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[[File:Front pareto.svg|thumb|300px|Example of a [[Pareto frontier]] (in red), the set of Pareto optimal solutions (those that are not dominated by any other feasible solutions). The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence do lie on the frontier.]]
If some objective function is to be maximized, it is equivalent to minimize its negative or its inverse. We denote <math>Y \subseteq \mathbb R^k</math> the image of <math>X</math>; <math>x^*\in X</math> a [[feasible solution]] or '''feasible decision'''; and <math>z^* = f(x^*) \in \mathbb R^k</math>an '''objective vector''' or an '''outcome'''.
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For example, [[portfolio optimization]] is often conducted in terms of [[Modern portfolio theory|mean-variance analysis]]. In this context, the efficient set is a subset of the portfolios parametrized by the portfolio mean return <math>\mu_P</math> in the problem of choosing portfolio shares to minimize the portfolio's variance of return <math>\sigma_P</math> subject to a given value of <math>\mu_P</math>; see [[Mutual fund separation theorem#Portfolio separation in mean-variance analysis|Mutual fund separation theorem]] for details. Alternatively, the efficient set can be specified by choosing the portfolio shares to maximize the function <math>\mu_P - b \sigma_P </math>; the set of efficient portfolios consists of the solutions as <math>b</math> ranges from zero to infinity.
Some of the above scalarizations involve invoking the [[minimax]] principle, where always the worst of the different objectives is optimized.<ref>Xu, J., Tao, Z. (2011). Rough Multiple Objective Decision Making. Vereinigtes Königreich: CRC Press., Page 67 https://books.google.com/books?id=zwDSBQAAQBAJ&dq=the%20minimax%20multi%20objective%20-game&pg=PA67</ref>
== A posteriori methods ==
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