I suggest you to read the following paper to understand the reason, http://www.maths-in-industry.org/miis/233/1/fmipw1-6.pdf

good luck 

Because the right alternative is not known to those people. You do not need to do so. You can optimize all objectives simultaneously using evolutionary multi- or many- objective optimization. In this case, the final solution is not a single optimum point rather a set of alternative solutions known as Pareto solutions. To know more, search evolutionary multi objective optimization literature.

The most simple answer to your question is that with a scalarization technique you reduce the problem to something that we know well how to solve - namely standard optimization problems. :-) 

There are some interesting alternatives to using heuristics and which I have tested to great advantage. One particularly nice one is the "Pareto navigator", in which you interactively can rule out bad alternatives based on setting lower and upper bounds on the objective values, thereby influencing the method. Here is one basic paper (and there are plenty of others, too) on the subject, motivated by cancer treatment - in which cure and side effects are natural conflicting goals: 

http://iopscience.iop.org/0031-9155/53/4/011

Multiple objective optimization is difficult because of the several conflicting objectives. Scalarization means that the problem is converted into one single or a family of single objective optimization problems. The solution process usually requires the participation of a human decision maker (DM) who can give preference information related to conflicting goals. See for further discussion:

Miettinen, K. (2001, January). Some methods for nonlinear multi-objective optimization. In Evolutionary Multi-Criterion Optimization (pp. 1-20). Springer Berlin Heidelberg.

Coming back to the second comment, I would not say that when building the Pareto front, all objectives are optimized simultaneously. Each point of the Pareto from corresponds indeed to the solving of a single objective optimization problem, whereby the objective function is a specific combination of the initial objective functions. Typically a weighted sum is used and the weights are varied between 0 and 1. In my opinion, the Pareto front cannot be qualified as a set of solutions, but rather as a set of points where none of the costs is dominated by the others. A Pareto front is thus more a set of compromise points between all the objectives, and one could say that none of the objective functions is indeed optimized. 

You construct a matrix with the objectives as columns in response to different alternative solutions and the alternatives as rows. If you read then the matrix horizontally you need weights to normalize the different units. if you read the matrix  vertically  and aggegrate after a certain procedure you get dimensionless measures. See methods like TOPSIS and MULTIMOORA, with prefererence to MULTIMOORA as it aggregates 3 different methods. Therefore see my bibliography.

Most people suggest to use weights with each objective to generate alternative solutions for Pareto frontier. If the Pareto frontier is not continuous and convex/concave, this method may not provide the true Pareto frontier. So try evolutionary multiobjective optimization that is a general methodology for all type of function properties.

There are several reasons why scalarization is often used for solving multi-objective problems, and whether it is useful or not depends on your application and the structure of your multiobjective problem.

We also need to discuss what solving a multiobjective problem actually means. This also depends on your application. First, it is clear that a point that is dominated by another point (i.e. every objectives at the second point is smaller than the corresponding objective of the first point) cannot be a solution of the multiobjective problem. The remaining points form the Pareto frontier. That is the reason why many multiobjective algorithms try to identify the Pareto frontier, and it is a good reason to call a point only then a solution if it is Pareto optimal.

However, for an application it is not necessarily important to find the whole Pareto frontier. Often, it is enough to find the one point on the Pareto frontier that "is preferred". Usually, it is difficult to find a proper mathematical model for that preference. However, you can try to use a priori preference information (i.e. model the preferences), you can choose a posteriori (this is when you need to find (most of) the Pareto frontier), and you can choose interactively (has been mentioned already by Michael Patriksson). Many methods to find one (or a few) of the preferred Pareto optimal points use scalarization to do so.

One more thing: I would be very careful to say that the "right alternative" is to use evolutionary algorithms to solve the problem. Evolutionary algorithms lack convergence theory, they often work not well unless the number of function evaluations is really high, and at no point in time while the solution algorithm works you have the faintest idea whether you have all of the Pareto frontier, just parts of it, or even a single Pareto optimal point. They usually do not even work fast in the single objective case (see various comparisons, e.g. by Rios and Sahinidis). If the evolutionary algorithm produces some "Pareto set" there is not even a guarantee that this set contains a single Pareto optimal point. Just in the black-box case when absolutely nothing is known about the objective functions except for a computationally cheap method to compute function values the evolutionary algorithms can usually expected to provide good solutions.

If gradients of the functions can be computed, descent like methods can be used, and optimality conditions can be checked to rule out non-Pareto-optimal points, etc. Usually, if something is know about the structure of the objective functions, algorithms can be used that make use of more mathematical theory, and those algorithms usually come up with better solution in less time.

@Hermann Schichl 

"If scalarization is performed carefully, the solution of the scalarized problem will be a Pareto optimal point"

Sorry, what exactly means to perform "carefully" the scalarization?

if you use weights you in fact reduces the problem to a mono objective function. Many methods keep the Original units of each objective without any normalization.

I still get confused about the 'non-dominated' concept about Pareto solutions. Does this mean that they are better in at least an objective when compared to others?

Thanks khalid. Your explanation was very helpful. 

Thank you.

Scalarization helps simplify the problem and there are methods to get a Pareto Optimal with even non-convex problems. Choice of weights is key because they can significantly affect the final solution.  

Use evolutionary multi-objective algorithms - it does not matter whether the problem is convex or non-convex. You do not need to worry about choice of weights if you use those algorithms.

Dear Mr Sarker

Excuse me but I do not agree with your last answer. In a weighted sum approach,  weights values do matter even for approximate optimization approaches.  Indeed, for non-convex problems some good solutions (according to the preference defined by the weights) can be outperformed by poor solutions. We cannot rely on the fact that evolutionary approaches do not ensure optimality and hence will stop on those poor (according to the weighted sum objective function) solutions that are actually good ones.    

Dear Menour: I believe you did not get my point - multi objective evolutionary algorithms do not require any wrights to use. I am not talking about weighted sum methods.

For others: Evolutionary algorithms do not ensue optimality - there is no doubt about it. Now we have two options: either approximate the problems and then solve using the methods we are comfortable with or solve the actual problem using a method that will provide approximate solutions such as evolutionary algorithms. That is your choice and always there are personal biases. However most practical problems cannot be represented with standard well behaved functions without making assumptions - that means your formulation can be quite different from the actual problem. The practical problems are often non-convex, non-differentiable, discontinuous, multi-modal, etc. what methods are available to us to deal with such problems? If you deal with standard and well behaved functions and problems, like textbook type examples or artificially created test problems, my answers are not really for you.

@Ruhul Sakar

I see that you are advertising evolutionary multi-objective optimization again for this problem class (see your answer from 17 July 2014). However, I still need to emphasize that the class of evolutionary optimization algorithms hardly has any provable convergence properties, and that they in general do not perform well in comparisons unless the number of function evaluations is really high (in the millions). Furthermore, the computed solution set does not necessarily consist of Pareto-optimal points, it is not even guaranteed to contain a single Pareto-optimal point. Evolutionary algorithms are pure heuristics. They can be fast and can produce quite stable and good solutions, but often the solution points are not optimal.

I can understand that ones own research deserves some advertisement. However, I think that in this forum a thorough explanation including advantages and disadvantages of solution methods is appropriate, because then the other participants in the forum get optimal information for choosing the method that is most useful for their applications.

What I said about evolutionary algorithms are absolutely correct. If someone has different opinion he or she has the right to provide arguments. But the use of non-academic arguments shows not only the lack of respects to fellow academics but also an expression of lack of knowledge on the topic under discussion. Note that I am always reluctant to interact with arrogant people. Sorry to everyone else to write such an unpleasant reply before I close the chapter of this topic.

About comparison of different types of optimization algorithms see Rios and Sahinidis, Journal of Global Optimization, July 2013, Volume 56, Issue 3, pp 1247–1293, there you can draw your own conclusions.

@Ruhul Sarker. Actually you are the arrogant person. I am quoting your post from July 2014: "Because the right alternative is not known to those people". I have seldom read a more impolite and exaggerated sentence on Research Gate. But if you are so extremely convinced, please provide the following information about the evolutionary algorithm you are mentioning to contradict my academic arguments and prove my lack of knowledge:

1. Proof of convergence against a Pareto-optimal solution.

2. Proof of approximation of the Pareto-frontier.

3. A thorough comparison of your algorithm against non-heuristic code

4. A comparison to other derivative free codes if the number of function evaluations is limited to

@Ruhul Sarker

I apologize for my latest post, because your answer from 17.6 did not show up on my account in contrast to your response from 7h ago, and I did not get any notification about it - which I think is quite strange from RG.

I think that the explanation from 17.6. correctly describes the merits of evolutionary algorithms, and now I can a bit understand your post from 7h ago. If you found that I am arrogant, then please also try to read your own earlier posts from the viewpoint of somebody not working in EA...

@Khalid Khawaja

You are right, that a lack of convergence theory is a widespread criticism of EA. There are other algorithms as well that lack convergence theory. For many of them there are hundreds of studies that show empirically that they converge to global optima - on selected problems. There are also several studies that show that the global optima are not found. This is often just a matter of properly selecting the test problems.

However, there is a second criticism, which is also emphasized by your description: EA is extremely expensive w.r.t. function evaluations. First, you need to fine-tune the parameters for every problem class you consider, and if you want to do it properly you need some sort of cross-validation procedure to do so. Then you need several runs of thousands of generations with population sizes also in the thousands. That results in millions of function evaluations. Unfortunately, in many cases, a fine-tuning of the parameters is not possible, e.g., because the problem is not one of a larger class. Then you need even more EA runs with different parameter sets, or you just use the default set - which often results in bad convergence. Consequently, lots of function evaluations need to be invested.

However, as you pointed out, EA algorithms should mostly be used in applications where the functions are difficult. But that usually means that those come from measurements or from simulations (via complex algorithms). Then the functions usually are expensive to evaluate, and that basically rules out millions of function evaluations. If the algorithms for the function evaluation are small/easy, in most cases algorithmic differentiation can provide high quality (sub)gradients, and then (sub)gradient algorithms can be applied that converge, at most using several 10000 function evaluations. For the class of discontinuous functions the global optimization problem is algorithmically unsolvable, so you can use whatever algorithm you like...

Summarizing: EA algorithms are great for complex problems, where the function evaluations are not very expensive (so that millions of them can be spent) and where algorithmic differentiation is not applicable. In those situations EA algorithms are among the best (in general better than multiple random start - and not many heuristics have this property), and the (in most situations) best EA algorithms are ones that evolve local models of the functions (e.g. CMA-ES) instead of pure point populations.

What I was missing from your descriptions of EA algorithms is one of their important advantages: They often produce robust solutions - not always globally optimal but most of the time with a significant robustness. This is often very valuable in real-life applications, and finding good solutions with high robustness is in general a difficult problem.

Regardless of the characteristics of an optimization problem, solving it using a scalarization approach requires multiple runs to obtain the Pareto-front (each run considers a different weight value), while an evolutionary algorithm obtains the entire trade-off solutions in a single run. There is numerous research works those demonstrate that the usefulness of using an evolutionary optimization algorithm for solving different real-world complex bi-objective/multi-objective problems over the traditional scalarization method.

Hello everybody,

 As in this forum, we are discussing lots about the bi-objective/multi-objective problems, can I seek some help for my problem. The details of the problem are discussed below.

 I am currently solving a bi-objective dynamic economic and emission (DEED) problem which is a non-linear, non-convex, non-smooth and multi-modal constrained optimization problem. The objectives of the problem are to simultaneously minimize the operating cost (unit is dollar) and gas emission (unit, pound/ton) of the given electrical generators. I have solved the problem using different EAs (e.g., NSGA-II, MODE, MOPSO, etc.). Now, for the comparison purpose, I’d like to solve the problem using a weighted sum approach. Can anyone suggest me, how I can scalarize those two objective functions in which their units are different? I understand, I can normalize the functions, if so, what I optimize (i.e., dollar or pound). Even after optimizing the normalized function, how I can obtain the true Pareto-front. Are the obtained solutions biased partially assuming their degrees are different?

My second concern is to set a weight value of each run. Since the problem has a piecewise objective function, dynamically increasing or decreasing the values of weight is not the best option, I believe.

I know some of you are very expert in the weighted sum approach, can anyone please solve the problem using their method and provide me the Pareto-front. The details of the problem could be found in the following paper:

-          An Evolutionary Framework for the Bi-Objectives Dynamic Economic and Environmental Dispatch Problems (Please use equations (11) to (20)).

Thank you in advance. I hope to hear back from you. If you need further information, please ask.

@Khalid Khawaja

Your post is so typical for scientists from some communities (and you have just added the EA community to that list), who brag that their own research is the absolute greatest stuff in the world, and are convinced that every discussion about their claims is outrageous. If then somebody is not immediately awestruck and dares to list limitations of the approach and asks the community not to overhype their research, he is being bashed with personal insults. Bravo!

Everybody in the optimization world knows the "No Free Lunch Theorem", and that is exactly the reason, why a proper explanation of advantages and disadvantages of algorithms is so essential. But this apparently is not in the sense of the EA community.

If you had read the whole thread properly, you would have realized that I did not advocate scalarization methods, I have rather provided an answer to a question. Then @Ruhul Sarker said that "the right alternative is not known to those people" (namely multi-objective EA) without further comment. In my response to this overhype I have just listed some disadvantages of EA algorithms. I have cited the scientific comparison study that supports my answer.

From the EA side there has not been a single paper citation that corroborates their claims. Just mentioning an algorithm without proper comparison is not enough to justify the claim that EA is "the right alternative".

Finally, about the end of your post: I can only cite a Latin phrase: "Si tacuisses philosophus mansisses". It is really sad that posts never go away, and that your future employers can read what you post in forums as this one. What they will see is a PhD candidate who insults other scientists, who does not read thoroughly, and who does not understand the difference between a mathematical model and a mathematical proof.

Having red all arguments, I can say that EA's are not the only heuristics that are available and admittedly, even if they are heuristics, they work well in multi-objective scenarios. The problem is of "scale".