A numerical implementation of an interior point method for semidefinite programming

Abstract

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.

Numerical Experiments with a Primal-Dual Algorithm for Solving Quadratic Problems

“…In (Crouzeix and Merikhi, 2008), they propose functions θ for which the step-size optimal solution is explicitly obtained. For more details about this procedure see (Chouzenoux et al, 2009), (Crouzeix and Merikhi, 2008), (Benterki and Merikhi, 2001) and (Benterki at al., 2003). In this paper we apply this procedure to compute the step length in the Primal-Dual Interior-Point Algorithm of SDPA.…”

“…Their feasible sets involving the cone of positive semidefinite matrices, a non polyhedral convex cone and they are called linear semidefinite programs. Such problems are the object of a particular attention since the papers by Alizadeh (Alizadeh, 1995) and (Alizadeh at al., 1994), as well on a theoretical or an algorithmical aspect, see for instance the following references (Alizadeh and Haberly, 1998;Benterki et al, 2003;Jarre, 1993;and Nesterov and Nemirovskii, 1990. SDP is not only an extension of LP but also includes convex quadratic optimization problems and some other convex optimization problems.…”

A relaxed logarithmic barrier method for semidefinite programming

“…The following examples are taken from the literature see for instance [2,23]. The tests were done using Matlab 10.…”

Logarithmic Barrier Method Via Minorant Function for Linear Semidefinite Programming

“…A priory, one of the advantages of the problem (1) with respect to its dual problem (2) is that variable of the objective function is a vector instead to be a matrix in the type problem (2). Furthermore, under certain convenient hypothesis, the resolution of the problem (1) is equivalent to the problem (2) in the sense that the optimal solution of one of the two problems can be reduced directly from the other through the application of the theorem of the slackness complementary, see for instance [1,6,8].…”