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cutting_plane.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\language english
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\index Index
\shortcut idx
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\end_index
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\end_header
\begin_body
\begin_layout Section
Cutting Plane Methods
\begin_inset CommandInset label
LatexCommand label
name "sec:Cutting-Plane-Methods"
\end_inset
\end_layout
\begin_layout Standard
The goal of this chapter is to present some polynomial-time algorithms for
convex optimization.
These algorithms use knowledge of the function in a minimal way, essentially
by querying the function value and weak bounds on its support/range.
\end_layout
\begin_layout Standard
Given a continuously differentiable convex function
\begin_inset Formula $f$
\end_inset
, Theorem
\begin_inset CommandInset ref
LatexCommand ref
reference "thm:first_order"
\end_inset
shows that
\begin_inset Formula
\begin{equation}
f(y)\geq f(x)+\left\langle \nabla f(x),y-x\right\rangle \text{ for all }y.\label{eq:sub_grad_def}
\end{equation}
\end_inset
Let
\begin_inset Formula $x^{*}$
\end_inset
be any minimizer of
\begin_inset Formula $f$
\end_inset
.
Replacing
\begin_inset Formula $y$
\end_inset
with
\begin_inset Formula $x^{*}$
\end_inset
, we have that
\begin_inset Formula
\[
f(x)\geq f(x^{*})\geq f(x)+\left\langle \nabla f(x),x^{*}-x\right\rangle .
\]
\end_inset
Therefore, we know that
\begin_inset Formula $\left\langle \nabla f(x),x^{*}-x\right\rangle \leq0$
\end_inset
.
Namely,
\begin_inset Formula $x^{*}$
\end_inset
lies in a halfspace
\begin_inset Formula $H$
\end_inset
with normal vector
\begin_inset Formula $-\nabla f(x)$
\end_inset
.
Roughly speaking, this shows that each gradient computation cuts the set
of possible solutions in half.
In one dimension, this allows us to do a binary search to minimize convex
functions.
\end_layout
\begin_layout Standard
It turns out that in
\begin_inset Formula $\Rn$
\end_inset
, binary search still works.
In this chapter, we will cover several ways to do this binary search.
All of them follow the same framework, called the
\emph on
cutting plane method
\emph default
.
In this method, the convex set or function of interest is given by an oracle,
typically a separation oracle: for any
\begin_inset Formula $x\notin K\subseteq\R^{n}$
\end_inset
, the oracle finds a vector
\begin_inset Formula $g(x)\in\R^{n}$
\end_inset
such that
\begin_inset Formula
\[
g(x)^{\top}(y-x)\leq0\text{ for all }y\in K.
\]
\end_inset
\end_layout
\begin_layout Standard
Cutting plane methods address the following class of problems.
\end_layout
\begin_layout Problem
\begin_inset CommandInset label
LatexCommand label
name "prob:abstract_convex_problem"
\end_inset
\begin_inset Argument 1
status open
\begin_layout Plain Layout
Finding a point in a convex set
\end_layout
\end_inset
Given
\begin_inset Formula $\epsilon>0,R>0,$
\end_inset
and a convex set
\begin_inset Formula $K\subseteq R\cdot B^{n}$
\end_inset
specified by a separation oracle, find a point
\begin_inset Formula $y\in K$
\end_inset
or conclude that
\begin_inset Formula $\vol K\leq\epsilon^{n}$
\end_inset
.
The complexity of an algorithm is measured by the number of calls to the
oracle and the number of arithmetic operations.
\end_layout
\begin_layout Remark*
To minimize a convex function, we set
\begin_inset Formula $g(x)=\nabla f(x)$
\end_inset
and
\begin_inset Formula $K$
\end_inset
to be the set of (approximate) minimizers of
\begin_inset Formula $f$
\end_inset
.
In Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:vol2val"
\end_inset
, we relate the problem of proving that
\begin_inset Formula $\vol(K)$
\end_inset
is small to the problem of finding an approximate minimizer of
\begin_inset Formula $f$
\end_inset
.
\end_layout
\begin_layout Standard
In this framework, we maintain a convex set
\begin_inset Formula $E^{(k)}$
\end_inset
that contains the set
\begin_inset Formula $K$
\end_inset
.
Each iteration, we choose some
\begin_inset Formula $x^{(k)}$
\end_inset
based on
\begin_inset Formula $E^{(k)}$
\end_inset
and query the oracle for
\begin_inset Formula $g(x^{(k)})$
\end_inset
.
The guarantee for
\begin_inset Formula $g(.)$
\end_inset
implies that
\begin_inset Formula $K$
\end_inset
lies in the halfspace
\begin_inset Formula
\[
H^{(k)}=\{y:g(x^{(k)})^{\top}(y-x^{(k)})\leq0\}
\]
\end_inset
and hence
\begin_inset Formula $K\subset H^{(k)}\cap E^{(k)}.$
\end_inset
The algorithm continues by choosing
\begin_inset Formula $E^{(k+1)}$
\end_inset
to be a convex set that contains
\begin_inset Formula $H^{(k)}\cap E^{(k)}$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{algorithm2e}[H]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
caption{
\end_layout
\end_inset
\begin_inset Formula $\mathtt{CuttingPlaneFramework}$
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
}
\end_layout
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "alg:cutting_plane"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
SetAlgoLined
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\series bold
Input:
\series default
Initial set
\begin_inset Formula $E^{(0)}\subseteq\Rn$
\end_inset
containing
\begin_inset Formula $K$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
For{
\end_layout
\end_inset
\begin_inset Formula $k=0,\cdots$
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
}{
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Choose a point
\begin_inset Formula $x^{(k)}\in E^{(k)}$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
lIf{
\end_layout
\end_inset
\begin_inset Formula $E^{(k)}$
\end_inset
is
\begin_inset Quotes eld
\end_inset
small enough
\begin_inset Quotes erd
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
}{
\backslash
Return
\end_layout
\end_inset
\begin_inset Formula $x^{(k)}$
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Find
\begin_inset Formula $E^{(k+1)}\supset E^{(k)}\cap H^{(k)}$
\end_inset
where
\begin_inset Formula
\begin{equation}
H^{(k)}\defeq\{x\in\Rn\text{ such that }g(x^{(k)})^{\top}(x-x^{(k)})\leq0\}.\label{eq:Hk}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{algorithm2e}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
To analyze the algorithm, the main questions we need to answer are:
\end_layout
\begin_layout Enumerate
How do we choose
\begin_inset Formula $x^{(k)}$
\end_inset
and
\begin_inset Formula $E^{(k+1)}$
\end_inset
?
\end_layout
\begin_layout Enumerate
How do we measure progress?
\end_layout
\begin_layout Enumerate
How quickly does the method converge?
\end_layout
\begin_layout Enumerate
How expensive is each step?
\end_layout
\begin_layout Standard
Progress on the cutting plane method is shown in the next table.
\end_layout
\begin_layout Standard
\begin_inset Float table
placement h
wide false
sideways false
status open
\begin_layout Plain Layout
\noindent
\align center
\size footnotesize
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Year
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $E^{(k)}$
\end_inset
and
\begin_inset Formula $x^{(k)}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Iter
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Cost/Iter
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
1965
\begin_inset CommandInset citation
LatexCommand cite
key "levin1965algorithm,newman1965location"
literal "true"
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Center of gravity
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $n$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $n^{n}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
1979
\begin_inset CommandInset citation
LatexCommand cite
key "yudin1976evaluation,shor1977cut,khachiyan1980polynomial"
literal "true"
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Center of ellipsoid
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $n^{2}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $n^{2}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
1988
\begin_inset CommandInset citation
LatexCommand cite
key "khachiyan1988inscribed"
literal "true"
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
Center of John ellipsoid
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $n$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\size footnotesize
\begin_inset Formula $n^{2.878}$
\end_inset
\end_layout
\end_inset
</cell>
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<row>
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1989
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1995
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2004
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