On piecewise quadratic Newton and trust region problems

Some recent algorithms for nonsmooth optimization require solutions to certain piecewise quadratic programming subproblems. Two types of subproblems are considered in this paper. The first type seeks the minimization of a continuously differentiable and strictly convex piecewise quadratic function subject to linear equality constraints. We prove that a nonsmooth version of Newton’s method is globally and finitely convergent in this case. The second type involves the minimization of a possibly nonconvex and nondifferentiable piecewise quadratic function over a Euclidean ball. Characterizations of the global minimizer are studied under various conditions. The results extend a classical result on the trust region problem. Partially supported by National University of Singapore under grant 930033.     

GLOBAL CONVERGENCE OF QPFTH METHOD FOR LARGE-SCALE NONLINEAR SPARSE CONSTRAINED OPTIMIZATION

1.IntroductionIn[6],aQPFTHmethodwasproposedforsolvingthefollowingnonlinearprogrammingproblemwherefunctionsf:R"-- RIandgi:R"-- R',jeJaretwicecontinuouslydifferentiable.TheQPFTHalgorithmwasdevelopedforsolvingsparselarge-scaleproblem(l.l)andwastwo-stepQ-quadraticallyandR-quadraticallyconvergent(see[6]).Theglobalconvergenceofthisalgorithmisdiscussedindetailinthispaper.Forthefollowinginvestigationwerequiresomenotationsandassumptions.TheLagrangianofproblem(1.1)isdefinedbyFOundationofJiangs…     

Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation

We consider the Sylvester equation AX?XB+C=0 where the matrix C∈?^n×m is of low rank and the spectra of A∈?^n×n and B∈?^m×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε∈(0,1) there exists a matrix X? of rank k=O(log(1/ε)) such that ∥X?X?∥₂?ε∥X∥₂. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62 : 89–108). The blockwise rank of the approximation is again proportional to log(1/ε). Copyright © 2004 John Wiley & Sons, Ltd.     

实分片代数曲线的拓扑结构

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.By analyzing the topology of real algebraic curves on the triangles,a practi-caUy algrithm for analyzing the topology of piecewise algebraic curves is given.The algrithm produces a planar graph which is topologically equivalent to the piecewise algebraic curve.     

Incremental Norm Estimation for Dense and Sparse Matrices

We present an incremental approach to 2-norm estimation for triangular matrices. Our investigation covers both dense and sparse matrices which can arise for example from a QR, a Cholesky or a LU factorization. If the explicit inverse of a triangular factor is available, as in the case of an implicit version of the LU factorization, we can relate our results to incremental condition estimation (ICE). Incremental norm estimation (INE) extends directly from the dense to the sparse case without needing the modifications that are necessary for the sparse version of ICE. INE can be applied to complement ICE, since the product of the two estimates gives an estimate for the matrix condition number. Furthermore, when applied to matrix inverses, INE can be used as the basis of a rank-revealing factorization.     

A Novel Energy-from-Waste Approach for Electrical Energy Production by Galvano–Fenton Process

A novel approach allowing the production of electrical energy by an advanced oxidation process is proposed to eliminate organic micropollutants (MPs) in wastewaters. This approach is based on associating the Galvano–Fenton process to the generation of electrical power. In the previous studies describing the Galvano–Fenton (GF) process, iron was directly coupled to a metal of more positive potential to ensure degradation of organic pollutants without any possibility of producing electrical energy. In this new approach, the Galvano–Fenton process is constructed as an electrochemical cell with an external circuit allowing recovering electrons exchanged during the process. In this study, Malachite Green (MG) dye was used as a model of organic pollutant. Simultaneous MG degradation and electrical energy production with the GF method were investigated in batch process. The investigation of various design parameters emphasis that utilization of copper as a low-cost cathode material in the galvanic couple, provides the best treatment and electrical production performances. Moreover, these performances are improved by increasing the surface area of the cathode. The present work reveals that the GF process has a potential to provide an electrical power density of about 200 W m⁻². These interesting performances indicate that this novel Energy-from-Waste strategy of the GF process could serve as an ecological solution for wastewater treatment.     

Integrated electrochemical transistor as a fast recoverable gas sensor

A new design of conductometric chemical sensors based on conducting polymers as chemosensitive elements was suggested. The sensor includes six electrodes. Four inner electrodes coated by chemosensitive polymer are used for simultaneous two- and four-point resistance measurements thus providing information on the bulk polymer resistance and on the resistance of the polymer/electrode contacts. Two outer electrodes wired to inner electrodes by polymeric electrolyte are used for electrical control of redox state of the chemosensitive polymer. The outer electrodes are connected to potentiostat as reference and counter electrodes. It allows us to control redox state of the inner (working) electrodes. This new measurement configuration, resembling chemosensitive electrochemical transistors, provides an internal test of the sensor integrity and an electrically driven sensor regeneration. It was tested as a sensor for the detection of nitrogen dioxide. Polythiophene or polyaniline was used as receptors. Cyclic voltammograms of these polymers on the sensor surface measured in air atmosphere were very similar to that measured in aqueous electrolyte. A control of conductivity of these chemosensitive polymers by electrical potential applied vs. incorporated reference electrode was demonstrated. This effect was used for the regeneration of the chemosensitive material after exposure to nitrogen dioxide: in comparison to usual chemiresistors displaying an irreversible behavior in such test even in the time scale of hours, a completely reversible sensor regeneration within few minutes was observed.     

Determination of N-methylcarbamate insecticides in water samples using dispersive liquid–liquid microextraction and HPLC with the aid of experimental design and desirability function

A rapid and simple method for the extraction and preconcentration of N-methylcarbamates (NMCs) (carbofuran, carbaryl and promecarb) in water samples using dispersive liquid–liquid microextraction (DLLME) using chemometrics was developed. Influence variables such as volume of extracting (CHCl₃) and dispersing solvents (ACN), pH and ionic strength, extraction time and centrifugation time and speed were screened in a 2^7–4 Plackett–Burman design was investigated. The significant variables were optimized by using a central composite design (CCD) combined with desirability function (DF). At optimum conditions values of variables set as 126 μL chloroform, 1.5 mL acetonitrile, 1 min extraction time, 10 min centrifugation at 4000 rpm min⁻¹, natural pH, 4.7% (w/v) NaCl, the separation was reached in less than 14 min using a C₁₈ column and an isocratic binary mobile phase (acetonitrile: water (50:50, v/v)) with flow rate of 1.0 mL min⁻¹. At optimum conditions method has linear response over 0.001–10 μg mL⁻¹ with detection limit between 0.0001 and 0.0005 μg mL⁻¹ with relative standard deviations (RSDs) in the range 2.18–5.06% (n = 6).     

Piecewise testable languages via combinatorics on words

A regular language L over an alphabet A is called piecewise testable if it is a finite Boolean combination of languages of the form A^∗a₁A^∗a₂A^∗…A^∗a_?A^∗, where a₁,…,a_?∈A, ?≥0. An effective characterization of piecewise testable languages was given in 1972 by Simon who proved that a language L is piecewise testable if and only if its syntactic monoid is J-trivial. Nowadays, there exist several proofs of this result based on various methods from algebraic theory of regular languages. Our contribution adds a new purely combinatorial proof.     

A piecewise polynomial approach to analyzing interpolatory subdivision

The four-point interpolatory subdivision scheme of Dubuc and its generalizations to irregularly spaced data studied by Warren and by Daubechies, Guskov, and Sweldens are based on fitting cubic polynomials locally. In this paper, we analyze the convergence of the scheme by viewing the limit function as the limit of piecewise cubic functions arising from the scheme. This allows us to recover the regularity results of Daubechies et al. in a simpler way and to obtain the approximation order of the scheme and its first derivative.