Talk:Multi-objective optimization

message: As with the MCDA-article the Wikipedia article on MCDA, we have been discussing the idea of making contributions to the article on multi-objective optimization in Wikipedia in the lists of the International Society on MCDM and INFORMS Section on MCDM. According to our opinion, the current version is somewhat hard to follow and missing some essential parts of multi-objective optimization. Thus, we would like to propose that we will change the contents of the article to the following:
# Multi-objective optimization: Problem and Definitions
#* Definition of the problem and other definitions
# Multiobjective optimization applications
#* Applications of multiobjective optimization
#* This section would extend the current Applications-section
# Solving a multiobjective optimization problem
#* Different definitions of what it means to solve a multiobjective optimization problem
#* According to different researchers, this could to be to
#*# find a preferred Pareto optimal solution,
#*# find a Pareto front approximation, or
#*# generate the whole Pareto front.
#* In this section, it is clarified how this affects how the problem is approached
#* This section would also include the classification of multiobjective optimization methods to no-preference, a priori, a posteriori and interactive methods
# Scalarization
#* Different methods for scalarizing multiobjective optimization problems
#* Having this section avoids having to repeat scalarizations in further sections
# No-preference methods
#* Some no-preference methods
# A priori methods
#* Some A priori methods
# A posteriori methods
#* Some a posteriori methods
#* This section partly extends the current Solution methods -section
# Interactive methods
#* Some interactive methods
# Hybrid Methods
#* Methods combining methods from different classes
# Approximation methods in multiobjective optimization
#* This section would include many of the methods surveyed in S. Ruzika and M. Wiecek. Approximation Methods in Multiobjective Programming, J Optimiz Theory App 126, p. 473-501 and some other methods
# Visualizations
#* Methods for visualizing solutions to a multiobjective optimization problem
# Software
#* Multiobjective optimization software
We could make a collaborative effort in writing the first draft of the proposed article and then everybody could improve it. At the same time, we would like to start Wikipedia articles on Linear and discrete multiobjective optimization and evolutionary multiobjective optimization. We would then briefly describe within the article how these fields relate and differ.
Please, comment and improve our proposition.
--Markus Hartikainen (talk) 12:11, 14 December 2011 (UTC)
thread_title: Revision of the Multi-objective optimization -article

message: I will be adding quite a few examples and methods from my experience of multiobjective optimization. Please contribute by adding more stuff/fixing any incorrect things. --Mullur1729 03:21, 30 March 2007 (UTC)
thread_title: Comment by Mullur in 2007

message: I have seen many selection methods that appear to be missing from this page. I am going to list some methods from some class notes of mine. I know some of these are already listed, but I feel like making a complete list here.
1) Sequential Optimization: Optimize the most important objective first. Then optimize the next most important variable, subject to the additional constraint that the previous objective function remains the same. Repeat for all the objectives. This is essentially already in the article.
2) Method of weights - same as in the article.
3) Distance based methods (using norms). (A) Minimize the distance to some desired point. (B) Maximize the distance from the nadir (worst case scenario)
4) Direction-based methods. (A) Take the status quo or worst point and move in the direction of improvement (need not be quickest improvement, but sometimes is). (B) Pick some ideal infeasible point and a relaxation vector to move in until constraints are satisfied.
5) Modified Nash Bargaining (Nash product - has its roots in game theory). I am going to slightly adapt this to make more sense in the context of multiobjective optimization. Maximize (f1-f1*)(f2-f2*), where f1* is the best possible outcome if optimized last using sequential optimization. Let f2* be similarly defined (in game theory f1* and f2* are the payoffs without cooperation). Similarly, axiomatic bargaining.
6) Area monotonic solution. This approach is easiest explained in two dimensions, but it can work in higher dimensions. Let f1* and f2* be the status quo outcomes for the first and second objective functions. Imagine a graph of the Pareto front with dashed lines marking f1* and f2* along the two axes. Bisect the Pareto front such that the area left of the bisection line is equal to the area right of the bisection line. More specifically, let g(f1) describe the optimal second objective in terms of the first objective. Then the basic idea is to set the two areas A1 and A2 equal. Where A1 = integral(g(t)dt, f1*, f) - (f-f1*)f2* - (f-f1*)(g(f)-f2*)/2 and A2 = integral(g(t)dt, f, g_inverse(f2*)) + (f-f1*)(g(f)+f2*)/2 - (g_inverse(f2*) - f1*)f2*.
7) Equal sacrifice: Use the point along the Pareto curve which provides an equal sacrifice for all objectives. If one objective when optimized alone, could achieve 15 and another could achieve 20, a point that achieves 10 and 15 would have an equal sacrifice (5 for each).
8) Kalai Smorodinsky solution. This is like the direction based methods except that the ideal point is determined as (f1*, f2*, etc.). The starting point is the status quo.  — Preceding unsigned comment added by 150.135.222.234 (talk) 23:32, 18 March 2013 (UTC)
thread_title:  Link to/create to more selection criterion 

message: See https://link.springer.com/article/10.1007/s11047-018-9685-y#citeas Biggerj1 (talk) 10:54, 5 January 2024 (UTC)
thread_title:  Pareto Order is missing