Local Superlinear Convergence of Polynomial-Time Interior-Point Methods for Hyperbolicity Cone Optimization Problems

2016 ◽  
Vol 26 (1) ◽  
pp. 139-170 ◽  
Author(s):  
Yu. Nesterov ◽  
L. Tunçel
Keyword(s):  
Polynomial Time ◽  
Interior Point ◽  
Superlinear Convergence ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Local Superlinear Convergence ◽  
Hyperbolicity Cone

scholarly journals Hyperbolic Polynomials and Convex Analysis

2001 ◽  
Vol 53 (3) ◽  
pp. 470-488 ◽  
Author(s):  
Heinz H. Bauschke ◽  
Osman Güler ◽  
Adrian S. Lewis ◽  
Hristo S. Sendov
Keyword(s):  
Convex Function ◽  
Interior Point ◽  
Convex Analysis ◽  
Symmetric Functions ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Real Polynomial ◽  
Hyperbolic Polynomials ◽  
Real Roots ◽  
Univariate Polynomial

AbstractA homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ⟼ p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Gårding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

scholarly journals A new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methods

2007 ◽  
Vol 49 (2) ◽  
pp. 259-270 ◽  
Author(s):  
Keyvan Aminis ◽  
Arash Haseli
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Point Of View ◽  
Theory And Practice ◽  
Theoretical Point ◽  
Barrier Functions ◽  
Iteration Bound ◽  
Linear Optimization Problems

AbstractInterior-Point Methods (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of log n. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos el al. by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods.

Finding improved local minima of power system optimization problems by interior-point methods

2003 ◽  
Vol 18 (1) ◽  
pp. 238-244 ◽  
Author(s):  
J. Riquelme Santos ◽  
A. Troncoso Lora ◽  
A. Gomez Expisito ◽  
J.L. Martinez Ramos
Keyword(s):  
Power System ◽  
Interior Point ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
System Optimization ◽  
Local Minima ◽  
Power System Optimization

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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