Class Of Barrier Functions Research Articles

On the Central Paths in Symmetric Cone Programming

This paper is devoted to the study of optimal solutions of symmetric cone programs by means of the asymptotic behavior of central paths with respect to a broad class of barrier functions. This class is, for instance, larger than that typically found in the literature for semidefinite positive programming. In this general framework, we prove the existence and the convergence of primal, dual and primal---dual central paths. We are then able to establish concrete characterizations of the limit points of these central paths for specific subclasses. Indeed, for the class of barrier functions defined at the origin, we prove that the limit point of a primal central path minimizes the corresponding barrier function over the solution set of the studied symmetric cone program. In addition, we show that the limit points of the primal and dual central paths lie in the relative interior of the primal and dual solution sets for the case of the logarithm and modified logarithm barriers.

Asymptotic Expansion for the Solutions of Control Constrained Semilinear Elliptic Problems with Interior Penalties

In this work we consider the optimal control problem of a semilinear elliptic partial differential equation (PDE) with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated family of penalized problems, parametrized by $\varepsilon>0$, whose solutions define a central path converging to the solution of the original problem. Our aim is to obtain an asymptotic expansion for the solutions of the penalized problems around the solution of the original problem. This approach allows us to recover some known error bounds for the logarithmic barrier approximation and to obtain some new ones for a rather general class of barrier functions. In this manner, we generalize the results of [F. Alvarez et al., Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems, Math. Program. Ser. A, to appear], which were obtained in the ordinary differential equation (ODE) framework.

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

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.

Proximal Methods in View of Interior-Point Strategies

This paper deals with regularized penalty-barrier methods for convex programming problems. In the spirit of an iterative proximal regularization approach, an interior-point method is constructed, in which at each step a strongly convex function has to be minimized and the prox-term can be scaled by a variable scaling factor. The convergence of the method is studied for an axiomatically given class of barrier functions. According to the results, a wide class of barrier functions (in particular, logarithmic and exponential functions) can be applied to design special algorithms. For the method with a logarithmic barrier, the rate of convergence is investigated and assumptions that ensure linear convergence are given.