Specify objective as:

• minimize scalar convex expression
• maximize scalar concave expression

and constraints:

• convex expr <= concave expr
• concave expr >= convex expr
• affine expr = affine expr

curvatures of all expressions are DCP certified. We do this because then you can just subtract the expressions and you’ll have a good time.

you certify DCP based on general composition rule that preserve convexity

DCP is sufficient, not necessary

Consider:

\begin{equation} f\qty(x) = \sqrt{1+x^{2}} \end{equation}

A geometric program:

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

where \(f_{i}\) is posynomial, and \(h_{i}\) monomial. Notice that taking a log of this thing transforms the monomial into an affine function in \(\log \qty(x)\), and into a logsumexp for posynomials. This also implies that solving for optimal \(\log \qty(x)\), which is same as solving for \(x\) for positive \(x\), is convex problem.

Want 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

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