Bahman Kalantari

An algorithm for the traveling salesman problem with pickup and delivery customers

Abstract The paper extends the branch and bound algorithm of Little, Murty, Sweeney, and Karel to the traveling salesman problem with pickup and delivery customers, where each pickup customer is required to be visited before its associated delivery customer. The problems considered include single and multiple vehicle cases as well as infinite and finite capacity cases. Computational results are reported.

On the complexity of nonnegative-matrix scaling

An n × n nonnegative matrix A is said to be (doubly stochastic) scalable if there exist two positive diagonal matrices X and Y such that XAY is doubly stochastic. We derive an upper bound on the norms of the scaling factors X and Y and give a polynomial-time complexity bound on the problem of computing the scaling factors to a prescribed accuracy.

A basic family of iteration functions for polynomial root finding and its characterizations

Abstract Let p(x) be a polynomial of degree n⩾2 with coefficients in a subfield K of the complex numbers. For each natural number m⩾2, let Lm(x) be the m×m lower triangular matrix whose diagonal entries are p(x) and for each j=1,…,m−1, its jth subdiagonal entries are p j (x) j! . For i=1,2, let Lmi)(x) be the matrix obtained from Lm(x) by deleting its first i rows and its last i columns. L1(1)(x)≡1. Then, the function Bm(x)=x−p(x) det (L m−1 (1) (x)) det (L m (1) (x)) defines a fixed-point iteration function having mth order convergence rate for simple roots of p(x). For m=2 and 3, Bm(x) coincides with Newtons and Halleys, respectively. The function Bm(x) is a member of S(m,m+n−2), where for any M⩾m, S(m,M) is the set of all rational iteration functions g(x) ∈ K(x) such that for all roots θ of p(x), then g(x)=θ+∑i=mMγi(x)(θ−x)i, with γi(x) ∈ K(x) and well-defined at any simple root θ. Given g ∈ S(m,M), and a simple root θ of p(x), gi(θ)=0, i=1, …, m−1 and the asymptotic constant of convergence of the corresponding fixed-point iteration is γ m (θ) = (−1)g m (θ) m! . For Bm(x) we obtain γ m (θ)= (−1) m det (L m+1 (2) (θ)) det (L m (1)(θ)) . If all roots of p(x) are simple, Bm(x) is the unique member of S(m,m + n − 2). By making use of the identity 0 = ∑ i=0 n [p (i) (x) i!] (θ − x) i , we arrive at two recursive formulas for constructing iteration functions within the S(m,M) family. In particular, the family of Bm(x) can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schroder, whose mth order member belongs to S(m,mn), m>2. The iteration functions within S(m,M) can be extended to any arbitrary smooth function f, with the uniform replacement of p(j) with f(j) in g as well as in γm(θ).

Generalization of Taylor's theorem and Newton's method via a new family of determinantal interpolation formulas and its applications

Abstract The general form of Taylors theorem for a function f : K→K , where K is the real line or the complex plane, gives the formula, f=Pn+Rn, where Pn is the Newton interpolating polynomial computed with respect to a confluent vector of nodes, and Rn is the remainder. Whenever f′≠0, for each m=2,…,n+1, we describe a “determinantal interpolation formula”, f=Pm,n+Rm,n, where Pm,n is a rational function in x and f itself. These formulas play a dual role in the approximation of f or its inverse. For m=2, the formula is Taylors and for m=3 is a new expansion formula and a Pade approximant. By applying the formulas to Pn, for each m⩾2, P m,m−1 ,…,P m,m+n−2 is a set of n rational approximations that includes Pn, and may provide a better approximation to f, than Pn. Hence each Taylor polynomial unfolds into an infinite spectrum of rational approximations. The formulas also give an infinite spectrum of rational inverse approximations, as well as a fundamental k-point iteration function Bm(k), for each k⩽m, defined as the ratio of two determinants that depend on the first m−k derivatives. Application of our formulas have motivated several new results obtained in sequel papers: (i) theoretical analysis of the order of B m (k) , k=1,…,m, proving that it ranges from m to the limiting ratio of generalized Fibonacci numbers of order m; (ii) computational results with the first few members of Bm(k) indicating that they outperform traditional root finding methods, e.g., Newtons; (iii) a novel polynomial rootfinding method requiring only a single input and the evaluation of the sequence of iteration functions Bm(1) at that input. This amounts to the evaluation of special Toeplitz determinants that are also computable via a recursive formula; (iv) a new strategy for general root finding; (v) new formulas for approximation of π,e, and other special numbers.

Diagonal Matrix Scaling and Linear Programming

A positive semidefinite symmetric matrix either has a nontrivial nonnegative zero or can be scaled by a positive diagonal matrix into a doubly quasi-stochastic matrix. This paper describes a simple path-following Newton algorithm of the complexity

Approximating the diameter of a set of points in the Euclidean space

O(\sqrt{n} L)

A greedy heuristic for a minimum-weight forest problem

iterations to either scale an

High order iterative methods for approximating square roots

n \times n

ON EXTRANEOUS FIXED-POINTS OF THE BASIC FAMILY OF ITERATION FUNCTIONS ∗

matrix or give a nontrivial nonnegative zero. The latter problem is well known to be equivalent to linear programming.

Newton's method and generation of a determinantal family of iteration functions

Abstract Given a set P with n points in R k, its diameter dp is the maximum of the Euclidean distances between its points. We describe an algorithm that in m⩽n iterations obtains r 1 2 m ⩽d p ⩽ min { 3 r 1 , 5−2 3 r m } . For k fixed, the cost of each iteration is O(n). In particular, the first approximation r1 is within 3 of dp, independent of the dimension k.