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PrimalDual Interior Point Algorithms for Convex Quadratically Constrained and Semidefinite Optimization Problems
 

Summary: Primal­Dual Interior Point Algorithms for Convex Quadratically
Constrained and Semidefinite Optimization Problems
I. Adler \Lambda F. Alizadeh y
May 13 1994 revised August 1994
1 introduction
It has been observed that many of the techniques used to derive and analyze interior point methods
for linear programming may be extended, in a sense ``word by word'', to more general domains.
For instance in [Ali91] Ye's methods based on Todd--Ye potential reduction function were extended
to semidefinite programming (SDP). In [MN93] Karmarkar's original algorithm was extended to
optimization problems over the ``ice cream cone'' (see below for definition); optimization over
such cones is equivalent to convex quadratically constrained quadratic programming. On the
other hand Nesterov and Nemirovskii have laid out a general theory of interior point methods
based on the concept of p--self--concordant barrier functions [NN94]. In this important work the
authors have shown that one can maximize a linear function over any convex set endowed with
such barriers in time proportional to O( p
p) iterations.
In this work we are concerned with extension of primal--dual methods of the type originally
proposed by Kojima et al [KMY89]. Such methods are proposed for linear programming and
define the primal--dual central path as
f(x; y; z)j Ax = b; A T y + z = c; and x i z i = ¯g

  

Source: Alizadeh, Farid - Rutgers Center for Operations Research, Rutgers University

 

Collections: Mathematics