Narendra Karmarkar

A new polynomial-time algorithm for linear programming

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n3.5L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n2.5). We prove that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containingP′ to the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time.

A continuous approach to inductive inference

In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean functionℱ:{0, 1}n → {0, 1} using outputs obtained by applying a limited number of random inputs to the hidden function. Given this input—output sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used for defining the search region is dynamically modified. Computational results on 8-, 16- and 32-input, 1-output functions are presented. Our implementation successfully identified the majority of hidden functions in the experiment.

Data Structures and Programming Techniques for the Implementation of Karmarkar's Algorithm

This paper describes data structures and programming techniques used in an implementation of Karmarkars algorithm for linear programming. Most of our discussion focuses on applying Gaussian elimination toward the solution of a sequence of sparse symmetric positive definite systems of linear equations, the main requirement in Karmarkars algorithm. Our approach relies on a direct factorization scheme, with an extensive symbolic factorization step performed in a preparatory stage of the linear programming algorithm. An interpretative version of Gaussian elimination makes use of the symbolic information to perform the actual numerical computations at each iteration of algorithm. We also discuss ordering algorithms that attempt to reduce the amount of fill-in in the LU factors, a procedure to build the linear system solved at each iteration, the use of a dense window data structure in the Gaussian elimination method, a preprocessing procedure designed to increase the sparsity of the linear programming coefficient matrix, and the special treatment of dense columns in the coefficient matrix. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Approximate polynomial greatest common divisors and nearest singular polynomials

The problem of computing the greatest common divisor (gcd) of two polynomials ~, g ~ A[z], A being a unique factorization domain, is well understood and there area number of efficient algorithms for computing polynomial gcds beginning with the the work of Collins and Brown [3, 4, 9]. In this paper, we investigate the problem of finding approximate gcds. Given a pair of polynomials ~, g with real/complex coefficients, we wish to determine a small perturbation of the coefficients of ~, g such that the perturbed polynomials have a non-trivial gtd. We treat several variations of this problem and apply our techniques to the problem of finding a polynomial having multiple roots clo~seto a given polynomial. In this paper, F[a, b] denotes the polynomial ring in a, b over the field ~ and F(a, b) denotes the field of rational functions in a, b over f. C denotes the field of complex numbers and ‘R denotes the field of real numbers. For a polynomial j = f.z” + j.–lz”–l + . . . + fo ~ C[x], 11~11denotes

A Monte-Carlo algorithm for estimating the permanent

Let A be an

Computational results of an interior point algorithm for large scale linear programming

n \times n

An interior point algorithm to solve computationally difficult set covering problems

matrix with 0-1 valued entries, and let

An interior point approach to Boolean vector function synthesis

{\operatorname{per}}(A)

Calculator of matrix products

be the permanent of A. This paper describes a Monte-Carlo algorithm that produces a “good in the relative sense” estimate of

Preconditioned conjugate gradient system

{\operatorname{per}}(A)