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A derivation of Lovász’ theta via augmented Lagrange duality

Mustapha Ç. Pinar (2003)

RAIRO - Operations Research - Recherche Opérationnelle

A recently introduced dualization technique for binary linear programs with equality constraints, essentially due to Poljak et al. [13], and further developed in Lemaréchal and Oustry [9], leads to simple alternative derivations of well-known, important relaxations to two well-known problems of discrete optimization: the maximum stable set problem and the maximum vertex cover problem. The resulting relaxation is easily transformed to the well-known Lovász θ number.

A Derivation of Lovász' Theta via Augmented Lagrange Duality

Mustapha Ç. Pinar (2010)

RAIRO - Operations Research

A recently introduced dualization technique for binary linear programs with equality constraints, essentially due to Poljak et al. [13], and further developed in Lemaréchal and Oustry [9], leads to simple alternative derivations of well-known, important relaxations to two well-known problems of discrete optimization: the maximum stable set problem and the maximum vertex cover problem. The resulting relaxation is easily transformed to the well-known Lovász θ number.

A logarithm barrier method for semi-definite programming

Jean-Pierre Crouzeix, Bachir Merikhi (2008)

RAIRO - Operations Research

This paper presents a logarithmic barrier method for solving a semi-definite linear program. The descent direction is the classical Newton direction. We propose alternative ways to determine the step-size along the direction which are more efficient than classical line-searches.

A new barrier for a class of semidefinite problems

Erik A. Papa Quiroz, Paolo Roberto Oliveira (2006)

RAIRO - Operations Research

We introduce a new barrier function to solve a class of Semidefinite Optimization Problems (SOP) with bounded variables. That class is motivated by some (SOP) as the minimization of the sum of the first few eigenvalues of symmetric matrices and graph partitioning problems. We study the primal-dual central path defined by the new barrier and we show that this path is analytic, bounded and that all cluster points are optimal solutions of the primal-dual pair of problems. Then, using some ideas from semi-analytic...

A numerical feasible interior point method for linear semidefinite programs

Djamel Benterki, Jean-Pierre Crouzeix, Bachir Merikhi (2007)

RAIRO - Operations Research

This paper presents a feasible primal algorithm for linear semidefinite programming. The algorithm starts with a strictly feasible solution, but in case where no such a solution is known, an application of the algorithm to an associate problem allows to obtain one. Finally, we present some numerical experiments which show that the algorithm works properly.

A Polynomial-time Interior-point Algorithm for Convex Quadratic Semidefinite Optimization

Y. Q. Bai, F. Y. Wang, X. W. Luo (2010)

RAIRO - Operations Research

In this paper we propose a primal-dual interior-point algorithm for convex quadratic semidefinite optimization problem. The search direction of algorithm is defined in terms of a matrix function and the iteration is generated by full-Newton step. Furthermore, we derive the iteration bound for the algorithm with small-update method, namely, O( n log n ε ), which is best-known bound so far.

A sensitivity result for quadratic second-order cone programming and its application

Qi Zhao, Wenhao Fu, Zhongwen Chen (2021)

Applications of Mathematics

In this paper, we present a sensitivity result for quadratic second-order cone programming under the weak form of second-order sufficient condition. Based on this result, we analyze the local convergence of an SQP-type method for nonlinear second-order cone programming. The subproblems of this method at each iteration are quadratic second-order cone programming problems. Compared with the local convergence analysis done before, we do not need the assumption that the Hessian matrix of the Lagrangian...

An interior-point algorithm for semidefinite least-squares problems

Chafia Daili, Mohamed Achache (2022)

Applications of Mathematics

We propose a feasible primal-dual path-following interior-point algorithm for semidefinite least squares problems (SDLS). At each iteration, the algorithm uses only full Nesterov-Todd steps with the advantage that no line search is required. Under new appropriate choices of the parameter β which defines the size of the neighborhood of the central-path and of the parameter θ which determines the rate of decrease of the barrier parameter, we show that the proposed algorithm is well defined and converges...

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