Lagrangian Mechanics
Last edited: January 1, 2026Want mechanics? No. You get energy.
First, recall the stationary-action principle. To define a system in Lagrangian Mechanics, we define a smooth function \(L\), called the “Lagrangian”, and some configuration space (axis) \(M\).
By convention, \(L=T-V\). \(T\) is the kinetic energy in the system, and \(V\) is the potential energy in the system.
By the stationary-action principle, then, we require \(L\) to remain at a critical point (max, min, saddle.) This fact allows us to calculate the equations of motion by hold \(L\) at such a point, and evolving the \((T,V)\) pair to remain at that point.
multicriterion optimization
Last edited: January 1, 2026multicriterion 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.
problem transformation
Last edited: January 1, 2026change 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 \(\tilde h_{i}\qty(z) = h_{i}\qty(\phi\qty(z))\).
SU-CS361 APR302024
Last edited: January 1, 2026Multi-Objective Optimization
- identify non-dominated individuals (individuals, for which in the multi-objective, is not dominated); this forms the “pareto frontier”
- create all combinations of input parameters, and create a pareto frontier for them
- identify a weighting between the variations you desire, and identify the elements which align with the Pareto frontier
Pareto Optimiality
Pareto Frontier
A Pareto frontier is the entire set of pareto optimal points—i. the set that’s not dominated.