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In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality constrained optimization when a nearly feasible point is known. In particular, we first propose a new Newton-conjugate gradient (Newton-CG) method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a substantially better complexity than the Newton-CG method in [C. W. Royer, M. O'Neill, and S. J. Wright, Math. Program., 180 (2020), pp. 451-488]. We then propose a Newton-CG based augmented Lagrangian (AL) method for finding an approximate SOSP of nonconvex equality constrained optimization, in which the proposed Newton-CG method is used as a subproblem solver. We show that under a generalized linear independence constraint qualification (GLICQ), our AL method enjoys a total inner iteration complexity of O(ε7/peration complexity of O(ε7/2 min\{ n, ε3/4\}) for finding an (ε, ε)-SOSP of nonconvex equality constrained optimization with high probability, which are significantly better than the ones achieved by the proximal AL method in [Y. Xie and S. J. Wright, J. Sci. Comput., 86 (2021), pp. 1-30]. In addition, we show that it has a total inner iteration complexity of O(ε11/2) and an operation complexity of O(ε11/2 min\{ n, ε5/4\}) when the GLICQ does not hold. To the best of our knowledge, all the complexity results obtained in this paper are new for finding an approximate SOSP of nonconvex equality constrained optimization with high probability. Preliminary numerical results also demonstrate the superiority of our proposed methods over the other competing algorithms. [ABSTRACT FROM AUTHOR]

The Laplace Transform is frequently encountered in mathematics, physics, engineering, and other fields. However, the spectral properties of the Laplace Transform tend to complicate its numerical treatment; therefore, the closely related "Truncated" Laplace Transforms are often used in applications. We have constructed efficient algorithms for the evaluation of the Singular Value Decomposition of Truncated Laplace Transforms; in the current paper, we introduce algorithms for the evaluation of the right singular functions and singular values of Truncated Laplace Transforms. Algorithms for the computation of the left singular functions will be introduced separately in an upcoming paper. The resulting algorithms are applicable to all environments likely to be encountered in applications, including the evaluation of singular functions corresponding to extremely small singular values (e.g., 10-1000). [ABSTRACT FROM AUTHOR]

For matrices with displacement structure, basic operations like multiplication, inversion, and linear system solving can all be expressed in terms of the following task: evaluate the product AB, where A is a structured n × n matrix of displacement rank α, and B is an arbitrary n × α matrix. Given B and a so-called generator of A, this product is classically computed with a cost ranging from O(α² M (n)) to O(α²M (n) log(n)) arithmetic operations, depending on the type of structure of A; here, M is a cost function for polynomial multiplication. In this paper, we first generalize classical displacement operators, based on block diagonal matrices with companion diagonal blocks, and then design fast algorithms to perform the task above for this extended class of structured matrices. The cost of these algorithms ranges from O(αω-1 M (n)) to O(αω-1 M (n) log(n)), with ω such that two n × n matrices can be multiplied using O(nω) ring operations. By combining this result with classical randomized regularization techniques, we obtain faster Las Vegas algorithms for structured inversion and linear system solving. [ABSTRACT FROM AUTHOR]

We present in this paper two different classes of general multiple-splitting algorithms for solving finite-dimensional convex optimization problems. Under the assumption that the function being minimized can be written as the sum of K convex functions, each of which has a Lipschitz continuous gradient, we prove that the number of iterations needed by the first class of algorithms to obtain an ε-optimal solution is O((K - 1)L/ε), where L is an upper bound on all of the Lipschitz constants. The algorithms in the second class are accelerated versions of those in the first class, where the complexity result is improved to O(√(K - 1)L/ε) while the computational effort required at each iteration is almost unchanged. To the best of our knowledge, the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction-type methods. Moreover, all algorithms proposed in this paper are parallelizable, which makes them particularly attractive for solving certain large-scale problems. [ABSTRACT FROM AUTHOR]

This paper presents some guidance for choosing multigrid preconditioners for systems of partial differential equations (PDEs). Rather than developing a black-box algorithm for arbitrary systems of PDEs, the strategy in this paper is to classify systems with multigrid preconditioners. From this classification, a new preconditioner is developed, and it is shown to be suitable for a particular class of strongly cross-coupled systems, solving the whole cross-coupled system almost as effectively as solving just the diagonal system. Also, this paper gives an explanation of why block diagonal preconditioners are somewhat effective for certain strongly cross-coupled systems that at first glance appear to be intractable for multigrid preconditioning. [ABSTRACT FROM AUTHOR]

A common thread in all of the recent results concerning the testing of dense graphs is the use of Szemerédi's regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was first raised by Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653-750] in the paper that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable. [ABSTRACT FROM AUTHOR]

In this paper, we propose iterative algorithms for solving image restoration problems. The iterative algorithms are based on decoupling of deblurring and denoising steps in the restoration process. In the deblurring step, an efficient deblurring method using fast transforms can be employed. In the denoising step, effective methods such as the wavelet shrinkage denoising method or the total variation denoising method can be used. The main advantage of this proposal is that the resulting algorithms can be very efficient and can produce better restored images in visual quality and signalto-noise ratio than those by the restoration methods using the combination of a data-fitting term and a regularization term. The convergence of the proposed algorithms is shown in the paper. Numerical examples are also given to demonstrate the effectiveness of these algorithms. [ABSTRACT FROM AUTHOR]

The Lloyd algorithm originated in the context of optimal quantization and represents a fixed point iteration for computing an optimal quantizer. Reducing average distortion at every step, it constructs a Voronoi partition of the domain and replaces each generator with the centroid of the corresponding Voronoi cell. Optimal quantization is obtained in the case of a centroidal Voronoi tessellation (CVT), which is a special Voronoi tessellation of a domain Ω ϵ ℝd having the property that the generators of the Voronoi diagram are also the centers of mass, with respect to a given density function ? ⩾ 0, of the corresponding Voronoi cells. The Lloyd iteration is currently the most popular and elegant algorithm for computing CVTs and optimal quantizers, but many questions remain about its convergence, especially in d-dimensional spaces (d > 1). In this paper, we prove that any limit point of the Lloyd iteration in any dimensional spaces is nondegenerate provided that Ω is a convex and bounded set and ? belongs to L¹(Ω) and is positive almost everywhere. This ensures that the fixed point map remains closed and hence the standard theory of descent methods guarantees weak global convergence of the Lloyd iteration to the set of nondegenerate fixed point quantizers. While previously only conjectured, the convergence properties of the Lloyd iteration are rigorously justified under such minimal regularity assumptions on the density functional. The results presented in this paper go beyond existing convergence theories for CVTs and optimal quantization related algorithms and should be of interest to both the mathematical and engineering communities. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available. We introduce the mesh adaptive direct search (MADS) class of algorithms which extends the generalized pattern search (GPS) class by allowing local exploration, called polling, in an asymptotically dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained. The main GPS convergence result is to identify limit points x0302;, where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions span the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for GPS. The main result of this paper is that the general MADS framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the Clarke derivatives are nonnegative. We propose an instance of MADS for which the refining directions are dense in the hypertangent cone at x0302; with probability 1 whenever the iterates associated with the refining directions converge to a single x0302;. The instance of MADS is compared to versions of GPS on some test problems. We also illustrate the limitation of our results with examples. [ABSTRACT FROM AUTHOR]

In this paper, an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in [Y. Li and J. Zhou, SIAM J. Sci. Comput. 23 (2001), pp. 840--865; Y. Li and J. Zhou, SIAM J. Sci. Comput., 24 (2002), pp. 840--865]. First an invariant space is defined in a more general sense and a principle of invariant criticality is proved for the generalization. Then the orthogonal projection to the invariant space is used to preserve the invariance and to reduce computational error across iterations. Simple averaging formulas are used for the orthogonal projections. Numerical computations of examples with various symmetries, of which some can and others cannot be characterized by a compact group of linear isomorphisms, are carried out to confirm the theory and to illustrate applications. The mathematical features of various problems demonstrated in these examples fall into two categories: nodal solutions of saddle-point type with large Morse indices and nonradial positive solutions via symmetry breaking in radially symmetric elliptic problems. The new numerical algorithm generates these rather unstable solutions in an efficient and stable way. The existence of many unstable solutions and their behavior found in this paper remain to be investigated. [ABSTRACT FROM AUTHOR]

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X½ SX½ = vW, where W is a fixed positive definite matrix and v>0 is a parameter, under the assumption that the problem has a strictly complementary primal-dual optimal solution. It is shown that a weighted central path as a function of √v can be extended analytically beyond 0 and hence that the path converges as v↓0. Characterization of the limit points of the path and its normalized first-order derivatives are also provided. As a consequence, it is shown that a weighted central path can have two types of behavior: it converges either as &Tetha;(v) or asT &Tetha;(√v) depending on whether the matrix W on a certain scaled space is block diagonal or not, respectively. We also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primal-dual optimal solutions. Finally, in light of the results of this paper, we give a characterization of a sufficient condition proposed by Potra and Sheng which guarantees the superlinear convergence of a class of primal-dual interior-point SDP algorithms. [ABSTRACT FROM AUTHOR]

In this paper we present a new iteration-complexity bound for the Mizuno--Todd--Ye predictor-corrector (MTY P-C) primal-dual interior-point algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program min {cTχ: Aχ = b, &chi ≥ 0 } with decision variable ..., we show that the MTY P-C algorithm, started from a well-centered interior-feasible solution with duality gap nμ0 finds an interior-feasible solution with duality gap less than nn in O (T(μ0/η)+n3.5 log (Χ*A)) iterations, where T(t ) =min {n² log (log t), log t} for all t>0 and Χ*A is a scaling invariant condition number associated with the matrix A. More specifically,Χ*A is the infimum of all the conditions numbers XAD, where D varies over the set of positive diagonal matrices. Under the setting of the Turing machine model, our analysis yields an O (n3.5 LA + min {n² log L, L}) iteration-complexity bound for the MTY P-C algorithm to find a primal-dual optimal solution, where LA and L are the input sizes of the matrix A and the data (A,b,c ), respectively. This contrasts well with the classical iteration-complexity bound for the MTY P-C algorithm, which depends linearly on L instead of log L. [ABSTRACT FROM AUTHOR]

Let M be a 2 × 2 matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or finite-impulse-response filters) u1, u2, v1, v2 with real coefficients and symmetry such that [This symbol cannot be presented in ASCII format] and [Su1](z)[Sv2](z) = [Su2](z)[Sv1](z), where [Sp](z) = p(z)/p(1/z) for a nonzero Laurent polynomial p. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters u1, u2, v1, v2. As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric multiresolution analysis tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal denoising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two highpass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some B-spline filters. [ABSTRACT FROM AUTHOR]

This paper contains a new convergence analysis for the Lewis and Torczon generalized pattern search (GPS) class of methods for unconstrained and linearly constrained optimization. This analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems. Specifically, even if the objective function is discontinuous or extended-valued, the methods find a limit point with some minimizing properties. Simple examples show that the strength of the optimality conditions at a limit point depends not only on the algorithm, but also on the directions it uses and on the smoothness of the objective at the limit point in question. The contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the defining directions for the algorithm, and it gives previous results as corollaries. [ABSTRACT FROM AUTHOR]

Given a directed graph G and an arc weight function w : E(G) → R+, the maximum directed cut problem (max dicut) is that of finding a directed cut δ(X) with maximum total weight. In this paper we consider a version of max dicut--max dicut with given sizes of parts or max dicut with gsp--whose instance is that of max dicut plus a positive integer p, and it is required to find a directed cut δ(X) having maximum weight over all cuts δ(X) with |X| = p. Our main result is a 0.5-approximation algorithm for solving the problem. The algorithm is based on a tricky application of the pipage rounding technique developed in some earlier papers by two of the authors and a remarkable structural property of basic solutions to a linear relaxation. The property is that each component of any basic solution is an element of a set {0,δ, 1/2, 1 - δ, 1}, where δ is a constant that satisfies 0 < δ < 1/2 and is the same for all components. [ABSTRACT FROM AUTHOR]

In this paper we study the inversion of an analytic matrix valued function A(z). This problem can also be viewed as an analytic perturbation of the matrix A0 = A(0). We are mainly interested in the case where A0 is singular but A(z) has an inverse in some punctured disc around z = 0. It is known that A-1 (z) can be expanded as a Laurent series at the origin. The main purpose of this paper is to provide efficient computational procedures for the coefficients of this series. We demonstrate that the proposed algorithms are computationally superior to symbolic algebra when the order of the pole is small. [ABSTRACT FROM AUTHOR]

The implicitly shifted (bulge-chasing) QZ algorithm is the most popular method for solving the generalized eigenvalue problem Av = λBv. This paper explains why the QZ algorithm functions well even in the presence of infinite eigenvalues. The key to rapid convergence of QZ (and QR) algorithms is the effective transmission of shifts during the bulge chase. In this paper the mechanism of transmission of shifts is identified, and it is shown that this mechanism is not disrupted by the presence of infinite eigenvalues. Both the QZ algorithm and the preliminary reduction to Hessenberg-triangular form tend to push the infinite eigenvalues toward the top of the pencil. Thus they should be deflated at the top. [ABSTRACT FROM AUTHOR]

Dynamical systems can often be decomposed into loosely coupled subsystems. The system of ordinary differential equations (ODEs) modelling such a problem can then be partitioned corresponding to the subsystems, and the loose couplings can be exploited by special integration methods to solve the problem using a parallel computer or just solve the problem more efficiently than by standard methods. This paper presents accuracy analysis of methods for the numerical integration of stiff partitioned systems of ODEs. The discretization formulas are based on the implicit Euler formula and the second order implicit backward differentiation formula (BDF2). Each subsystem of the partitioned problem is discretized independently, and the couplings to the other subsystems are based on solution values from previous time steps. Applied this way, the discretization formulas are called decoupled. The stability properties of the decoupled implicit Euler formula are well understood. This paper presents error bounds and asymptotic error expansions to be used in controlling step size, relaxation between subsystems and the validity of the partitioning. The decoupled BDF2 formula is analyzed within the same framework. Finally, the analysis is used in the design of a decoupled numerical integration algorithm with variable step size to control the local error and adaptive selection of partitionings. Two versions of the algorithm with decoupled implicit Euler and BDF2, respectively, are used in examples where a realistic problem is solved. The examples compare the results from the decoupled implicit Euler and BDF2 formulas, and compare them with results from the corresponding classical formulas. [ABSTRACT FROM AUTHOR]

Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a well-known ill-posed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and complete the mathematical theory of these relationships and show how they may be applied to the staircase algorithm. This paper is a continuation of our Part I paper on versal deformations, but it may also be read independently. [ABSTRACT FROM AUTHOR]

This paper analyzes the process of merging groups of many Gaussian basis functions into a single basis function in vortex simulations. Analysis of the equations governing this process yields fundamental parameters and uniform estimates of the merger-induced error. In a merging event, the uniform error bound depends only on relationships between nearby computational elements permitting fast and effective merging schemes. In this paper, one such algorithm is proposed and demonstrated. [ABSTRACT FROM AUTHOR]

It is known that the First-Fit algorithm for partitioning a poset P into chains uses relatively few chains when P does not have two incomparable chains each of size k. In particular, if P has width w, then Bosek, Krawczyk, and Szczypka [SIAM J. Discrete Math., 23 (2010), pp. 1992-1999], proved an upper bound of ckw² on the number of chains used by First-Fit for some constant c, while Joret and Milans [Order, 28 (2011), pp. 455-464] gave one of ck²w. In this paper we prove an upper bound of the form ckw. This is most possible up to the value of c. [ABSTRACT FROM AUTHOR]

Cops and robber is a classical pursuit-evasion game on undirected graphs, where the task is to identify the minimum number of cops sufficient to catch the robber. In this paper, we investigate the changes in problem's complexity and combinatorial properties with constraining the following natural game parameters: fuel, the number of steps each cop can make; cost, the total sum of steps along edges all cops can make; and time, the number of rounds of the game. [ABSTRACT FROM AUTHOR]

The Nelder-Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalar-valued function f of n real variables using only function values, without any derivative information. Each Nelder-Mead iteration is associated with a nondegenerate simplex defined by n + 1 vertices and their function values; a typical iteration produces a new simplex by replacing the worst vertex by a new point. Despite the method's widespread use, theoretical results have been limited: for strictly convex objective functions of one variable with bounded level sets, the algorithm always converges to the minimizer; for such functions of two variables, the diameter of the simplex converges to zero but examples constructed by McKinnon show that the algorithm may converge to a nonminimizing point. This paper considers the restricted Nelder-Mead algorithm, a variant that does not allow expansion steps. In two dimensions we show that for any nondegenerate starting simplex and any twice-continuously differentiable function with positive definite Hessian and bounded level sets, the algorithm always converges to the minimizer. The proof is based on treating the method as a discrete dynamical system and relies on several techniques that are nonstandard in convergence proofs for unconstrained optimization. [ABSTRACT FROM AUTHOR]

In this paper, a method for solving constrained convex optimization problems is introduced. The problem is cast equivalently as a parametric unconstrained one, the (single) parameter being the optimal value of the original problem. At each stage of the algorithm the parameter is updated, and the resulting subproblem is only approximately solved. A linear rate of convergence of the parameter sequence is established. Using an optimal gradient method due to Nesterov [Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543 547] to solve the arising subproblems, it is proved that the resulting gradient-based algorithm requires an overall of O(log(1/ε)/√ε) inner iterations to obtain an ε-optimal and feasible solution. An image deblurring problem is solved, demonstrating the capability of the algorithm to solve large-scale problems within reasonable accuracy. [ABSTRACT FROM AUTHOR]

To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century, when Carl Friedrich Gauss developed the method of least squares of a system of linear equations--its solutions can be viewed as fixed points of averaged projections onto hyperplanes. A powerful generalization of this problem is to find fixed points of averaged resolvents (i.e., firmly nonexpansive mappings). This paper concerns the relationship between the set of fixed points of averaged resolvents and certain fixed point sets of compositions of resolvents. It partially extends recent work for two mappings on a question of C. Byrne. The analysis suggests a reformulation in a product space. Algorithmic consequences are also presented. [ABSTRACT FROM AUTHOR]

This paper develops an optimization-based theory for the existence and uniqueness of equilibria of a noncooperative game wherein the selfish players' optimization problems are nonconvex and there are side constraints and an associated price clearance to be satisfied by the equilibria. A new concept of equilibrium for such a noncouvex game, which we term a "quasi-Nash equilibrium" (QNE), is introduced as a solution of the variation inequality (VI) obtained by aggregating the first order optimality conditions of the players' problems while retaining the convex constraints (if any) ill the defining set of the VI. Under a second-order sufficiency condition from nonlinear programming, a QNE becomes a local Nash equilibrium of the game. Uniqueness of a QNE is established using a degree-theoretic proof. Under a key bounded ness property of the Karush-Kuhn-Tucker multipliers of the nonconvex constraints and the positive definiteness of the Hessians of the players' Lagrangian functions, we establish the single-valuedness of the players' best-response maps, from which the existence of a Nash equilibrium (NE) of the nonconvex game follows. We also present a distributed algorithm for computing an NE of such a game and provide a matrix-theoretic condition for the convergence of the algorithm. An application is presented that pertains to a special multi-leader follower game wherein the nonconvexity is due to the followers' equilibrium conditions in the leaders' optimization problems. Another application to a cognitive radio paradigm in a signal processing game is described in detail in [G. Scutari and J.S. Pang, IEEE Trans. Inform. Theory, submitted; J.S. Pang and G. Scutari, Joint IEEE Trans. Signal Process, submitted]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the half-integral disjoint paths packing. For a graph G and k pairs of vertices (s1, t1), (s2, t2), …, (sk, tk) in G, the objective is to find paths P1, …, Pk in G such that Pi joins si and ti for i = 1, 2, …, k, and in addition, each vertex is on at most two of these paths. We give a polynomial-time algorithm to decide the feasibility of this problem with k = O((log n/log log n)1/7). This improves a result by Kleinberg [Proceedings of the 30th ACM Symposium on Theory of Computing, 1998, pp 530-539] who proved the same conclusion when k = O((log log n)2/15). Our algorithm still works for several problems related to the bounded unsplittable flow. These results can all carry over to problems involving edge capacities. Our main technical contribution is to give a "crossbar" of a polynomial size of the tree width of the graph. [ABSTRACT FROM AUTHOR]

In the (k, λ)-subgraph problem, we are given an undirected graph G = (V, E) with edge costs and two positive integers k and λ, and the goal is to find a minimum cost simple λ-edge-connected subgraph of G with at least k nodes. This generalizes several classical problems, such as the minimum cost k-spanning tree problem, or k-MST (which is a (k, 1)-subgraph), and the minimum cost λ-edge-connected spanning subgraph (which is a (|V(G)|, λ-subgraph). The only previously known results on this problem [L. C. Lau, J. S. Naor, M. R. Salavatipour, and M. Singh, SIAM J. Comput., 39 (2009), pp. 1062-1087], [C. Chekuri and N. Korula, in Proceedings of the IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Bangalore, India, LIPIcs 2, Schloss Dagstuhl--Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2008, pp. 119-130] show that the (k, 2)-subgraph problem has an O(log² n)-approximation (even for 2-node-connectivity) and that the (k, λ)-subgraph problem in general is almost as hard as the densest k-subgraph problem. In this paper we show that if the edge costs are metric (i.e., satisfy the triangle in equality), like in the k-MST problem, then there is an O(1)-approximation algorithm for the (k, λ)-subgraph problem. This essentially generalizes the k-MST constant factor approximability to higher connectivity. [ABSTRACT FROM AUTHOR]

Compressed sensing (CS) seeks to recover an unknown vector with N entries by making far fewer than N measurements; it posits that the number of CS measurements should be comparable to the information content of tile vector, not simply N. CS combines directly the important task of compression with the measurement task. Since its introduction in 2004 there have been hundreds of papers on CS, a large fraction of which develop algorithms to recover a signal from its compressed measurements. Because of the paradoxical nature of CS--exact reconstruction from seemingly undersampled measurements--it is crucial for acceptance of an algorithm that rigorous analyses verify the degree of undersampling the algorithm permits. The restricted isometry property (RIP) has become the dominant tool used for the analysis in such cases. We present here an asymmetric form of RIP that gives tighter bounds than the usual symmetric one. We give the best known bounds on the RIP constants for matrices from the Gaussian ensemble. Our derivations illustrate the way in which the combinatorial nature of CS is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners. We also document the extent to which RIP gives precise information about the true performance limits of CS, by comparison with approaches from high-dimensional geometry. [ABSTRACT FROM AUTHOR]

In this paper, we give the first approximation algorithm for the problem of max-min fair allocation of indivisible goods. An instance of this problem consists of a set of k people and m indivisible goods. Each person has a known linear utility function over the set of goods which might be different from the utility functions of other people. The goal is to distribute the goods among the people and maximize the minimum utility received by them. The approximation ratio of our algorithm is Ω(1/√k log3 k). As a crucial part of our algorithm, we design and analyze an iterative method for rounding a fractional matching on a tree which might be of independent interest. We also provide better bounds when we are allowed to exclude a small fraction of the people from the problem. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a weighted undirected graph having nonnegative edge weights. An estimate δˆ(u, v) of the actual distance d(u, v) between u, v ∈ V is said to be of stretch t if and only if δ(u, v) ≤ δˆ(u, v) = t · δ(u, v). Computing all-pairs small stretch distances efficiently (both in terms of time and space) is a well-studied problem in graph algorithms. We present a simple, novel, and generic scheme for all-pairs approximate shortest paths. Using this scheme and some new ideas and tools, we design faster algorithms for all-pairs t-stretch distances for a whole range of stretch t, and we also answer an open question posed by Thorup and Zwick in their seminal paper [J. ACM, 52 (2005), pp. 1-24]. [ABSTRACT FROM AUTHOR]

This paper shows how to compute O(√log n)-approximations to the Sparsest Cut and Balanced Separator problems in Õ(n²) time, thus improving upon the recent algorithm of Arora, Rao, and Vazirani [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222-231]. Their algorithm uses semidefinite programming and requires Õ (n9.5) time. Our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs. The existence of expander flows was also established by Arora, Rao, and Vazirani [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222-231]. [ABSTRACT FROM AUTHOR]

We study compression that preserves the solution to an instance of a problem rather than preserving the instance itself. Our focus is on the compressibility of NP decision problems. We consider NP problems that have long instances but relatively short witnesses. The question is whether one can efficiently compress an instance and store a shorter representation that maintains the information of whether the original input is in the language or not. We want the length of the compressed instance to be polynomial in the length of the witness and polylog in the length of original input. Such compression enables succinctly storing instances until a future setting will allow solving them, either via a technological or algorithmic breakthrough or simply until enough time has elapsed. In this paper, we first develop the basic complexity theory of compression, including reducibility, completeness, and a stratification of NP with respect to compression. We then show that compressibility (say, of SAT) would have vast implications for cryptography, including constructions of one-way functions and collision resistant hash functions from any hard-on-average problem in NP and cryptanalysis of key agreement protocols in the "bounded storage model" when mixed with (time) complexity-based cryptography. [ABSTRACT FROM AUTHOR]

This paper studies the so-called biquadratic optimization over unit spheres Minx∈ℝn,y∈ℝm ∑1≤i,k≤n, 1≤j,l≤m bijklxiyjxkyl, subject to ||x|| = 1, ||y|| = 1. We show that this problem is NP-hard, and there is no polynomial time algorithm returning a positive relative approximation bound. Then, we present various approximation methods based on semidefinite programming (SDP) relaxations. Our theoretical results are as follows: For general biquadratic forms, we develop a 1/2max{m,n}² -approximation algorithm under a slightly weaker approximation notion; for biquadratic forms that are square-free, we give a relative approximation bound 1/nm; when min/{n,m} is a constant, we present two polynomial time approximation schemes (PTASs) which are based on sum of squares (SOS) relaxation hierarchy and grid sampling of the standard simplex. For practical computational purposes, we propose the first order SOS relaxation, a convex quadratic SDP relaxation, and a simple minimum eigenvalue method and show their error bounds. Some illustrative numerical examples are also included. [ABSTRACT FROM AUTHOR]

We study the maximum edge-disjoint paths problem in undirected planar graphs: given a graph G and node pairs (demands) s1t1, s2t2, … , sktk, the goal is to maximize the number of demands that can be connected (routed) by edge-disjoint paths. The natural multicommodity flow relaxation has an Ω(√n) integrality gap, where n is the number of nodes in G. Motivated by this, we consider solutions with small constant congestion c > 1, that is, solutions in which up to c paths are allowed to use an edge (alternatively, each edge has a capacity of c). In previous work we obtained an O(log n) approximation with congestion 2 via the flow relaxation. This was based on a method of decomposing into well-linked subproblems. In this paper we obtain an O(1) approximation with congestion 4. To obtain this improvement we develop an alternative decomposition that is specific to planar graphs. The decomposition produces instances that we call Okamura-Seymour (OS) instances. These have the property that all terminals lie on a single face. Another ingredient we develop is a constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation. [ABSTRACT FROM AUTHOR]

In this paper, an optimal linear-time algorithm is presented to solve the haplotype inference problem for pedigree data when there are no recombinations and the pedigree has no mating loops. The approach is based on the use of graphs to capture SNP, Mendelian, and parity constraints of the given pedigree. This representation allows us to capture the constraints as the edges in a graph, rather than as a system of linear equations as in previous approaches. Graph traversals are then used to resolve the parity of these edges, resulting in an optimal running time. [ABSTRACT FROM AUTHOR]

Suffix trees and suffix arrays are the most prominent full-text indices, and their construction algorithms are well studied. In the literature, the fastest algorithm runs in O(n) time, while it requires O(n log n)-bit working space, where n denotes the length of the text. On the other hand, the most space-efficient algorithm requires O(n)-bit working space while it runs in O(n log n) time. It was open whether these indices can be constructed in both o(n log n) time and o(n log n)- bit working space. This paper breaks the above time-and-space barrier under the unit-cost word RAM. We give an algorithm for constructing the suffix array, which takes O(n) time and O(n)-bit working space, for texts with constant-size alphabets. Note that both the time and the space bounds are optimal. For constructing the suffix tree, our algorithm requires O(n logϵ n) time and O(n)-bit working space for any 0 < ϵ < 1. Apart from that, our algorithm can also be adopted to build other existing full-text indices, such as compressed suffix tree, compressed suffix arrays, and FM-index. We also study the general case where the size of the alphabet Σ is not constant. Our algorithm can construct a suffix array and a suffix tree using optimal O(n log ∣Σ∣)-bit working space while running in O(n log log ∣Σ∣) time and O(n(logϵ n + log ∣Σ∣)) time, respectively. These are the first algorithms that achieve o(n log n) time with optimal working space. Moreover, for the special case where log ∣Σ∣ = O((log log n)1-ϵ), we can speed up our suffix array construction algorithm to the optimal O(n). [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of constructing a small arithmetic circuit for a polynomial for which we have oracle access. Our focus is on n-variate polynomials, over a finite field F, that have depth-3 arithmetic circuits (with an addition gate at the top) with two multiplication gates of degree at most d. We obtain the following results: 1. Multilinear case. When the circuit is multilinear (multiplication gates compute multilinear polynomials) we give an algorithm that outputs, with probability 1 - o(1), all the depth-3 circuits with two multiplication gates computing the polynomial. The running time of the algorithm is poly(n, ∣F∣). 2. General case. When the circuit is not multilinear we give a quasi-polynomial (in n, d, ∣F∣) time algorithm that outputs, with probability 1-o(1), a succinct representation of the polynomial. In particular, if the depth-3 circuit for the polynomial is not of small depth-3 rank (namely, after removing the g.c.d. (greatest common divisor) of the two multiplication gates, the remaining linear functions span a not too small linear space), then we output the depth-3 circuit itself. In the case that the rank is small we output a depth-3 circuit with a quasi-polynomial number of multiplication gates. ... Prior to our work there have been several interpolation algorithms for restricted models. However, all the techniques used there completely fail when dealing with depth-3 circuits with even just two multiplication gates. Our proof technique is new and relies on the factorization algorithm for multivariate black-box polynomials, on lower bounds on the length of linear locally decodable codes with two queries, and on a theorem regarding the structure of identically zero depth-3 circuits with four multiplication gates. [ABSTRACT FROM AUTHOR]

In this paper, based on Patel's algorithm (1993), we propose a structure-preserving algorithm for solving palindromic quadratic eigenvalue problems (QEPs). We also show the relationship between the structure-preserving algorithm and the URV-based structure-preserving algorithm by Schröder (2007). For large sparse palindromic QEPs, we develop a generalized T-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi algorithm for solving the resulting T-skew-Hamiltonian pencils. Numerical experiments show that our proposed structure-preserving algorithms perform well on the palindromic QEP arising from a finite element model of high-speed trains and rails. [ABSTRACT FROM AUTHOR]

The truncated SPIKE algorithm is a parallel solver for linear systems which are banded and strictly diagonally dominant by rows. There are machines for which the current implementation of the algorithm is faster and scales better than the corresponding solver in ScaLAPACK (PDDBTRF/PDDBTRS). In this paper we prove that the SPIKE matrix is strictly diagonally dominant by rows with a degree no less than the original matrix. We establish tight upper bounds on the decay rate of the spikes as well as the truncation error. We analyze the error of the method and present the results of some numerical experiments which show that the accuracy of the truncated SPIKE algorithm is comparable to LAPACK and ScaLAPACK. [ABSTRACT FROM AUTHOR]

Minimal elimination orderings were introduced by Rose, Tarjan, and Lueker in 1976, and during the last decade they have received increasing attention. Such orderings have important applications in several different fields, and they were first studied in connection with minimizing fill in sparse matrix computations. Rather than computing any minimal ordering, which might result in fill that is far from minimum, it is more desirable for practical applications to start from an ordering produced by a fill-reducing heuristic and then compute a minimal fill that is a subset of the fill produced by the given heuristic. This problem has been addressed previously, and there are several algorithms for solving it. The drawback of these algorithms is that either there is no theoretical bound given on their running time, although they might run fast in practice, or they have a good theoretical running time, but they have never been implemented, or they require a large machinery of complicated data structures to achieve the good theoretical time bound. In this paper, we present an algorithm called MCS-ETree for solving the mentioned problem in O(nm A(m, n)) time, where m and n are, respectively, the number of edges and vertices of the graph corresponding to the input sparse matrix and A(m, n) is the very slowly growing inverse of Ackerman's function. A primary strength of MCS-ETree is its simplicity and its straightforward implementation details. We present run time test results to show that our algorithm is fast in practice. Thus our algorithm is the first that both has a provably good running time with easy implementation details and is fast in practice. [ABSTRACT FROM AUTHOR]

It is becoming increasingly apparent that probabilistic approaches can overcome conservatism and computational complexity of the classical worst-case deterministic framework and may lead to designs that are actually safer. In this paper we argue that a comprehensive probabilistic robustness analysis requires a detailed evaluation of the robustness function, and we show that such an evaluation can be performed with essentially any desired accuracy and confidence using algorithms with complexity that is linear in the dimension of the uncertainty space. Moreover, we show that the average memory requirements of such algorithms are absolutely bounded and well within the capabilities of today's computers. In addition to efficiency, our approach permits control over statistical sampling error and the error due to discretization of the uncertainty radius. For a specific level of tolerance of the discretization error, our techniques provide an efficiency improvement upon conventional methods which is inversely proportional to the accuracy level; i.e., our algorithms get better as the demands for accuracy increase. [ABSTRACT FROM AUTHOR]

Recently lexicographic breadth first search (LexBFS) has been shown to be a very powerful tool for the development of linear time, easily implementable recognition algorithms for various families of graphs. In this paper, we add to this work by producing a simple two LexBFS sweep algorithm to recognize the family of cographs. This algorithm extends to other related graph families such as P4-reducible, P4-sparse, and distance hereditary. It is an open question whether our cograph recognition algorithm can be extended to a similarly easy algorithm for modular decomposition. [ABSTRACT FROM AUTHOR]

In this paper, we consider the unilateral frictional contact problem of a hyperelastic body in the case of large displacements and small strains. In order to retain the linear elasticity framework, we decompose the deformation into a large global rotation and a small elastic displacement. This corotational approach is combined with a primal-dual active set strategy to tackle the contact problem. The resulting algorithm preserves both energy and angular momentum. [ABSTRACT FROM AUTHOR]

In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard LU or LDLT factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the Sherman-Morrison formula [R. Bru, J. Cerdán, J. Marín, and J. Mas, SIAM J. Sci. Comput., 25 (2003), pp. 701-715]. In contrast to the robust incomplete decomposition (RIF) algorithm [M. Benzi and M. Tůma, Numer. Linear Algebra Appl., 10 (2003), pp. 385-400] the direct and inverse factors here directly influence each other throughout the computation. Consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. For the symmetric positive definite case, we derive the theory and present an algorithm for computing the incomplete LDLT factorization, and we discuss experimental results. We call this new approximate LDLT factorization the balanced incomplete factorization (BIF). Our experimental results confirm that this factorization is very robust and may be useful in solving difficult ill conditioned problems by preconditioned iterative methods. Moreover, the internal coupling of the computation of direct and inverse factors results in much shorter setup times (times to compute approximate decomposition) than RIF, a method of a similar and very high level of robustness. We also derive and present the theory for the general nonsymmetric case, but do not discuss its implementation. [ABSTRACT FROM AUTHOR]

In this paper we derive alternating least squares algorithms for the computation of the block term decompositions introduced in Part II. We show that degeneracy can also occur for block term decompositions. [ABSTRACT FROM AUTHOR]

We describe an asynchronous parallel derivative-free algorithm for linearly constrained optimization. Generating set search (GSS) is the basis of our method. At each iteration, a GSS algorithm computes a set of search directions and corresponding trial points and then evaluates the objective function value at each trial point. Asynchronous versions of the algorithm have been developed in the unconstrained and bound-constrained cases which allow the iterations to continue (and new trial points to be generated and evaluated) as soon as any other trial point completes. This enables better utilization of parallel resources and a reduction in overall run time, especially for problems where the objective function takes minutes or hours to compute. For linearly constrained GSS, the convergence theory requires that the set of search directions conforms to the nearby boundary. This creates an immediate obstacle for asynchronous methods where the definition of nearby is not well defined. In this paper, we develop an asynchronous linearly constrained GSS method that overcomes this difficulty and maintains the original convergence theory. We describe our implementation in detail, including how to avoid function evaluations by caching function values and using approximate lookups. We test our implementation on every CUTEr test problem with general linear constraints and up to 1000 variables. Without tuning to individual problems, our implementation was able to solve 95% of the test problems with 10 or fewer variables, 73% of the problems with 11-100 variables, and nearly half of the problems with 100-1000 variables. To the best of our knowledge, these are the best results that have ever been achieved with a derivative-free method for linearly constrained optimization. Our asynchronous parallel implementation is freely available as part of the APPSPACK software. [ABSTRACT FROM AUTHOR]

We discuss in this paper the numerical approximation of fluid-structure interaction (FSI) problems dealing with strong added-mass effect. We propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressure-structure velocity system at each iteration, reducing the computational cost. We investigate explicit-implicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluid-structure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a blood-vessel system. [ABSTRACT FROM AUTHOR]

Differential repetitive processes are a distinct class of continuous-discrete two-dimensional linear systems of both systems theoretic and applications interest. These processes complete a series of sweeps termed passes through a set of dynamics defined over a finite duration known as the pass length, and once the end is reached the process is reset to its starting position before the next pass begins. Moreover the output or pass profile produced on each pass explicitly contributes to the dynamics of the next one. Applications areas include iterative learning control and iterative solution algorithms, for classes of dynamic nonlinear optimal control problems based on the maximum principle, and the modeling of numerous industrial processes such as metal rolling, long-wall cutting, etc. In this paper we develop substantial new results on optimal control of these processes in the presence of constraints where the cost function and constraints are motivated by practical application of iterative learning control to robotic manipulators and other electromechanical systems. The analysis is based on generalizing the well-known maximum and ϵ-maximum principles to them. [ABSTRACT FROM AUTHOR]

In 1996 Ivkovič and Lloyd [A fundamental restriction on fully dynamic maintenance of bin packing, Inform. Process. Lett., 59 (1996), pp. 229-232] gave the lower bound 4/3 on the asymptotic worst-case ratio for so-called fully dynamic bin packing algorithms, where the number of repackable items in each step is restricted by a constant. In this paper we improve this result to about 1.3871. We present our proof for a semionline case of the classical bin packing, but it works for fully dynamic bin packing as well. We prove the lower bound by analyzing and solving a specific optimization problem. The bound can be expressed exactly using the Lambert W function. [ABSTRACT FROM AUTHOR]