. We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interior-point, and other methods. Key words. linear programming -- history -- simplex method -- ellipsoid method -- interior-point methods 1. Introduction At the last Mathematical Programming Symposium in Lausanne, we celebrated the 50th anniversary of the simplex method. Here, we are at or close to several other anniversaries relating to linear programming: the sixtieth of Kantorovich's 1939 paper on "Mathematical Methods in the Organization and Planning of Production" (and the fortieth of its appearance in the Western literature) [55]; the fiftieth of the historic 0th Mathematical Programming Symposium that took place in Chicago in 1949 on Activity Analysis of Production and Allocation [64]; the forty-fifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twenty-fifth of the awarding of the 1975 Nobe...

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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 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...

A new class of continuation methods is presented which, in particular, solve linear complementarity problems with copositive-plus and L -matrices. Let a# b 2 R be nonnegativevectors. Weembed the complementarity problem with a continuously differentiable mapping f : R in an artificial system of F (x# y)=(a#ib) and (x# y) 0 # () where F : R is defined by F (x# y)=(x 1 y 1 # ...#x n y n # y ; f(x)) and 0 and i 0 are parameters. A pair (x# y) is a solution of the complementarity problem if and only if it solves ()for = 0 and i = 0. A general idea of continuation methods founded on the system () is as follows.

In this paper we study a first-order and a high-order algorithm for solving linear complementarity problems. These algorithms are implicitly associated with a large neighborhood whose size may depend on the dimension of the problems. The complexity of these algorithms depends on the size of the neighborhood. For the first order algorithm, we achieve the complexity bound which the typical large-step algorithms possess. It is well-known that the complexity of large-step algorithms is greater than that of short-step ones. By using high-order power series (hence the name high-order algorithm), the iteration complexity can be reduced. We show that the complexity upper bound for our high-order algorithms is equal to that for short-step algorithms. Key Words: Interior point algorithm, High-order power series, Large neighborhood, Large step, Complexity, Linear complementarity problem. Abbreviated Title: Interior point algorithms based on large neighborhoods AMS(MOS) subject classifications: 90...

In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method applied to convex optimization problems, which may lead to weaker results than Kelley's cutting plane method. Improvements to the analytic center cutting plane method are suggested. 1 Introduction In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method, and propose improvements. Cutting plane algorithms are designed to solve general convex optimization problems. They assume that the only information available around the current iterate takes the form of cutting planes, either supporting hyperplanes to the epigraph of the objective function, or separating hyperplanes from the feasible set. The two types of hyperplanes jointly define a linear programming, polyhedral, relaxation of the original convex optimization problem. The key issue in designing a specific cutting plane algorithm is the choice of a point in the current poly...

We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar’s condition number by Dunagan, Spielman and Teng

... In this paper, we survey the various theoretical and practical issues related to degeneracy in IPM's for linear programming. We survey results which for the most part already appeared in the literature. Roughly speaking, we shall deal with four topics: the effect of degeneracy on the convergence of IPM's, on the trajectories followed by the algorithms, the effect of degeneracy in numerical performance, and on finding basic solutions.

Abstract. We present a new algorithm for solving Simple Stochastic Games (SSGs). This algorithm is based on an exhaustive search of a special kind of positional optimal strategies, the f-strategies. The running time is O ( |VR|! · (|V ||E | + |p|)), where |V |, |VR|, |E | and |p | are respectively the number of vertices, random vertices and edges, and the maximum bit-length of a transition probability. Our algorithm improves existing algorithms for solving SSGs in three aspects. First, our algorithm performs well on SSGs with few random vertices, second it does not rely on linear or quadratic programming, third it applies to all SSGs, not only stopping SSGs.