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Equality constrained smooth optimization problem:

\begin{align} \min_{x}\quad & f\qty(x) \\ \textrm{s.t.} \quad & Ax = b \end{align}

for \(f\) convex, and twice differentiable; for \(A \in \mathbb{R}^{p\times n}\), rank \(p\).

additional information

equality constrained quadratic minimization

say its a quadratic:

\begin{align} f\qty(x) = \frac{1}{2} x^{T}P x + q^{T} x + r \end{align}

for \(P \in \mathbb{S}^{n}_{+}\)

We can form optimality via the KKT Conditions in a block:

\begin{align} \mqty(P & A^{T}\\ A & 0) \mqty(x^{*}\\v^{*}) = \mqty(-q \\ b) \end{align}

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constraint

recall constraint; our general constraints means that we can select \(f\) within a feasible set \(x \in \mathcal{X}\).

active constraint

an “active constraint” is a constraint which, upon application, changes the solution to be different than the non-constrainted solution. This is always true at the equality constraint, and not necessarily with inequality constraints.

types of constraints

We can write all types of optimization problems into two types of constraints; we will use these conventions EXACTLY:

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• equality constrained minimization

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