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!
If the dimension of the problem doesn't overcome some dozens of thousands of possible solutions the exhaustive search method is the best one and very easy for the programming work. To use the exhaustive search, your problem has to be described by discrete variables. The nonfulfillment of the restrictions is taken into consideration adding to the multiple-objective function a penalization function (in the must simple variant the magnitude of nonfulfillment of each restriction must to be multiplied for a very big number that avoids the solutions that not fulfil the restrictions appears in the first places in the solutions population). There are a group of approaches for multi-objective optimization I could to advise using the Tchebyshev distance between every evaluated solution with regard to the ideal (or desirable) one..
I think that with the above-explained will be not difficult for you solving this 3 problems.
Best wishes, José Arzola
First of all, a big thank you to the researchers who have put their input so far.
However, I would like to add that I am well aware of various methods for solving multi-objective problems and have tired implementing several different methods.
As such, I would like to clarify that I am only interested in trying to solve the above problem using the Exhaustive Search method for finding a Pareto front in the case of this post.
Thank you very much^^
Are your variable domains discreet and is the domain size small? If so, generate all possible combinations, calculate the objectives, and in the case of Problem 2 filter all solutions that violate the constraint. Then use a global Pareto filter (for example the one proposed by Messac et al.) to get rid of all non-Pareto and locally Pareto points.
If this is not the case, then you have to use standard scalarization techniques such as the mentioned weighted sum, epsilon-constraint, or augmented weighted Tchebyshev norm. See for example Ehrgott's discussion on scalarization methods.
https://messac.expressions.syr.edu/wp-content/uploads/2012/05/Messac_2004_EO_Pareto.pdf
http://link.springer.com/article/10.1007%2Fs10479-006-0074-z
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