@@ -14,9 +14,10 @@ corresponding to extreme values of some specific criteria. Problems

consisting of more than one criterion are called _multi-objective

optimization problems_. Multiple objectives arise naturally in many real

world optimization problems, such as portfolio optimization, design,

planning and many more [pgnl:06,zepv:00,gala:98,yrss:09,basi:05]. It is

important to stress that multi-objective problems are in general harder

and more expensive to solve than single-objective optimization problems.

planning and many more <<bib.pgnl:06>>, <<bib.zepv:00>>, <<bib.gala:98>>,

<<bib.yrss:09>>, <<bib.basi:05>>. It is important to stress that

multi-objective problems are in general harder and more expensive to solve

than single-objective optimization problems.

In this chapter we introduce multi-objective optimization problems and

discuss techniques for their solution with an emphasis on evolutionary

...

...

@@ -85,7 +86,7 @@ candidate solutions based on the concept of dominance. A solution is

said to dominate another solution if it is no worse than the other

solution in all objectives and if it is strictly better in at least one

objective. A more formal description of the dominance relation is given

in [deb:09].

in <<bib.deb:09>>.

The properties of the dominance relation include transitivity

...

...

@@ -108,7 +109,7 @@ non-dominated solutions.

The problem of deciding if a point truly belongs to the Pareto set is

NP-hard. As shown in <<fig_pareto-def>> there exist "weaker"

formulations of Pareto optimality. Of special interest is the result

shown in [paya:01], where the authors present a polynomial (in the input

shown in <<bib.paya:01>>, where the authors present a polynomial (in the input

size of the problem and latexmath:[1/\varepsilon]) algorithm for

finding an approximation, with accuracy latexmath:[\varepsilon], of

the Pareto set for database queries.

...

...

@@ -472,3 +473,29 @@ or to count all invalid simulations:

```

cat opt.trace.0 | grep invalid | wc -l

```

[[sec.optimiser.bibliography]]

=== References

anchor:bib.pgnl:06[[{counter:bib-cnt}\]]

<<bib.pgnl:06>> A. Persson et al., https://ieeexplore.ieee.org/document/4117810[_Simulation-based multi-objective optimization of a real-world scheduling problem_], in Proceedings of the 38th conference on Winter Simulation Conference (WSC’06), pp. 1757–1764 (Monterey, CA, USA, 2006).

anchor:bib.zepv:00[[{counter:bib-cnt}\]]

<<bib.zepv:00>> R. Zebulum, M. Pacheco, and M. Vellasco, _A novel multi-objective optimization methodology applied to the synthesis of cmos operational amplifiers_, J. Solid-State Dev. and Circ., pp. 10-15 (2000).

anchor:bib.gala:98[[{counter:bib-cnt}\]]

<<bib.gala:98>> M. Galante, https://onlinelibrary.wiley.com/doi/10.1002/(SICI)1097-0207(19960215)39:3%3C361::AID-NME854%3E3.0.CO;2-1[_Genetic algorithms as an approach to optimize real-world trusses_], Int. J. Numer. Methods Eng., 39, 361 (1998).

anchor:bib.yrss:09[[{counter:bib-cnt}\]]

<<bib.yrss:09>> L. Yang et al., https://www.sciencedirect.com/science/article/pii/S0168900209016040[_Global optimization of an accelerator lattice using multiobjective genetic algorithms_], Nucl. Instrum. Methods. Phys. Res. A, 609, 50 (2009).

anchor:bib.basi:05[[{counter:bib-cnt}\]]

<<bib.basi:05>> I. Bazarov and C. Sinclair, https://journals.aps.org/prab/pdf/10.1103/PhysRevSTAB.8.034202[_Multivariate optimization of a high brightness dc gun photoinjector_], Phys. Rev. ST Accel. Beams, 8, 034202 (2005).

anchor:bib.deb:09[[{counter:bib-cnt}\]]

<<bib.deb:09>> K. Deb, _Multi-Objective Optimization Using Evolutionary Algorithms_, Wiley (2009).

anchor:bib.paya:01[[{counter:bib-cnt}\]]

<<bib.paya:01>> C. Papadimitriou and M. Yannakakis, https://dl.acm.org/doi/10.1145/375551.375560[_Multiobjective query optimization_], in Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pp. 52–59 (Santa Barbara, CA, USA, 2001).