Intro to Multicriteria Optimization
blog.sigopt.com
Intro to Multicriteria Optimization
1–10 of 46 posts
Re: Intro to Multicriteria Optimization
#2I'm one of the co-founders of SigOpt (YC W15) and am happy to answer any questions about SigOpt or the methods we apply.
More info on our research (and examples) can be found at https://sigopt.com/research
Re: Intro to Multicriteria Optimization
#3I wrote this post and am also happy to comment. Hopefully we'll be following this up soon with a post on treating robustness and cost simultaneously in a multicriteria setting. Also, special thanks again to Devon Sigler at the University of Colorado Denver for his help editing this post.
Re: Intro to Multicriteria Optimization
#4I guess the novelty here is that constraint based optimization with no guarantees on convexity or biconvexity is a PITA?
Re: Intro to Multicriteria Optimization
#5> The reason this is a more complicated situation is that an ordering of vectors in RkRk does not exist... how would one order the vectors u = (1,2,3), v = (2,1,3), w = (3,2,1)?
It seems straightforward to order those vectors by first comparing the first component, then the second, then the third. The result is u,v,w. It's as if you wanted to sort a multi-column report in Excel. What am I missing?
Re: Intro to Multicriteria Optimization
#6I guess the novelty here is that constraint based optimization with no guarantees on convexity or biconvexity is a PITA?
First off, I am very hesitant to say anything about biconvex problems - I only see them in passing and they are definitely not in my wheelhouse. If anyone out there is an expert, or even just has a solid (basic) reference on biconvex problems, please feel free to drop some knowledge on me.
For this particular problem, which has only one input variable, yes the answer can be resolved with a good-old fashioned Plug-In-The-Answer strategy. For problems with more than one input variable, that will almost certainly not be the case.
Really, all I was trying to say there at the end is that converting the multicriteria problem to a constraint based problem has potential benefits over scalarization. Speaking only for myself, I always default to treating multicriteria problems in some sort of norm-scalarized sense: minimize ||g|| for some vector norm. I thought it was valuable to remind myself, and maybe others, that there are other ways to naturally rephrase multicriteria problems as scalar optimization problems. I'm definitely not saying anything about how easy it is to solve, as in general these constrained problems are going to be harder than the non-constrained linear (or norm) scalarization.
Re: Intro to Multicriteria Optimization
#7> The reason this is a more complicated situation is that an ordering of vectors in RkRk does not exist... how would one order the vectors u = (1,2,3), v = (2,1,3), w = (3,2,1)? It seems straightforward to order those vectors by first comparing the first component, then the second, then the third. The result is u,v,w. It's as if you wanted to sort a multi-column report in Excel. What am I missing?
this implies you're giving weight to the first element then the second.
let's say you were optimizing something to be pretty A, delicious B, soft C.
you tune the system and evaluate the prettiness, deliciousness and softness.
you tuned it several times and got three products (A, B, C) of (1,2,3), (2,1,3), (3,2,1) - concrete values are correct evaluations. how exactly do you choose the best one, is prettiness more important?
pareto efficiency come to mind - also discussed in the article. [1]
Re: Intro to Multicriteria Optimization
#8> The reason this is a more complicated situation is that an ordering of vectors in RkRk does not exist... how would one order the vectors u = (1,2,3), v = (2,1,3), w = (3,2,1)? It seems straightforward to order those vectors by first comparing the first component, then the second, then the third. The result is u,v,w. It's as if you wanted to sort a multi-column report in Excel. What am I missing?
You are absolutely correct that some sort of lexicographic ordering could exist: https://en.wikipedia.org/wiki/Lexicographical_order#Finite_s...
If such an ordering did exist, then we could certainly apply that ordering to sort results from the vector objective function so as to find the "answer" to the multicriteria problem. The Wikipedia article on multiobjective optimization discusses this strategy: https://en.wikipedia.org/wiki/Multi-objective_optimization#A.... On that note, lemme throw a shout out to the wonderful person who took the time to write that Wikipedia article - it is outstanding.
Given that, such an ordering may not be appropriate in all circumstances. Sorting objective vectors from the function suggested in this post would first sort by "time to destination" and then break ties in "time to destination" with "cost of trip". That would mean that (1, 1000) Does that help?
Re: Intro to Multicriteria Optimization
#9> The reason this is a more complicated situation is that an ordering of vectors in RkRk does not exist... how would one order the vectors u = (1,2,3), v = (2,1,3), w = (3,2,1)? It seems straightforward to order those vectors by first comparing the first component, then the second, then the third. The result is u,v,w. It's as if you wanted to sort a multi-column report in Excel. What am I missing?
If you change the basis of R_n (e.g., you rotate the axes, or you can do), you get a different result by this comparison. If there's no intrinsic reason for using a particular basis, you get different results with this ordering. In other words, you're more or less being completely arbitrary.
Re: Intro to Multicriteria Optimization
#10I'm one of the co-founders of SigOpt (YC W15) and am happy to answer any questions about SigOpt or the methods we apply. More info on our research (and examples) can be found at https://sigopt.com/research
Does your platform support more than one level of preemption / lexicographic goal setting? (Like the ε-constraint scalarization technique, but with a second, third, nth set of goals between the first set and the objective?)