Logarithmic Barrier Methods Research Articles

WITHDRAWN: Logarithmic barrier methods and linear connections on manifolds

This article presented several main stages of timeline of evolutionary history of logarithmic barrier functions along with interior point methods and affine connections on manifolds. The crucial idea is many optimization problems can be better stated on manifolds rather than Euclidean space, such as interior point methods, which in turns based on self-concordant functions (logarithmic barrier functions). This led us to the truth, that there is a logical evidence to study the relationships between self-concordant functions and Riemannian manifolds including affine connections.

Fluxo de potência ótimo reativo via método da função lagrangiana barreira modificada

Este artigo apresenta uma nova abordagem para a resolução do problema de Fluxo de Potência Ótimo Reativo. Nesta abordagem, as restrições de desigualdade são tratadas pela associação dos métodos de Barreira Modificada e Primal-Dual Barreira Logarítmica (PDBL). As restrições de desigualdade são transformadas em igualdades introduzindo variáveis de folga positivas e são relaxadas através do parâmetro de barreira. Uma função Lagrangiana é associada ao problema modificado. As condições necessárias de primeira ordem são aplicadas à função Lagrangiana, gerando um sistema de equações não-lineares, o qual é linearizado pelo método de Newton. A relaxação das variáveis de folga resulta na expansão da região factível do problema original, permitindo que os limites das restrições de desigualdade sejam atingidos. Testes numéricos utilizando os sistemas CESP-440kV e SUL-SUDESTE BRASILEIRO e um teste comparativo com o método PDBL indicam que a nova abordagem é eficiente na resolução do problema Fluxo de Potência Ótimo Reativo.

An Interior Point Heuristic for the Hamiltonian Cycle Problem via Markov Decision Processes

We consider the Hamiltonian cycle problem embedded in a singularly perturbed Markov decision process (MDP). More specifically, we consider the HCP as an optimization problem over the space of long-run state-action frequencies induced by the MDP's stationary policies. We show that Hamiltonian cycles (if any) correspond to the global minima of a suitably constructed indefinite quadratic programming problem over the frequency space. We show that the above indefinite quadratic can be approximated by quadratic functions that are `nearly convex' and as such suitable for the application of logarithmic barrier methods. We develop an interior-point type algorithm that involves an arc elimination heuristic that appears to perform rather well in moderate size graphs. The approach has the potential for further improvements.

APPLYING NEWTON ALGORITHMS WITHIN A SUPERVISED FEED FORWARD NEURAL NETWORK ARCHITECTURE TO FORECAST A MISSILE TRAJECTORY

A Neural Network is trained to forecast a moving trajectory. The neural network training is formulated as a nonlinear programming problem and a Newton method is used to find the optimal weights. The learning Algorithm is derived using a Recursive Prediction Error Method that approximates the inverse of the Hessian. Furthermore, box Constraints are added to the network weights to avoid network paralysis and a constraint nonlinear programming problem is formulated. Logarithmic Barrier methods which are a class of Interior Point Methods are presented. Interior point methods have good convergence properties because the weights move on a center path in the interior of the feasible weight. The logarithmic barrier method is combined with the Newton method to form a Newton-Barrier method. The moving missile trajectory is simulated using differential equations and the proposed algorithm is used to train the network in order to forecast the missile position at any given time.

A logarithmic barrier cutting plane method for convex programming

The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly non-smooth, semi-infinite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein, solve a linear approximation (localization) of the problem and then generate an additional cut to remove the linear program's optimal point. Other methods, like the “central cutting” plane methods of Elzinga-Moore and Goffin-Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. In contrast to these existing techniques, we develop a method which does not solve the linear relaxations to optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the effect of adding and deleting constraints in long-step logarithmic barrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the class of numerically difficult convex geometric and semi-infinite programming problems.

Feature Article—Interior Point Methods for Linear Programming: Computational State of the Art

A survey of the significant developments in the field of interior point methods for linear programming is presented, beginning with Karmarkar's projective algorithm and concentrating on the many variants that can be derived from logarithmic barrier methods. Full implementation details of the primal-dual predictor-corrector code OB1 are given, including preprocessing, matrix orderings, and matrix factorization techniques. A computational comparison of OB1 with a state-of-the-art simplex code using eight large models is given. In addition, computational results are presented where OB1 is used to solve two very large models that have never been solved by any simplex code INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A unified view of interior point methods for linear programming

The paper shows how various interior point methods for linear programming may all be derived from logarithmic barrier methods. These methods include primal and dual projective methods, affine methods, and methods based on the method of centers. In particular, the paper demonstrates that Karmarkar's algorithm is equivalent to a classical logarithmic barrier method applied to a problem in standard form.