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The article highlights two papers published within the issue titled "Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Algorithm," by Steven Brams, Michael Jones and Christian Klamler, and "Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples," by Rodrigo Platte, Lloyd Trefethen and Arno Kuijlaars. The first paper discussed the mathematical problem of proportional division. The second one talked about potential theory, matrix iterations in the form of Krylov space methods and quadrature formulae.

In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving a variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorization can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converge at the rate of $O(1/t)$ . Numerical illustrations are given to demonstrate the performances of the algorithms. [ABSTRACT FROM AUTHOR]

Highlights: [•] Aiming at the problem that how to classify T-splines this paper gives a mathematical analysis, and a sufficient condition of standard T-splines is also given. [•] Then this paper presents an algorithm of determining the T-spline classification, and we can get the inherently mathematical properties of T-splines by the algorithm. [ABSTRACT FROM AUTHOR]

The purpose of this paper is using Korpelevich's extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results. [ABSTRACT FROM AUTHOR]

Abstract: Recovering missing data from its partial samples is a fundamental problem in mathematics and it has wide range of applications in image and signal processing. While many such algorithms have been developed recently, there are very few papers available on their error estimations. This paper is to analyze the error of a frame based data recovery approach from random samples. In particular, we estimate the error between the underlying original data and the approximate solution that interpolates (or approximates with an error bound depending on the noise level) the given data that has the minimal norm of the canonical frame coefficients among all the possible solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the problem of clustering a set of n finite point-sets in d-dimensional Euclidean space. Different from the traditional clustering problem (called points clustering problem where the to-be-clustered objects are points), the point-sets clustering problem requires that all points in a single point-set be clustered into the same cluster. This requirement disturbs the metric property of the underlying distance function among point-sets and complicates the clustering problem dramatically. In this paper, we use a number of interesting observations and techniques to overcome this difficulty. For the k-center clustering problem on point-sets, we give an -time 3-approximation algorithm and an -time -approximation algorithm, where m is the total number of input points and k is the number of clusters. When k is a small constant, the performance ratio of our algorithm reduces to for any . For the k-median problem on point-sets, we present a polynomial time -approximation algorithm. Our approaches are rather general and can be easily implemented for practical purpose. [Copyright &y& Elsevier]

Abstract: The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well. [Copyright &y& Elsevier]

This paper proposes a simple closed-form formula of the ground-return impedance of horizontal parallel wires above a lossy ground plane. It is derived by a double logarithmic approximation of the integral term with a semi-infinite interval in Carson's ground-return impedance formula. The proposed double logarithmic approximation gives better accuracy compared to a similar approximation proposed by Pizarro and Eriksson, and this is achieved by an advanced optimization technique used for determining the coefficients in the double logarithmic approximation and also by incorporating an additional variable which minimizes errors. In this paper, it is shown, using practical examples, that the derived formula gives more accurate results compared with other existing approximate formulas. It is also shown that a graphical interpretation of the double logarithmic approximation is to replace a homogeneous lossy ground with a pair of perfectly conducting return planes located at different complex depths. [ABSTRACT FROM AUTHOR]

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schönhage in 1982. The main drawbacks of Pan's method are that it is quite involved 2 2 In Victor Pan's own words: “Our algorithms are quite involved, and their implementation would require a non-trivial work, incorporating numerous known implementation techniques and tricks”. In fact, we are not aware of any implementation of Pan's method. and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted A New Dsc , which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. A New Dsc can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task. [ABSTRACT FROM AUTHOR]

In this paper we establish approximation preserving reductions between scheduling problems in which jobs either consume some raw materials, or produce some intermediate products, and variants of the knapsack problem. Through the reductions, we get new approximation algorithms, as well as inapproximability results for the scheduling problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we discuss constrained approximation with arbitrary-level hanging nodes in adaptive higher-order finite element methods ( -FEM) for three-dimensional problems. This technique enables using highly irregular meshes, and it greatly simplifies the design of adaptive algorithms as it prevents refinements from propagating recursively through the finite element mesh. The technique makes it possible to design efficient adaptive algorithms for purely hexahedral meshes. We present a detailed mathematical description of the method and illustrate it with numerical examples. [Copyright &y& Elsevier]

Abstract: Let and be nontrivial involutions, i.e., and . A matrix is called -symmetric if . This paper presents a -symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint. The solvability condition of the constrained inverse eigenproblem is also derived. The existence, the uniqueness and the expression of the -symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved, respectively. An algorithm is presented to compute the -symmetric matrix solution to the best approximation problem. Two numerical examples are given to illustrate the effectiveness of our results. [Copyright &y& Elsevier]

The Douglas-Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it yields an algorithm for finding a point in the intersection of two convex sets. This method for solving feasibility problems has attracted a lot of attention due to its good performance even in nonconvex settings. In this paper, we consider the Douglas-Rachford algorithm for finding a point in the intersection of two subspaces. We prove that the method converges strongly to the projection of the starting point onto the intersection. Moreover, if the sum of the two subspaces is closed, then the convergence is linear with the rate being the cosine of the Friedrichs angle between the subspaces. Our results improve upon existing results in three ways: First, we identify the location of the limit and thus reveal the method as a best approximation algorithm; second, we quantify the rate of convergence, and third, we carry out our analysis in general (possibly infinite-dimensional) Hilbert space. We also provide various examples as well as a comparison with the classical method of alternating projections. [ABSTRACT FROM AUTHOR]

In this paper we offer an efficient controller synthesis algorithm for assume-guarantee specifications of the form $$\varphi _1 \wedge \varphi _2 \wedge \cdots \wedge \varphi _n \rightarrow \psi _1 \wedge \psi _2 \wedge \cdots \wedge \psi _m$$ . Here, $$\{\varphi _i,\psi _j\}$$ are all safety-MTL $$_{0, \infty }$$ properties, where the sub-formulas $$\{\varphi _i\}$$ are supposed to specify assumptions of the environment and the sub-formulas $$\{\psi _j\}$$ are specifying requirements to be guaranteed by the controller. Our synthesis method exploits the engine of Uppaal-Tiga and the novel translation of safety- and co-safety-MTL $$_{0, \infty }$$ properties into under-approximating, deterministic timed automata. Our approach avoids determinization of Büchi automata, which is the main obstacle for the practical applicability of controller synthesis for linear-time specifications. The experiments demonstrate that the chosen specification formalism is expressive enough to specify complex behaviors. The proposed approach is sound but not complete. However, it successfully produced solutions for all the experiments. Additionally we compared our tool with Acacia+ and Unbeast, state-of-the-art LTL synthesis tools; and our tool demonstrated better timing results, when we applied both tools to the analogous specifications. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the problem of computing a minimum bending energy path (or MinBEP) in a simple corridor. Given a simple 2D corridor C bounded by straight line segments and arcs of radius 2r, the MinBEP problem is to compute a path P inside C and crossing two pre-specified points s and t located at each end of C so that the bending energy of P is minimized. For this problem, we first show how to lower bound the bending energy of an optimal curve with bounded curvature, and then use this lower bound to design a -approximation algorithm for this restricted version of the MinBEP problem. Our algorithm is based on a number of interesting geometric observations and approximation techniques on smooth curves, and can be easily implemented for practical purpose. It is the first algorithm with a guaranteed performance ratio for the MinBEP problem. [Copyright &y& Elsevier]

Abstract: This paper introduces new and significantly fast algorithms to evaluate the price of double barrier options using binomial trees. To compute the price of double barrier options accurately, trees with large numbers of steps must be used, which is time consuming. In order to overcome this weakness, we develop new computational algorithms based on the spectral expansion method. The original idea of this method is coming from the eigenexpansion approach in PDEs. We show that this method enables us to compute double barrier options within 0.07 s, even if we use binomial trees with one billion steps. Moreover, this algorithm is easy to implement. In addition, the prices obtained by the proposed approach are always the same as those obtained by conventional binomial trees and show a good approximation to those by earlier studies. [Copyright &y& Elsevier]

Abstract: A semi-linear elliptic problem with variable coefficient is approximated by expanded mixed formulation based on the RTN and BDM mixed finite elements. In order to solve the nonlinear approximation system of equations efficiently, a two-grid algorithm is considered and discussed in this paper. The work includes a small nonlinear system on a coarse grid space with mesh size and a linear system on a fine grid space with mesh size . It follows from error estimates that asymptotically optimal accuracy can be obtained as long as the mesh sizes satisfy in the norm. Some numerical examples are presented to illustrate the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, a new definition of a reduced Padé approximant and an algorithm for its computation are proposed. Our approach is based on the investigation of the kernel structure of the Toeplitz matrix. It is shown that the reduced Padé approximant always has nice properties which the classical Padé approximant possesses only in the normal case. The new algorithm allows us to avoid the appearance of Froissart doublets induced by computer roundoff in the non-normal Padé table. [Copyright &y& Elsevier]

Under the condition that the values of mapping F are evaluated approximately, we propose a proximal analytic center cutting plane algorithm for solving variational inequalities. It can be considered as an approximation of the earlier cutting plane method, and the conditions we impose on the corresponding mappings are more relaxed. The convergence analysis for the proposed algorithm is also given at the end of this paper. [ABSTRACT FROM AUTHOR]

Very recently, Moudafi (2011) introduced an algorithm with weak convergence for the split common fixed-point problem. In this paper, we will continue to consider the split common fixedpoint problem. We discuss the strong convergence of the viscosity approximation method for solving the split common fixed-point problem for the class of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and extend the corresponding results announced by many others. [ABSTRACT FROM AUTHOR]

Abstract: This paper concerns systems with small coherence parameter. Simple greedy-type algorithms perform well on these systems, which are also useful in the construction of compressed sensing matrices. We discuss the following problems for both and . How large can a dictionary be, if we prescribe the coherence parameter? How small could the resulting coherence parameter be, if we impose a size on the dictionary? How could we construct such a system? Several fundamental results from different areas of mathematics shed light on these important problems with far-reaching implications in approximation theory. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a new boundary condition that can be used when reconstructing an image from observed blurred and noisy data. Our approach uses information from the observed image to enforce boundary conditions that continue image features such as edges and texture across the boundary. Because of its similarity to methods used in texture synthesis, we call our approach synthetic boundary conditions. We provide an efficient algorithm for implementing the new boundary condition, and provide a linear algebraic framework for the approach that puts it in the context of more classical and well known image boundary conditions, including zero, periodic, reflective, and anti-reflective. Extensive numerical experiments show that our new synthetic boundary conditions provide a more accurate approximation of the true image scene outside the image boundary, and thus allow for better reconstructions of the unknown, true image scene. [Copyright &y& Elsevier]

Currently, two-dimensional dyadic wavelet transform (2D-DWT) is habitually considered as the one presented by Mallat, which is defined by an approximation component, two detail components in horizontal and vertical directions. This paper is to introduce a new type of two-dimensional dyadic wavelet transform and its application so that dyadic wavelet can be studied and used widely furthermore. (1) Two-dimensional stationary dyadic wavelet transform (2D-SDWT) is proposed, it is defined by approximation coefficients, detail coefficients in horizontal, vertical and diagonal directions, which is essentially the extension of two-dimensional stationary wavelet transform for orthogonal/biorthogonal wavelet filters. (2) ε-decimated dyadic discrete wavelet transform (DDWT) is introduced and its relation with 2D-SDWT is given, where ε is a sequence of 0's and 1's. (3) Mallat decomposition algorithm based on dyadic wavelet is introduced as a special case of ε-decimated DDWT, and so a face recognition algorithm based on dyadic wavelet is proposed, and experimental results are given to show its effectiveness. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Gröbner bases via Buchberger’s Algorithm. In a nutshell, the idea is to extend the advantages of computing with homogeneous polynomials or vectors to the general case. When the input data are not homogeneous, we use as a main tool the procedure of a self-saturating Buchberger’s Algorithm. Another strictly related topic is treated later when a mathematical foundation is given to the sugar trick which is nowadays widely used in most of the implementations of Buchberger’s Algorithm. A special emphasis is also given to the case of a single grading, and subsequently some timings and indicators showing the practical merits of our approach. [Copyright &y& Elsevier]

Abstract: Symmetric rank-one (SR1) is one of the competitive formulas among the quasi-Newton (QN) methods. In this paper, we propose some modified SR1 updates based on the modified secant equations, which use both gradient and function information. Furthermore, to avoid the loss of positive definiteness and zero denominators of the new SR1 updates, we apply a restart procedure to this update. Three new algorithms are given to improve the Hessian approximation with modified secant equations for the SR1 method. Numerical results show that the proposed algorithms are very encouraging and the advantage of the proposed algorithms over the standard SR1 and BFGS updates is clearly observed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new algorithm to solve general linear fifth-order boundary value problems (BVPs) in the reproducing kernel space . Representation of the exact solution is given in the reproducing kernel space. Its approximate solution is obtained by truncating the -term of the exact solution. Some examples are displayed to demonstrate the computational efficiency of the method. [Copyright &y& Elsevier]

Abstract: Borodin, Nielsen and Rackoff  introduced the class of priority algorithms as a framework for modeling deterministic greedy-like algorithms. In this paper we address the effect of randomization in greedy-like algorithms. More specifically, we consider approximation ratios within the context of randomized priority algorithms. As case studies, we prove inapproximation results for two well-studied optimization problems, namely facility location and makespan scheduling. [Copyright &y& Elsevier]

Abstract: Many real problems can be modeled to the problems with a hierarchical structure, and bilevel programming is a useful tool to solve the hierarchical optimization problems. So the bilevel programming is widely applied, and numerous methods have been proposed to solve this programming. In this paper, we propose an approximate programming algorithm to solve bilevel nonlinear programming problem. Finally, the example illustrates the feasibility of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: Let be a simple graph without isolated vertices. A vertex set is a paired-dominating set if every vertex in has a neighbor in and the induced subgraph has a perfect matching. In this paper, we investigate the approximation hardness of paired-domination in graphs. For weighted paired-domination, an approximation algorithm in general graphs and an exact dynamic programming style algorithm in trees are also given. [Copyright &y& Elsevier]

Abstract: By introducing a bivariate matrix-valued linear functional on the scalar polynomial space, a general two-dimensional (2-D) matrix Padé-type approximant () in the inner product space is defined in this paper. The coefficients of its denominator polynomials are determined by taking the direct inner product of matrices. The remainder formula is developed and an algorithm for the numerator polynomials is presented when the generating polynomials are given in advance. By means of the Hankel-like coefficient matrix, a determinantal expression of is presented. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. At the end the method of is applied to partial realization problems of 2-D linear systems. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new algorithm for computing local extrema by modifying and combining algorithms in symbolic and numerical computation. This new algorithm improves the classical steepest descent method that may not terminate, by combining a Sturm’s theorem based separation method and a sufficient condition on infeasibility. In addition, we incorporate a grid subdivision method into our algorithm to approximate all local extrema. The complexity of our algorithm is polynomial in a newly defined condition number, and singly exponential in the number of variables. [Copyright &y& Elsevier]

Abstract: The problem of recognizing approximate graph products arises in theoretical biology. This paper presents an algorithm that recognizes a large class of approximate graph products. The main part of this contribution is concerned with a new, local prime factorization algorithm that factorizes all strong products on an extensive class of graphs that contains, in particular, all products of triangle-free graphs on at least three vertices. The local approach is linear for graph with fixed maximal degree. [Copyright &y& Elsevier]

Abstract: This paper deals with the maximum triangle packing problem. For this problem, Hassin and Rubinstein gave a randomized polynomial-time approximation algorithm that achieves an expected ratio of for any constant . By modifying their algorithm, we obtain a new randomized polynomial-time approximation algorithm for the problem which achieves an expected ratio of for any constant . [Copyright &y& Elsevier]

Abstract: For a fixed family of graphs, an -packing in a graph is a set of pairwise vertex-disjoint subgraphs of , each isomorphic to an element of . Finding an -packing that maximizes the number of covered edges is a natural generalization of the maximum matching problem, which is just . In this paper we provide new approximation algorithms and hardness results for the -packing problem where . We show that already for the -packing problem is APX-complete, and, in fact, we show that it remains so even for graphs with maximum degree 4. On the positive side, we give an approximation algorithm with approximation ratio at most 2 for every fixed . For we obtain better approximations. For we obtain a simple -approximation, achieving a known ratio that follows from a more involved algorithm of Halldórsson. For , we obtain a -approximation, and for we obtain a -approximation. [Copyright &y& Elsevier]

In this paper, we consider algorithms for constructing n-terms approximations with nonnegative coefficients. The convergence theorem is proved for a “positive” analog of the Pure Greedy Algorithm. We establish a condition on the sequence of weakness coefficients which is sufficient for the convergence of the Positive Weak Greedy Algorithm. This condition is also necessary for the class of monotone sequences. [ABSTRACT FROM AUTHOR]

We investigate a practical variant of the well-known graph Steiner tree problem. For a complete graph G = ( V, E ) with length function l: E → R + and two vertex subsets R ⊂ V and R ′ ⊆ R, a partial terminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R ∖ R ′ belong to the leaves of this Steiner tree. The partial terminal Steiner tree problem is to find a partial terminal Steiner tree T whose total lengths ∑ (u,v ) ∈T l ( u, v ) is minimum. In this paper, we show that the problem is both NP-complete and MAX SNP-hard when the lengths of edges are restricted to either 1 or 2. We also provide an approximation algorithm for the problem. [ABSTRACT FROM AUTHOR]

This paper presents two recently developed algorithms for efficient model order reduction. Both algorithms enable the fast solution of continuous-time algebraic Riccati equations (CAREs) that constitute the bottleneck in the passivity-preserving balanced stochastic truncation (BST). The first algorithm is a Smith-method-based Newton algorithm, called Newton/Smith CARE, that exploits low-rank matrices commonly found in physical system modeling. The second algorithm is a project- and-balance scheme that utilizes dominant eigenspace projection, followed by a simultaneous solution of a pair of dual CAREs through completely separating the stable and unstable invariant subspaces of a Hamiltonian matrix. The algorithms can be applied individually or together. Numerical examples show the proposed algorithms offer significant computational savings and better accuracy in reduced-order models over those from conventional schemes. [ABSTRACT FROM AUTHOR]

We present a 0.7499-approximation algorithm for Max Set Splitting in this paper. The previously best known result for this problem is a 0.7240-approximation by Andersson and Engebretsen (Inform. Process. Lett. 65 (1998) 305), which is based on a semidefinite programming (SDP) relaxation. Our improvement is resulted from a strengthened SDP relaxation, an improved rounding method, and a tighter analysis compared with that in Andersson and Engebretsen (1998). [Copyright &y& Elsevier]