In this paper we present a new primal-dual affine scaling method for linear programming. The method yields a strictly complementary optimal solution pair, and also allows a polynomial-time convergence proof. The search direction is obtained by using the original idea of Dikin, namely by minimizing the objective function (which is the duality gap in the primal-dual case), over some suitable ellipsoid. This gives rise to completely new primal-dual affine scaling directions, having no obvious relation with the search directions proposed in the literature so far. The new directions guarantee a significant decrease in the duality gap in each iteration, and at the same time they drive the iterates to the central path. In the analysis of our algorithm we use a barrier function which is the natural primal--dual generalization of Karmarkar's potential function. The iteration bound is O(nL), which is a factor O(L) better than the iteration bound of an earlier primal-dual affine scaling meth...

We present a generalization of the Penalty/Barrier Multiplier algorithm for the semidefinite programming, based on a matrix form of Lagrange multipliers. Our approach allows to use among others logarithmic, shifted logarithmic, exponential and a very eective quadratic-logarithmic penalty/barrier functions. We present dual analysis of the method, based on its correspondence to a proximal point algorithm with nonquadratic distance-like function. We give computationally tractable dual bounds, which are produced by the Legendre transformation of the penalty function. Numerical results for large-scale problems from robust control, stable truss topology design and free material design demonstrate high eciency of the algorithm. 1 The research was supported in part by the grant # 9636-1-96 of the Israeli Ministry of Sciences and The Center for Absorption in Science, Ministry of Immigrant Absorption State of Israel 2 The research was supported in part by the Optimization Laboratory in the ...

We propose a strategy for building up the linear program while using a logarithmic barrier method. The method starts with a (small) subset of the dual constraints, and follows the corresponding central path until the iterate is close to (or violates) one of the constraints, which is in turn added to the current system. This process is repeated until an optimal solution is reached. If a constraint is added to the current system, the central path will, of course, change. We analyze the effect on the barrier function value if a constraint is added. More importantly, we give an upper bound for the number of iterations needed to return to the new path. We prove that in the worst case the complexity is the same as that of the standard logarithmic barrier method. In practice this build--up scheme is likely to save a great deal of computation. Key Words: interior point method, linear programming, logarithmic barrier function, polynomial algorithm, build--up variant. 1 Introduction Karmarkar...

In this paper we propose a large--step analytic center method for smooth convex programming. The method is a natural implementation of the classical method of centers. It is assumed that the objective and constraint functions fulfil the socalled Relative Lipschitz Condition, with Lipschitz constant M ? 0. A great advantage of the method, above the existing path--following methods, is that the steps can be made long by performing linesearches. In our method we do linesearches along the Newton direction with respect to a strictly convex potential function if we are far away from the central path. If we are sufficiently close to this path we update a lower bound for the optimal value. We prove that the number of iterations required by the algorithm to converge to an ffl--optimal solution is O((1 +M 2 ) p nj ln fflj) or O((1 +M 2 )nj ln fflj), dependent on the updating scheme for the lower bound.

In this paper we propose a long--step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations is O( p nL) or O(nL), dependent on how the barrier parameter is updated. Key Words: convex quadratic programming, interior point method, logarithmic barrier function, polynomial algorithm. 1 Introduction Karmarkar's [14] invention of the projective method for linear programming has given rise to active research in interior point algorithms. At this moment, the variants can roughly be categorized into four classes: projective, affine scaling, path--following and potential reduction methods. Researchers have also extended interior point methods to other problems, including convex qu...

In this paper we will deal with primal-dual interior point methods for solving the linear programming problem. We present a short step and a long step path-following primal-dual method and derive polynomial-time bounds for both methods. The iteration bounds are as usual in the existing literature, namely O( p nL) iterations for the short step, and O(nL) for the long step variant. In the analysis of both variants we use a new proximity measure, which is closely related to the Euclidean norm of the scaled search direction vectors. The analysis of the long step method strongly depends on the fact that the (usual) search directions form a descent direction for the so-called primal-dual logarithmic barrier function.

The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...

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In the recent interior point methods for linear programming much attention has been given to the logarithmic barrier method. In this paper we will analyse the class of inverse barrier methods for linear programming, in which the barrier is P x \Gammar i , where r ? 0 is the rank of the barrier. There are many similarities with the logarithmic barrier method. The minima of an inverse barrier function for different values of the barrier parameter define a 'central path' dependent on r, called the r--path of the problem. For r # 0 this path coincides with the central path determined by the logarithmic barrier function. We introduce a metric to measure the distance of a feasible point to a point on the path. We prove that in a certain region around a point on the path the Newton process converges quadratically. Moreover, outside this region, taking a step into the Newton direction decreases the barrier function value at least with a constant. We will derive upper bounds for the total ...

A generalization of the weighted central path--following method for convex quadratic programming is presented. This is done by uniting and modifying the main ideas of the weighted central path--following method for linear programming and the interior point methods for convex quadratic programming. By means of the linear approximation of the weighted logarithmic barrier function and weighted inscribed ellipsoids, `weighted' trajectories are defined. Each strictly feasible primal dual point pair define such a weighted trajectory. The algorithm can start in any strictly feasible primal-dual point pair that defines a weighted trajectory, which is followed through the algorithm. This algorithm has the nice feature, that it is not necessary to start the algorithm close to the central path and so additional transformations are not needed. In return, the theoretical complexity of our algorithm is dependent on the position of the starting point. Polynomiality is proved under the usual mild cond...