Generalization of Primal—Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

2001 ◽  
Vol 1 (3) ◽  
pp. 229-254 ◽  
Author(s):  
Levent Tunçel
Keyword(s):  
Convex Optimization ◽  
Interior Point ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Convex Optimization Problems ◽  
Primal Dual ◽  
Conic Form

scholarly journals Novel Interior Point Algorithms for Solving Nonlinear Convex Optimization Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Sakineh Tahmasebzadeh ◽  
Hamidreza Navidi ◽  
Alaeddin Malek
Keyword(s):  
Convex Optimization ◽  
Objective Function ◽  
Interior Point ◽  
Optimization Problems ◽  
Numerical Algorithms ◽  
Optimal Solution ◽  
Linear Constraints ◽  
Convex Optimization Problems ◽  
Novel Algorithms ◽  
Interior Point Technique

This paper proposes three numerical algorithms based on Karmarkar’s interior point technique for solving nonlinear convex programming problems subject to linear constraints. The first algorithm uses the Karmarkar idea and linearization of the objective function. The second and third algorithms are modification of the first algorithm using the Schrijver and Malek-Naseri approaches, respectively. These three novel schemes are tested against the algorithm of Kebiche-Keraghel-Yassine (KKY). It is shown that these three novel algorithms are more efficient and converge to the correct optimal solution, while the KKY algorithm fails in some cases. Numerical results are given to illustrate the performance of the proposed algorithms.

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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