scholarly journals Implementation of interior point methods for mixed semidefinite and second order cone optimization problems

2002 ◽  
Vol 17 (6) ◽  
pp. 1105-1154 ◽  
Author(s):  
Jos F. Sturm
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Second Order ◽  
Second Order Cone

scholarly journals New complexity analysis of full Nesterov-Todd step infeasible interior point method for second-order cone optimization

2018 ◽  
Vol 28 (1) ◽  
pp. 21-38
Author(s):  
Behrouz Kheirfam
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Interior Point Methods ◽  
Complexity Analysis ◽  
Second Order ◽  
Interior Point Algorithm ◽  
Complexity Bound ◽  
Second Order Cone ◽  
Infeasible Interior Point Method ◽  
Barrier Parameter

We present a full Nesterov-Todd (NT) step infeasible interior-point algorithm for second-order cone optimization based on a different way to calculate feasibility direction. In each iteration of the algorithm we use the largest possible barrier parameter value ?. Moreover, each main iteration of the algorithm consists of a feasibility step and a few centering steps. The feasibility step differs from the feasibility step of the other existing methods. We derive the complexity bound which coincides with the best known bound for infeasible interior point methods.

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.

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