multicriterion optimization

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

objective is the vector \(f_{0}\qty(x) \in \mathbb{R}^{q}\), essentially brings together \(q\) different objectives \(F_{i}, …, F_{q}\).

models of optimality

for the set of achievable points:

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

• feasible \(x\) is optimal if \(f_{0}\qty(x)\) is the minimum value of \(O\)
• feasible \(x\) is Pareto optimal if \(f_{0}\qty(x)\) is a minimal value of \(O\)

non-competing optimality

\(x^{*}\) optimal means \(f_{0}\qty(x^{*}) \preceq f_{0} \qty(y^{* })\) 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}\qty(x) \\ \textrm{s.t.} \quad & f_{i}\qty(x)\leq 0, i= 1\dots m, h_{i}\qty(x) = 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.