convex-concave problems

Heuristic method for solving a specific type of non-convex problem. Solves a small sequence of convex problems.

difference-of-convex function

For:

\begin{equation} h\qty(x) = f\qty(x) - g\qty(x) \end{equation}

for convex \(f\qty(x)\) and \(g\qty(x)\).

some examples

• a convex quadratic, except the \(P\) in \(\qty(\frac{1}{2}) x^{T}Px\) is not PSD; we express this in terms of \(P = P_{\text{psd}} - P_{\text{nsd}}\). And thus we can get: \(\qty(\frac{1}{2})x^{T}P_{\text{psd}} x - \qty(\frac{1}{2})x^{T}P_{\text{nsd}}x\)

majorization

Taylor approximation:

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Lecture

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if we are within the feasible set already, we can do these to prevent us form getting out:

inequality constrained optimization

Generally things are of t]shape:

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

• convex, tie differentiable
• \(p^{*}\) is finite and attained
• strictly feasible

These include… LP, QP, QCQP, geometric program.

indicator barrier

Reformulate:

\begin{align} \min_{x}\quad & f_{0}\qty(x) + \sum_{i=1}^{m} I_{-}\qty(f_{i}\qty(x)) \\ \textrm{s.t.} \quad & Ax = b \end{align}