A Combinatorial Active Set Algorithm for Linear and Quadratic Programming

We propose an algorithm for linear programming, which we call the Sequential Projection algorithm. This new approach is a primal improvement algorithm that keeps both a feasible point and an active set, which uniquely define an improving direction. Like the simplex method, the complexity of this algorithm need not depend explicitly on the size of the numbers of the problem instance. Unlike the simplex method, however, our approach is not an edge-following algorithm, and the active set need not form a row basis of the constraint matrix. Moreover, the algorithm has a number of desirable properties that ensure that it is not susceptible to the simple pathological examples (e.g., the Klee-Minty problems) that are known to cause the simplex method to perform an exponential number of iterations. We also show how to randomize the algorithm so that it runs in an expected time that is on the order of mn^{2 log n} for most LP instances. This bound is strongly subexponential in the size of the problem instance (i.e., it does not depend on the size of the data, and it can be bounded by a function that grows more slowly than 2^m, where m is the number of constraints in the problem). Moreover, to the best of our knowledge, this is the fastest known randomized algorithm for linear programming whose running time does not depend on the size of the numbers defining the problem instance. Many of our results generalize in a straightforward manner to mathematical programs that maximize a concave quadratic objective function over linear constraints (i.e., quadratic programs), and we discuss these extensions as well

A Survey of Linear Programming in Randomized Subexponential Time

"It is remarkable to see how different paths have led to rather similar results so close in time." -- Kalai, 1992 ([8]). Three papers were published in 1992, each providing a combinatorial, randomized algorithm solving linear programming in subexponential expected time. Bounds on independent algorithms were proven, one by Kalai, and the other by Matousek, Sharir, and Welzl. Results by Gartner combined techniques from these papers to solve a much more general optimization problem in similar time bounds. Although the algorithms by Kalai and Sharir--Welzl seem remarkably different in style and evolution, this paper demonstrates that one of the variants of Kalai's algorithm is identical (although dual) to the algorithm of Sharir--Welzl. Also the implication of Gartner's framework on future improvements is examined more carefully. 1 Introduction Linear programming has long been an important problem in computer science. Since 1950, when the simplex method was introduced by Dantzig [4], thou..