A unified semilocal convergence analysis of a family of iterative algorithms for computing all zeros of a polynomial simultaneously

In this paper, we first present a family of iterative algorithms for simultaneous determination of all zeros of a polynomial. This family contains two well-known algorithms: Dochev-Byrnev’s method and Ehrlich’s method. Second, using Proinov’s approach to studying convergence of iterative methods for polynomial zeros, we provide a semilocal convergence theorem that unifies the results of Proinov (Appl. Math. Comput. 284: 102–114, 2016) for Dochev-Byrnev’s and Ehrlich’s methods.     

Numeric vs. symbolic homotopy algorithms in polynomial system solving: a case study

We consider a family of polynomial systems which arises in the analysis of the stationary solutions of a standard discretization of certain semi-linear second-order parabolic partial differential equations. We prove that this family is well-conditioned from the numeric point of view, and ill-conditioned from the symbolic point of view. We exhibit a polynomial-time numeric algorithm solving any member of this family, which significantly contrasts the exponential behavior of all known symbolic algorithms solving a generic instance of this family of systems.     

Modified policy iteration algorithms are not strongly polynomial for discounted dynamic programming

This note shows that the number of arithmetic operations required by any member of a broad class of optimistic policy iteration algorithms to solve a deterministic discounted dynamic programming problem with three states and four actions may grow arbitrarily. Therefore any such algorithm is not strongly polynomial. In particular, the modified policy iteration and $\lambda$-policy iteration algorithms are not strongly polynomial.     

Convergence and conditioning of a Nyström method for Stokes flow in exterior three-dimensional domains

Convergence and conditioning results are presented for the lowest-order member of a family of Nyström methods for arbitrary, exterior, three-dimensional Stokes flow. The flow problem is formulated in terms of a recently introduced two-parameter, weakly singular boundary integral equation of the second kind. In contrast to methods based on product integration, coordinate transformation and singularity subtraction, the family of Nyström methods considered here is based on a local polynomial correction determined by an auxiliary system of moment equations. The polynomial correction is designed to remove the weak singularity in the integral equation and provide control over the approximation error. Here we focus attention on the lowest-order method of the family, whose implementation is especially simple. We outline a convergence theorem for this method and illustrate it with various numerical examples. Our examples show that well-conditioned, accurate approximations can be obtained with reasonable meshes for a range of different geometries.     

An infinite family of homogeneous polynomial self-maps of spheres

We provide an infinite family of homogeneous polynomial self-maps of spheres. Furthermore, we identify the gradient map of the Cartan–Münzner polynomial as a member of this infinite family and thus supply it with a geometric meaning.     

On Early Stopping in Gradient Descent Learning

In this paper we study a family of gradient descent algorithms to approximate the regression function from reproducing kernel Hilbert spaces (RKHSs), the family being characterized by a polynomial decreasing rate of step sizes (or learning rate). By solving a bias-variance trade-off we obtain an early stopping rule and some probabilistic upper bounds for the convergence of the algorithms. We also discuss the implication of these results in the context of classification where some fast convergence rates can be achieved for plug-in classifiers. Some connections are addressed with Boosting, Landweber iterations, and the online learning algorithms as stochastic approximations of the gradient descent method.     

Correlation decay and the absence of zeros property of partition functions

Absence of (complex) zeros property is at the heart of the interpolation method developed by Barvinok for designing deterministic approximation algorithms for various graph counting and related problems. An earlier method used for the same problem is one based on the correlation decay property. Remarkably, the classes of graphs for which the two methods apply often coincide or nearly coincide. In this article we show that this is not a coincidence. We establish that if the interpolation method is valid for a family of graphs, then this family exhibits a form of the correlation decay property which is asymptotic strong spatial mixing at superlogarithmic distances. Our proof is based on a certain graph polynomial representation of the associated partition function. This representation is at the heart of the design of the polynomial time algorithms underlying the interpolation method itself. We conjecture that our result holds for all, and not just amenable graphs. Indeed this conjecture was recently confirmed by Regts. See the body of the article for details.     

On the accelerated iterative method for estimating the optimum overrelaxation parameter

This paper is concerned with the iterative method for estimating the optimum overrelaxation parameter. The improved power method (IP method) with the greatest rate of convergence is derived and compared with the Chebyshev polynomial iterative method (CP method) and the other iterative methods. Two algorithms (algorithms A and B) based on the IP method are presented. Some numerical results are shown.     

Error analysis of efficient evaluation algorithms for tensor product surfaces

Backward stability of the Casteljau algorithm and two more efficient algorithms for polynomial tensor product surfaces with interest in CAGD is shown. The conditioning of the corresponding bases are compared. These algorithms are also compared with the corresponding Horner algorithm and their higher accuracy is shown. A running error analysis of the algorithms is also carried out providing algorithms which calculate “a posteriori” sharp error bounds simultaneously to the evaluation of the surface without increasing significantly the computational cost.     

A polynomial algorithm for minimum quadratic cost flow problems

Network flow problems with quadratic separable costs appear in a number of important applications such as; approximating input-output matrices in economy; projecting and forecasting traffic matrices in telecommunication networks; solving nondifferentiable cost flow problems by subgradient algorithms. It is shown that the scaling technique introduced by Edmonds and Karp (1972) in the case of linear cost flows for deriving a polynomial complexity bound for the out-of-kilter method, may be extended to quadratic cost flows and leads to a polynomial algorithm for this class of problems. The method may be applied to the solution of singly constrained quadratic programs and thus provides an alternative approach to the polynomial algorithm suggested by Helgason, Kennington and Lall (1980).     

Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.     

Scheduling UET task systems with concurrency on two parallel identical processors

Problems with unit execution time tasks and two identical parallel processors have received a great deal of attention in scheduling theory. In contrast to the conventional models, where each task requires only one processor, we consider a situation when a task may require both processors simultaneously. For problems without precedence constraints we present several polynomial time algorithms which complement recent results of Lee and Cai. We also show that the introduction of precedence constraints leads to NP-hardness results for maximum lateness and mean flow time objective functions. For the maximum lateness problem, a family of algorithms, based upon the idea of modified due dates, is considered. The worst case behaviour of these algorithms is analysed, and it is shown that the same upper bound is tight for each algorithm of this family.     

Polynomial Algorithms for a Class of Discrete Minmax Linear Programming Problems

We consider the problem of obtaining integer solutions to a minmax linear programming problem. Although this general problem is NP-complete, it is shown that a restricted version of this problem can be solved in polynomial time. For this restricted class of problems two polynomial time algorithms are suggested, one of which is strongly polynomial whenever its continuous analogue and an associated linear programming problem can be solved by a strongly polynomial algorithm. Our algorithms can also be used to obtain integer solutions for the minmax transportation problem with an inequality budget constraint. The equality constrained version of this problem is shown to be NP-complete. We also provide some new insights into the solution procedures for the continuous minmax linear programming problem.     

Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions

In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semidefinite programming. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the context of SOCP, that is they have an O( logε^-1) iteration-complexity to reduce the duality gap by a factor of ε, where n is the number of second-order cones. Since the MZ-type family studied in this paper includes an analogue of the Alizadeh, Haeberly and Overton pure Newton direction, we establish for the first time the polynomial convergence of primal-dual algorithms for SOCP based on this search direction. Received: June 5, 1998 / Accepted: September 8, 1999?Published online April 20, 2000     

Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data

Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer.     

Simple linkage: Analysis of a threshold-accepting global optimization method

In a recent paper the authors introduced an infinite class of global optimization algorithms based upon random sampling from the feasible region and local searches started from selected sample points, based upon an acceptance/rejection criterion. All of the algorithms of that class possess strong theoretical properties.Here we analyze a member of that family, which, although being significantly simpler to implement and more efficient than the well known Multi-Level Single-Linkage algorithm, enjoys the same theoretical properties. It is shown here that, with very high probability, our method is able to discover from which points Multi-Level Single-Linkage will decide to start local search.     

How good are the proximal point algorithms?

Proximal point algorithms are applicable to a variety of settings in optimization. See Rockafellar, R.T. (1976), and Spingarn, J.E. (1981) for examples. We consider a simple idealized proximal point algorithm using gradient minimization on C² convex functions. This is compared to the direct use of the same gradient method with an appropriate mollifier. The comparison is made by determining estimates of the costrequired to reduce the function to a given precision E. Our object is to assess the potential efficiency of these algorithms even if we do not know how to realize this potential.

We find that for distant starting values, proximal point algorithms are considerably less laborious than a direct method. However there is no essential improvement in the complexity - only in the numerical factors. This negative conclusion holds for the entire family of proximal point algorithms based on the gradient methods of this paper.

The algorithms considered may be important for large scale optimization problems. In applications, the precision e that is desired is usually fixed. Assume this is the case and assume that one is given a family of problems parameterized by the dimension

n. Suppose further that for all n, the condition number Q (defined below) is bounded. Then it will be seen below that for all n sufficiently large our algorithms will require a smaller number of steps than a polynomial algorithm with cost n |Ine|     

A Structured Family of Clustering and Tree Construction Methods

A cluster A is an Apresjan cluster if every pair of objects within A is more similar than either is to any object outside A. The criterion is intuitive, compelling, but often too restrictive for applications in classification. We therefore explore extensions of Apresjan clustering to a family of related hierarchical clustering methods. The extensions are shown to be closely connected with the well-known single and average linkage tree constructions. A dual family of methods for classification by splits is also presented. Splits are partitions of the set of objects into two disjoint blocks and are widely used in domains such as phylogenetics. Both the cluster and split methods give rise to progressively refined tree representations. We exploit dualities and connections between the various methods, giving polynomial time construction algorithms for most of the constructions and NP-hardness results for the rest.     

Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.     

The effectiveness of finite improvement algorithms for finding global optima

This paper investigates the effectiveness of using finite improvement algorithms for solving decision, search, and optimization problems. Finite improvement algorithms operate in a finite number of iterations, each taking a polynomial amount of work, where strict improvement is required from iteration to iteration. The hardware, software, and way of measuring complexity found in the polynomial setting are modified to identify the concept of repetition and define the new classes of decision problems,FI andNFI. A firstNFI-complete problem is given using the idea ofFI-transformations. Results relating these new classes toP, NP, andNP-complete are given. It is shown that if an optimization problem in a new classPGS isNP-hard, thenNP=co-NP. TwoPGS problems are given for which no polynomial algorithms are known to exist.