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This paper presents a logarithmic barrier method for solving a linear programming problem. We are interested in computation of the direction by the Newton's method and in computation of the displacement step using majorant functions instead line search methods in order to reduce the computation cost. This purpose is confirmed by numerical experiments, showing the efficiency of our approach, which are presented in the last section of this paper.

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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.

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This paper is concerned with an algorithm proposed by Alizadeh for linear semidefinite programming. The proof of convergence given by Alizadeh relies on a wrong inequality, we correct the proof. At each step, the algorithm uses a line search. To be efficient, such a line search needs the value of the derivative, we provide this value. Finally, a few numerical examples are treated.Keywords: semidefinite programming; interior point methods. ResumoEste artigo considera um algoritmo proposto por Alizadeh para programação semidefinida linear. A prova de convergência apresentada por Alizadeh baseia-se numa inequação errada, corrigimos a demonstração. Em cada passo, o algoritmo utiliza uma busca linear. Para ser eficiente, esta busca linear precisa do valor da derivada, apresentamos este valor. Finalmente, alguns exemplos numéricos são tratados.Palavras-chave: programação semidefinida, métodos de pontos interiores.

Kernel function plays an important role in defining new search directions for primaldual interior point algorithm for solving linear optimization problems. This problem has attracted the attention of many researchers for some years. The goal of their works is to find kernel functions that improve algorithmic complexity of this problem. In this paper, we introduce a real parameter p > 0 to generalize the kernel function (5) given by Bai et al. in [Y.Q. Bai, M El Ghami and C. Roos, SIAM J. Optim. 15 (2004) 101-128.], and give the corresponding primal-dual interior point methods for linear optimization. This parameterized kernel function yields the similar complexity bound given in [Y.Q. Bai, M El Ghami and C. Roos, SIAM J. Optim. 15 (2004) 101-128.] for both large-update and small-update methods.

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