multicriterion optimization

\begin{align} \min_{x}\quad & f_{0}\left(x\right) = \left(F_{1}\left(x\right), \dots, F_{q}\left(x\right)\right) \\ \textrm{s.t.} \quad & f_{i}\left(x\right) \leq 0, i = 1 \dots m, Ax = b \end{align}

objective is the vector f_{0}\left(x\right) \in \mathbb{R}^{q}, essentially brings together q different objectives F_{i}, …, F_{q}. models of optimality for the set of achievable points:

\begin{equation} O = \left\{f_{0}\left(x\right) \mid x \text{ feasible}\right\} \end{equation}

feasible x is optimal if f_{0}\left(x\right) is the minimum value of O feasible x is Pareto optimal if f_{0}\left(x\right) is a minimal value of O non-competing optimality x^{*} optimal means f_{0}\left(x^{*}\right) \preceq f_{0} \left(y^{* }\right) for all feasible y^{*}. x^{*} simultaneously minimizes each F_{i}, which means the objectives are non-competing. Pareto Optimality At a pareto optimal point, increasing one objective value decreases another. that is, a pareto optimal point is not dominated. A point is Pareto Optimal if its not dominated by any feasible point. dominate one point x dominates another x’ if:

\begin{align} f_{i}(x) \leq f_{i}(x’) \forall i \\ f_{i}(x) < f_{i}(x’)\ \text{for some}\ i \end{align}

scalarization Choose \lambda \succ 0 which are your weights, then you can find pareto optimal points based on:

\begin{align} \min_{x}\quad & \lambda^{T}f_{0}\left(x\right) \\ \textrm{s.t.} \quad & f_{i}\left(x\right)\leq 0, i= 1\dots m, h_{i}\left(x\right) = 0, i = 1 \dots p \end{align}

you can then find one particular Pareto optimal point based on your choices. For a convex problem, you can find (almost) all Pareto optimal points by varying \lambda \succ 0. The only points you can’t get is things where \lambda_{j} = \infty because its on the edge of a pareto set.