Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization

1995 ◽  
Vol 5 (1) ◽  
pp. 13-51 ◽  
Author(s):  
Farid Alizadeh
Keyword(s):  
Combinatorial Optimization ◽  
Semidefinite Programming ◽  
Interior Point ◽  
Interior Point Methods

scholarly journals On the Embed and Project Algorithm for the Graph Bandwidth Problem

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2030
Author(s):  
Janez Povh
Keyword(s):  
Semidefinite Programming ◽  
Interior Point ◽  
Main Part ◽  
Interior Point Methods ◽  
Linear Constraints ◽  
Bundle Method ◽  
Hard Problem ◽  
Semidefinite Programming Relaxation ◽  
Semidefinite Programming Problem ◽  
Np Hard Problem

The graph bandwidth problem, where one looks for a labeling of graph vertices that gives the minimum difference between the labels over all edges, is a classical NP-hard problem that has drawn a lot of attention in recent decades. In this paper, we focus on the so-called Embed and Project Algorithm (EPA) introduced by Blum et al. in 2000,which in the main part has to solve a semidefinite programming relaxation with exponentially many linear constraints. We present several theoretical properties of this special semidefinite programming problem (SDP) and a cutting-plane-like algorithm to solve it, which works very efficiently in combination with interior-point methods or with the bundle method. Extensive numerical results demonstrate that this algorithm, which has only been studied theoretically so far, in practice gives very good labeling for graphs with n≤1000.

Interior Point Methods for Combinatorial Optimization

1998 ◽  
pp. 189-297 ◽  
Author(s):  
John E. Mitchell ◽  
Panos M. Pardalos ◽  
Mauricio G. C. Resende
Keyword(s):  
Combinatorial Optimization ◽  
Interior Point ◽  
Interior Point Methods

Exploiting sparsity in primal-dual interior-point methods for semidefinite programming

1997 ◽  
Vol 79 (1-3) ◽  
pp. 235-253 ◽  
Author(s):  
Katsuki Fujisawa ◽  
Masakazu Kojima ◽  
Kazuhide Nakata
Keyword(s):  
Semidefinite Programming ◽  
Interior Point ◽  
Interior Point Methods ◽  
Primal Dual

Eigenvalue optimization

Acta Numerica ◽  
1996 ◽  
Vol 5 ◽  
pp. 149-190 ◽  
Author(s):  
Adrian S. Lewis ◽  
Michael L. Overton
Keyword(s):  
Semidefinite Programming ◽  
Interior Point ◽  
Duality Theory ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Lyapunov Theory ◽  
Conjugate Duality ◽  
Problem Class ◽  
Primal Dual ◽  
Matrix Norms

Optimization problems involving eigenvalues arise in many different mathematical disciplines. This article is divided into two parts. Part I gives a historical account of the development of the field. We discuss various applications that have been especially influential, from structural analysis to combinatorial optimization, and we survey algorithmic developments, including the recent advance of interior-point methods for a specific problem class: semidefinite programming. In Part II we primarily address optimization of convex functions of eigenvalues of symmetric matrices subject to linear constraints. We derive a fairly complete mathematical theory, some of it classical and some of it new. Using the elegant language of conjugate duality theory, we highlight the parallels between the analysis of invariant matrix norms and weakly invariant convex matrix functions. We then restrict our attention further to linear and semidefinite programming, emphasizing the parallel duality theory and comparing primal-dual interior-point methods for the two problem classes. The final section presents some apparently new variational results about eigenvalues of nonsymmetric matrices, unifying known characterizations of the spectral abscissa (related to Lyapunov theory) and the spectral radius (as an infimum of matrix norms).

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