am trying to find a set of Pareto-optimal solutions that gives an overview of the trade-off between the two objectives. MPL for Windows 4.2 is used as the modelling language with CPLEX 10.0 as the solver.Here's a simplified mathematical formulation of the problem:

Min z1 = Σi Σj C

_{ij}Q_{ij}Min z2 = Σi Σj R

_{ij}Q_{ij}Subject to:

Σj Q

_{ij}<= A_{i}Σi Q

_{ij}>= B_{j}Q

_{ij}>= 0How do I mathematically treat one objective as a constraint while the other remains the objective function? The goal is to develop a Pareto front of non-dominated solutions.

*Comments, feedback and alternative recommendations are appreciated.*

What are the requirements for carrying out Multi-Objective Optimization? Is it possible to do the same if I have all non-beneficial criteria in my problem

What is the best method to solve Multiobjective Optimization , weight, bounded, goal....etc ?

I have a question regarding the Hypervolume indicator in multi-objective optimization:

As far as I know, the Hypervolume is the volume of the trade-off space covered by the Pareto-front. I.e., finding the best possible front implies maximizing the Hypervolume.

Does this definition imply that we want to minimize our multiple objectives? What happens to the sign of the Hypervolume when we are maximizing one or more of the objectives? In other words, are we always trying to maximize the Hypervolume, or are there cases where we want to minimize?

In reference point-based multi-objective optimization, the reference points are determined by DM.

What are the methods that can be used to calculate or determine reference points for a multi-objective optimization problem?

I want to get optimized value using Optimization Trials in HEC-HMS, but it didn't work, the optimized value is same as initial value, and the objective is zero, please give me some advice about how to make it.

- Using Mixed integer liner programming.

- I need to determine the optimal timing to invest(maximizing Net Present Value) in district heating power plants and at the same time minimizaing Carbon emissions.

- Main constraint: coverage of a given heat demand.

- Investment decision through mixed integer linear programming.

- Investment optimization as extension of unit commitment. (schedule optimization)

- Deterministic approach.

I want to calculate inverted generational distance (IGD) to evaluate the performance of a multi-objective optimization algorithm. I have the approximate Pareto fronts. But, I could not find the true Pareto front for the structural engineering problems, such as, welded beam, spring, gear design problems,... etc. Any one has the data of true PF for them?

I am working on a project in which I need to optimize the scheduling of heat generating power plants.

I want to Maximize Net Present Value and at the same time minimize Carbon emission.

Mixed integer liner programming will be used through out the modelling.

I'm using optimization tool box in Matlab to solve multi-objective optimization, I have linear and nonlinear constraint, after running the optimization, I got Pareto front (see the file attached in this message), increasing the population size give me the same result.

what do you think ?

In multiobjective optimization, some algorithms were designed to obtain a set of solutions that approximate the whole Pareto front (PF), while the others were designed to approximate a partial and preferred PF, known as, preference-based algorithms.

Performance indicators, such as HV and IGD, are widely used to compare between their performance. However, IGD is designed to measure the distance between whole true PF and approximate PF, rather than partial PF. Therefore, it is not suitable to evaluate the performance of preference-based algorithms.

Some studies proposed performance indicators to evaluate the performance of preference-based algorithms. However, these performance indicators can not be used with non-preference -based algorithms.

In this case, how to compare the performance of multiobjective optimization algorithms that are used different approaches (non-preference -based and preference -based)?

The figures below are the result of R-NSGA-II algorithm in solving DTLZ2 problem with 3 objectives. The green rhomboids represent the reference points, (0.2 0.5 0.6) and (0.7 0.8 0.5).

the result in Figure 1 looks good. However, I do not know what is wrong with the Inverted generational distance (IGD) curve.

To calculate the IGD values I have used this Matlab code.

maxGen = 200;

for gen =1:maxGen

Distance = min(pdist2(PF,popObj),[],2);

IGD_values(gen) = mean(Distance);

end

Hi Everyone,

I am applying constrained NSGA-2 multi objective optimization method. In my problem, there are 2108 variables and there are three objective functions which need to be minimized. Can you please tell me what is the suitable population size and number of generations for applying constrained NSGA-2?

Thanks,

Karan

In multiobjective optimization, what does the distance exactly means, is it:

**1)**The distance from reference point (V) to an individual (Xi) (candidate solution) in the population (decision space).

Euclidean Distance = d(Xi,V)

**OR**

**2)**The distance from reference point (V) to the objective vector

*f(X)*in the objective space, where

*f(Xi)*=

*f*(

*f*1(Xi),...

*f*m(Xi)).

*m*is the number of objectives.

Euclidean Distance = d(f(Xi),V)

I have two objective functions. i want to minimize it using GA in MATLAB. One objective function is more important than other.

One way is to use weighted objective function and use different weights for finding the solution.

Other method is to use multiobjective GA function in matlab and find the solution from Pareto graph.

Among these two method, which one is advantageous?

Hello everyone,

i'm applying gamultiobj (Multiobjective optimization using genetic algorithm) from optimization tool of matlab

* how do i export the results frome and to command window?

** is it possible to apply gamultiobj for fitness function who has 3 objective functions?

plz it's urgent .. i need answers ^^

In the multi-objective optimization problems we often say that the objective functions are conflicting in nature. In what sense the objective functions are said to be conflicting with each other? Also, how it could be proved numerically that the objective functions in our multi-objective problem are conflicting in nature?

Recently I've been studying dCOEA (https://ieeexplore.ieee.org/abstract/document/4553723/) and trying to implement it for a project. However I got stuck in the EA process. While the parent individuals of each subpopulation have already been evaluated in the cooperative/competitive process and their respective Pareto rank and niche count calculated, after the crossover and mutation operations on each subpopulation,

**how are the child individuals evaluated?**Since an individual needs to be combined with representatives from other subpopulations to form a valid solution that can be evaluated. I've been thinking on simply combining the child individuals with those representatives to evaluate them. However, I don't know if that is the correct form to perform this process.Also, since there is a possibility that the individual defined as the current subpopulation's representative might not be selected from the mating individual pool for the next generation,

**how is the new representative defined?**I've been thinking on using a random selection but that will definitely affect the quality of the solutions obtained and I know that answering the first question might answer this one as well.I am using multi-objectve GA toolbox in Matlab to optimize 3 objective function. I can plot pareto two objective each time, but I am unable to plot the pareto fronts of 3 objectives together.

Thanks in advance

I would like to calculate Spread , HyperVolume and Generational Distance values to measure the performance of multi -objective optimization problems. I know how to calculate the performance in case true pareto front is available. However, for the test data available I don't have the true pareto front but I have the minimum and maximum values for each objective function. I would like to know the methodology or some one can share the code to find out the spread, GD and HV values of multi-objective optimization problem. Really appreciate the help.

Pareto solutions are found below true PF in case of 3D DTLZ1 problem. Whereas they are found above true PF in case of 2D DTLZ1 problem.

I have checked and confirmed problem formulation and variable bound. ( variables used are 3).

Algorithm converges on DTLZ-2, 3, 4.

What might be the cause or area of improvement in MO algorithm.

Reference: Deb K., Thiele L., Laumanns M., Zitzler E. (2005) Scalable Test Problems for Evolutionary Multiobjective Optimization. In: Abraham A., Jain L., Goldberg R. (eds) Evolutionary Multiobjective Optimization. Advanced Information and Knowledge Processing. Springer, London

I am looking for examples of the combination of ABM, MO optimization, and game theory, preferably the ones that have been used for practical purposes.

My solution set in the Cartesian system placed in range x=0.5 and y=0.3 as diagonal line, but pareto set is in range placed in range x=0.5 and y=0.5 as diagonal line.

Is it possible the solution set pass from the pareto set ?

or Is it possible the solution set being bether than the pareto set ?

Given a number of objectives, is there any quantitative method through which I can determine what objectives are more crucial than others

How exactly can i choose the best solution among so many values ?

How to find the pareto front for maximizing a fxn. and minimizing another fxn. at the same time ?

I am using this toolbox :

Can a search method that optimizes one criteria and uses other as a tie-breaker be called multi-objective? If yes, is there a subcategory in which it falls? If not, what is it called?

Thank you.

here, CMOEAs is an extension of EAs. However, for example DE is time consuming when deals with more than three objectives.

Please guide for selection of many objective optimization algorithm.

Suppose I am optimizing ZDT-1 2 objective test function. I want to stop the algorithm when there is no significant improvement in Pareto front. how can I achieve this?

If anyone know about NSGA|| evolutionary algorithm then please let me know why crowded distance is not used in initial condition for the generation of the offspring.?

My goal is to minimize an unimodal function using direct optimization method. According to the problem characteristics, I decide apply a single-based optimization rather than a population-based one. However, I am a newbie in this field and don't know which is the best feasible method. Hence, who can help me point out the fastest single-based optimization?

Thank you very much.