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.

change of variables

\begin{equation} \phi :\mathbb{R}^{n} \to \mathbb{R}^{n} \end{equation}

is a one-to-one mapping with \(\phi \qty(\text{dom } \phi) \subseteq \mathcal{D}\). We can have a possibly non-convex problem:

\begin{align} &\min f_{0}\qty(x) \\ &s.t.\ f_{i}\qty(x) \leq 0\\ &h_{i}\qty(x) = 0 \end{align}

We can change variable \(x =\phi\qty(z)\).

\begin{align} &\min \tilde f_{0}\qty(z)\\ &s.t.\ \tilde f_{i}\qty(z) \leq 0,\quad i = 1,\dots,m\\ &\tilde h_{i}\qty(z) = 0,\quad i = 1,\dots,p \end{align}

where \(\tilde f_{i}\qty(z) = f_{i}\qty(\phi\qty(z))\) and \(\tilde h_{i}\qty(z) = h_{i}\qty(\phi\qty(z))\).

Multi-Objective Optimization

1. identify non-dominated individuals (individuals, for which in the multi-objective, is not dominated); this forms the “pareto frontier”
2. create all combinations of input parameters, and create a pareto frontier for them
3. identify a weighting between the variations you desire, and identify the elements which align with the Pareto frontier

Pareto Optimiality

see Pareto Optimality

Pareto Frontier

A Pareto frontier is the entire set of pareto optimal points—i. the set that’s not dominated.

Key Sequence

Notation

New Concepts

• convex relaxation
• DCP
• CVXPY
• geometric programming
• quasiconvex optimization

Important Results / Claims

• Convex Relaxation with Boolean LP
• bisection method for quasiconvex optimization

Questions

Interesting Factoids

Diminished State Capacity

1. China’s central government has very few officials
2. Uses a lot of meetings to communicate information from top to bottom.

China is generally more more understaffed because lack of people.

Information Control in China

Information control is required to ensure cohesion in a super decentralized environment; and thus messages deemed divisive or destabilizing are restricted.

China’s restrictions are highly unusual since US trade partner + China + Russia didn’t do this.