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https://github.com/nitsas/pareto-approximator

Approximate the Pareto set using the chord algorithm. (multiobjective optimization, c++)
https://github.com/nitsas/pareto-approximator

algorithm c-plus-plus multiobjective-optimization

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Approximate the Pareto set using the chord algorithm. (multiobjective optimization, c++)

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README 

        
Author:  Christos Nitsas

Date:    2012-2013



Approximate the Pareto set of a multiobjective optimization problem using 

the Chord and PGEN algorithms. (both algorithms based on the weighted 

sum method)



The software presented here is general and can be applied to any 

multicriteria optimization problem, as long as that problem has a COMB 

routine that optimizes (minimizes) linear combinations of the objectives.



Please read [1] for more info on the Chord algorithm. (biobjective case)



Please read [2] for more info on the PGEN algorithm. (3+ objectives)



In-code comments:

------------------------------

In-code comments follow the Doxygen QT style so if you don't understand some

symbol in the comments (comments contain symbols/commands similar to Latex) or 

are just curious about Doxygen please check here:

http://www.stack.nl/~dimitri/doxygen/docblocks.html

and here:

http://www.stack.nl/~dimitri/doxygen/manual.html



Requirements:

------------------------------

1. The Armadillo C++ linear algebra library is required (chiefly) for some 

   linear algebra operations inside the Facet class (and some less important 

   operations elsewhere). The earliest version I've tried is 3.6.1 and it 

   worked. 

   (you can find it on http://arma.sourceforge.net/)

2. The Qhull program (specifically "qconvex") is required by PGEN to compute 

   the convex hull of sets of points. 

   (you can find it on http://www.qhull.org/)



Usage:

------------------------------

Let's say that we have a specific instance S of a multicriteria optimization 

problem P and we want to use Chord or PGEN to approximate its Pareto set. We 

will need to create a class that inherits from BaseProblem, let's call it 

MyProblem, (we can put any problem specific data (e.g. graphs e.t.c.) inside 

MyProblem) and implement MyProblem's comb() function. comb() is declared 

virtual (and null) in BaseProblem and used by the Chord and PGEN algorithms 

to access the problem's Pareto set (i.e. generate Pareto points).



The comb() function is the implementation of the theoretical COMB routine. 

(see below for more info on the COMB routine)



Please check the examples (./examples/) and experiments 

(./experiments/vs_namoa_star/) for examples of how to use the software.



What is the COMB routine:

------------------------------

The COMB routine is a routine that optimizes (minimizes) linear combinations 

of the objective functions. We give it a set of weights w_{i} and it 

optimizes (minimizes) the combined objective:

\f$ w_{1} f_{1} + w_{2} f_{2} + ... + w_{n} f_{n} \f$,

where f_{i} is the ith objective function. The COMB routine is specific to 

the underlying problem (in this case Multicriteria Shortest Paths). Chord 

and PGEN use the COMB routine as a black-box to generate Pareto optimal 

points (on the convex hull of the Pareto set).



We have assumed (w.l.o.g.) that all objectives are supposed to be minimized. 

If some objectives are maximization objectives they can easily be converted 

to equivalent minimization objectives.



References:

------------------------------

[1] C. Daskalakis, I. Diakonikolas and M. Yannakakis: "How good is the Chord 

algorithm?". Proceedings of the Twenty-First Annual ACM-SIAM Symposium on 

Discrete Algorithms. Society for Industrial and Applied Mathematics, 2010.



[2] D. L. Craft, T. F. Halabi, H. A. Shih and T. R. Bortfeld: "Approximating 

convex Pareto surfaces in multiobjective radiotherapy planning". Med. Phys., 

33, pp. 3399-3407, 2006.