On the primal-duty geometry of level sets in linear and conic optimization

36 Pages·2001·1 MB·English

by  Freund, Robert Michael

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MIT Sloan School of Management
Sloan Working Paper 4176-01
July 2001
ON THE PRIMAL-DUAL GEOMETRY
OF LEVEL SETS IN LINEAR
AND CONIC OPTIMIZATION
Robert M. Freund
The paper can be downloaded without charge from the
Social Science Research Network Electronic Paper Collection:
http://papers.ssm.com/paper.taf?abstract_id=283690

ON THE PRIMAL-DUAL GEOMETRY OF LEVEL
SETS IN LINEAR AND CONIC OPTIMIZATION ^
Robert M. Freund"^
M.I.T.
July, 2001
Abstract
For a conic optimization problem
P minimize^ c^x
:
s.t. Ax — b
xeC
and its dual:
D supremum^ b^y
: 3
s.t. A^y + s = c
seC\
we present a geometric relationship between the maximum norms ofthe
level sets of the primal and the inscribed sizes of the level sets of the
dual (or the other way around).
AMS
Subject Classification: 90C, 90C05, 90C60
Keywords: Convex Optimization, Conic Optimization, Duality, Level Sets
^This research has been partially supported through the MIT-Singapore Alliance.
^MIT Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142-1347,
USA. email: [email protected]

PRIMAL-DUAL GEOMETRY OF LEVEL SETS
1
1 Primal-Dual Geometry of Level Sets for Lin-
ear Optimization
Consider the following dual pair of linear optimization problems:
LP minimize c^x
:
Ax —
s.t. b
X > 0,
and:
LD
maximize b'^y
:
s.t. A'^y + s = c
>
s
,
whose common optimal value is z*. For e > and <5 > 0, define the e- and
(5-level sets for the primal and dual problems as follows::
P, := [x Ax = b,x>0, Jx < 2* + e}
\
and
Di := Is A'^y + s = c, s > for some y satisfying b^y > z* — S\ .
I
Define:
:= max
i?e ||2:||i
Ax =
s.t. b
Jx < +
z* e
x >
and
rs := max minj{sj}
s.t. AJ'y + s = c
h^y >z* -6 (2)
>
s
The quantity R^ is simply the size of the largest vector x in the primal
level set P^, measured in the Li-norm. The quantity rs can be interpreted
as the positivity of the most positive vector s in the dual level set Ds, or
equivalently as the maximum distance to the boundary of the nonnegative