Localization and CuttingPlane Methods
S. Boyd and L. Vandenberghe
April 13, 2008
Contents
1
Cuttingplanes
2
2
Finding cuttingplanes
3
2.1
Unconstrained minimization
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
Feasibility problem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Inequality constrained problem
. . . . . . . . . . . . . . . . . . . . . . . . .
6
3
Localization algorithms
7
3.1
Basic cuttingplane and localization algorithm
. . . . . . . . . . . . . . . . .
7
3.2
Measuring uncertainty and progress
. . . . . . . . . . . . . . . . . . . . . . .
9
3.3
Choosing the query point
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
4
Some specific cuttingplane methods
12
4.1
Bisection method on
R
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.2
Center of gravity method
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
4.3
MVE cuttingplane method
. . . . . . . . . . . . . . . . . . . . . . . . . . .
15
4.4
Chebyshev center cuttingplane method
. . . . . . . . . . . . . . . . . . . . .
16
4.5
Analytic center cuttingplane method
. . . . . . . . . . . . . . . . . . . . . .
16
5
Extensions
16
5.1
Multiple cuts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
5.2
Dropping or pruning constraints
. . . . . . . . . . . . . . . . . . . . . . . . .
17
5.3
Nonlinear cuts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
6
Epigraph cuttingplane method
18
7
Lower bounds and stopping criteria
19
1
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In these notes we describe a class of methods for solving general convex and quasiconvex
optimization problems, based on the use of
cuttingplanes
, which are hyperplanes that sepa
rate the current point from the optimal points. These methods, called
cuttingplane methods
or
localization methods
, are quite different from interiorpoint methods, such as the barrier
method or primaldual interiorpoint method described in [BV04,
§
11]. Cuttingplane meth
ods are usually less efficient for problems to which interiorpoint methods apply, but they
have a number of advantages that can make them an attractive choice in certain situations.
•
Cuttingplane methods do not require differentiability of the objective and constraint
functions, and can directly handle quasiconvex as well as convex problems. Each itera
tion requires the computation of a subgradient of the objective or contraint functions.
•
Cuttingplane methods can exploit certain types of structure in large and complex
problems. A cuttingplane method that exploits structure can be faster than a general
purpose interiorpoint method for the same problem.
•
Cuttingplane methods do not require evaluation of the objective and all the constraint
functions at each iteration. (In contrast, interiorpoint methods require evaluating all
the objective and constraint functions, as well as their first and second derivatives.)
This can make cuttingplane methods useful for problems with a very large number of
constraints.
•
Cuttingplane methods can be used to decompose problems into smaller problems that
can be solved sequentially or in parallel.
To apply these methods to nondifferentiable problems, you need to know about subgra
dients, which are described in a separate set of notes. More details of the analytic center
cuttingplane method are given in another separate set of notes.
1
Cuttingplanes
The goal of cuttingplane and localization methods is to find a point in a convex set
X
⊆
R
n
, which we call the
target set
, or, in some cases, to determine that
X
is empty.
In an
optimization problem, the target set
X
can be taken as the set of optimal (or
ǫ
suboptimal)
points for the problem, and our goal is find an optimal (or
ǫ
suboptimal) point for the
optimization problem.
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 '09
 BOYD,S
 Linear Programming, Pk, Optimization, Convex function, Convex Optimization, query point

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