Multi-objective optimization: Difference between revisions

Revision as of 21:34, 12 November 2023

Revision as of 21:39, 12 November 2023

Line 161:

\begin{array}{ll}

\min & f_j(x)\\

\text{s.t. } & x \in X\\

& f_i(x)\leq \epsilon_i \text{ for }i\in\{1,\ldots,k\}\setminus\{j\},

\end{array}

</math>

Line 168:

Somewhat more advanced examples are the following:

* '''achievement scalarizing problems of Wierzbicki'''.<ref name="Wierzbicki1982">{{cite journal|last1=Wierzbicki|first1=A. P.|year=1982|title=A mathematical basis for satisficing decision making|journal=Mathematical Modelling|volume=3|issue=5|pages=391–405|doi=10.1016/0270-0255(82)90038-0|doi-access=free}}</ref>
One example of the achievement scalarizing problems can be formulated as

:<math>

\begin{array}{ll}

\min & \max_{i=1,\ldots,k} \left[ \frac{f_i(x)-\bar z_i}{z^{\text{nad}}_i-z_i^{\text{utopian}}}\right] + \rho\sum_{i=1}^k\frac{f_i(x)}{z_i^{nad}-z_i^{\text{utopian}}}\\

\text{subjects.t.} to }& x\in S,

\end{array}

</math>