Relative Interior

Oct 14 2016

This is a series of notes on convex optimization, covering technical details/backgrounds as I study it for the first time. The classic text Convex Optimization, will be heavily referenced.

The mathematical definition \(\textbf{relint }C = \{x \in C | B(x, r) ∩ \textbf{aff } C ⊆ C \text{ for some }r > 0\}\), stated in English, is the set of points that are the centers of norm balls with some positive radius such that these balls' intersections with \(\textbf{aff }C\) are always contained in \(C\). We then define the relative boundary of a set \(C\) as \(\textbf{cl } C \setminus \textbf{relint } C\), where \(\textbf{cl } C\) is the closure of \(C\). The notion of closure is in many ways dual to the notion of interior--the closure of \(C\) is the smallest closed set containing \(C\), whereas the interior of \(C\) is the biggest open set within \(C\). The boundary of \(C\) is the set of points in \(\textbf{cl }C\) not belonging to the interior of \(C\).

A point is in the \(\textit{interior}\) of a set if you can draw a small open ball around it which is itself contained in the set. The concept of the ''interior'' of a set can also be defined relative to the ''ambient space'' around it (more precisely, its affine hull)--that's the idea behind the relative interior. This is most useful when the interior of a set is empty, but its relative interior captures the points we want to work with. For example, the unit circle in \(\mathbb{R}^2\), \(A=\{x ∈ \mathbb{R}^2 | x_1^2 + x_2^2 = 1\}\), has the unit open disk \(B=\{x ∈ \mathbb{R}^2 | x_1^2 + x_2^2 < 1\}\) as its interior; but the unit circle sitting on x-y plane in \(\mathbb{R}^3\), i.e.,\(E=\{x ∈ \mathbb{R}^3 | x_1^2 + x_2^2 = 1, x_3=0\}\), has empty interior, by definition. The open disk \(D=\{x ∈ \mathbb{R}^3 | x_1^2 + x_2^2 < 1, x_3=0\}\) is not the interior of \(E\), since any norm ball with center located in \(D\) will necessarily capture some points of \(\mathbb{R}^3\) above and below \(D\) that are not in \(E\). But \(D\) is kind of an interior (and a far more interesting one than the empty interior)--we declare it the relative interior of \(E\), relative to \(\textbf{aff }E = \{x ∈ \mathbb{R}^3|x_3=0\}\); then we're back to the \(\mathbb{R}^2\) case of \(B\) being the interior of \(A\).

Also see book example 2.2, p.23 of a relative boundary.

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