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?
You surely use some heuristic to find an approximate Pareto front. You will not able to find the Pareto front. To approach better to it, in the supposition that you generate a solutions population, you should to simulate the real behavior of your process, for each solution in the population to recalculate the the multiple objective function value and to reorder the found solutions. You won't reach exactly the Pareto front but you will approach enough and to obtain very good close to the Pareto front solutions
Hello Shayma,
Do you really need the true Pareto front? The answer is NO, in my opinion.
All you need to prove for your 'new' multi-objective optimization algorithm is that it performs better than existing ones on these problems. There are many criteria for comparison, inverted generational distance is just one of them.
Can you prove that your new Pareto front is better that the old one(s)?
Best regards,
Bala
The classical way of getting around this is to generate a reference set R (a proxy of the true PF) using multiple multi-obj algorithms with generous function evaluations. For example:
1. Run NSGA-II with #FEs= 1e5 * dimension and compute the obtained approx front PF_1
2. Run MO-DIRECT with #FEs= 1e5 * dimension and compute the obtained approx front PF_2
3. Run DMS with #FEs= 1e5 * dimension and compute the obtained approx front PF_3
4. Run MO-SOO #FEs=1e5 * dimension and compute the obtained approx front PF_4
4. The true PF can be approximated with = ND(PF_1 U PF_2 U PF_3 U PF_4)
Both MO-SOO and MO-DIRECT are asymptotically consistent. I.e. they converge to the optimal PF given infinite function evaluation (which is not the case in practice, so you can give them ~1e6 FEs).
Check:
1. Dimo Brockhoff, Thanh-Do Tran, Nikolaus Hansen. Benchmarking Numerical Multiobjective Optimizers Revisited. A.I. Esparcia and S. Silva. Genetic and Evolutionary Computation Conference (GECCO 2015), Jul 2015, Madrid, Spain. pp.639-646, .
2. Al-Dujaili, Abdullah, and S. Suresh. "BMOBench: Black-Box Multi-Objective Optimization Benchmarking Platform." arXiv preprint arXiv:1605.07009 (2016).
3. Al-Dujaili, Abdullah, and Sundaram Suresh. "Dividing rectangles attack multi-objective optimization." Evolutionary Computation (CEC), 2016 IEEE Congress on. IEEE, 2016.
code: https://github.com/ash-aldujaili/modirect
4. A. L. Custódio, J. F. A. Madeira, A. I. F. Vaz, and L. N. Vicente, Direct multisearch for multiobjective optimization, SIAM Journal on Optimization, 21 (2011) 1109-1140
code: http://www.mat.uc.pt/dms/
5. Al-Dujaili, Abdullah, and Sundaram Suresh. "Multi-objective simultaneous optimistic optimization." Information Sciences424 (2018): 159-174.
code: https://github.com/ash-aldujaili/mosoo
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