This paper proposes two sets of rules Rule G and Rule P for controlling step lengths in a generic primal dual interior point method for solving the linear program ming problem in standard form and its dual Theoretically Rule G ensures the global convergence while Rule P which is a special case of Rule G ensures the O nL iteration polynomial time computational complexity Both rules depend only o...

The classical column generation is based on optimal solutions of the restricted master problems. This strategy frequently results in an unstable behaviour and may require an unnecessarily large number of iterations. To overcome this weakness, variations of the classical approach use interior points of the dual feasible set, instead of optimal solutions. In this paper, we address the primal-dual...

The central path is the most important concept in the theory of interior point methods. It is an analytic curve in the interior of the feasible set which tends to an optimal point at the boundary. The analyticity properties of the paths are connected with the analysis of the superlinear convergence of the interior point algorithms for semidefinite programming. In this paper we study the analyti...

This paper develops an algorithm for solving a standard-form linear program directly from an infeasible "warm start,"i.e., directly from a given infeasible solution x that satisfies A = b but x X 0. The algorithm is a potential function reduction algorithm, but the potential function is somewhat different than other interior-point method potential functions, and is given by F(x, B) = q In (ct x...

A lot of research has been done to 1nd a faster (polynomial) algorithm that can solve linear programming (LP) problems. The main branch of this research has been devoted to interior point methods (IPM). The IPM outperforms the Simplex method in large LPs. However, there is still much research being done in order to improve pivoting algorithms. In this paper, we present a new approach to the pro...

This paper proposes a hybrid computational method (DIPS method) which works as a simplex method for solving a standard form linear program, and, simultaneously, as an interior point method for solving its dual. The DIPS method generates a sequence of primal basic feasible solutions and a sequence of dual interior feasible solutions interdependently. Along the sequences, the duality gap decrease...

In this paper we present a class of polynomial-time primal-dual interior-point methods (IPMs) for semide nite optimization based on a new class of kernel functions. This class is fairly general and includes the class of nite kernel functions [1]: the corresponding barrier functions have a nite value at the boundary of the feasible region. They are not exponentially convex and also not strongly ...

All forms of the simplex method reach the optimum by traversing a series of basic solutions. Since each basic solution represents an extreme point of the feasible region, the track followed by the algorithm moves around the boundary of the feasible region. In the worst case, it may be necessary to examine most if not all of the extreme points. This can be cripplingly inefficient given that the ...

This paper is concerned with completely positive and semidefinite relaxations of quadratic programs with linear constraints and binary variables as presented in Burer [2]. It observes that all constraints of the relaxation associated with linear constraints of the original problem can be accumulated in a single linear constraint without changing the feasible set of either the completely positiv...

We present a new admissible pivot method for linear programming that works with a sequence of improving primal feasible interior points and dual feasible interior points. This method is a practicable variant of the short admissible pivot sequence algorithm, which was suggested by Fukuda and Terlaky. Here, we also show that this method can be modified to terminate in finite pivot steps. Finedly,...