On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian*

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. *Partially Supported by 2003 UniSA ITEE Small Research Grant Ero2. Supported by CAPES, Brazil, Grant No. 0664-02/2, during her visit to the School of Mathematics and Statistics, UniSA.     

Augmented Lagrangian Duality and Nondifferentiable Optimization Methods in Nonconvex Programming

In this paper we present augmented Lagrangians for nonconvex minimization problems with equality constraints. We construct a dual problem with respect to the presented here Lagrangian, give the saddle point optimality conditions and obtain strong duality results. We use these results and modify the subgradient and cutting plane methods for solving the dual problem constructed. Algorithms proposed in this paper have some advantages. We do not use any convexity and differentiability conditions, and show that the dual problem is always concave regardless of properties the primal problem satisfies. The subgradient of the dual function along which its value increases is calculated without solving any additional problem. In contrast with the penalty or multiplier methods, for improving the value of the dual function, one need not to take the penalty like parameter to infinity in the new methods. In both methods the value of the dual function strongly increases at each iteration. In the contrast, by using the primal-dual gap, the proposed algorithms possess a natural stopping criteria. The convergence theorem for the subgradient method is also presented.     

Singularly Perturbed Switching Diffusions: Rapid Switchings and Fast Diffusions

Motivated by many problems in optimization and control, this paper is concerned with singularly perturbed systems involving both diffusions and pure jump processes. Two models are treated. In the first model, the jump process changes very rapidly by comparison with the diffusion processes. In the second model, the diffusions change rapidly in comparison with the jump process. Asymptotic expansions are developed for the transition density vectors via a constructive method; justification of the asymptotic expansions and analysis of the remainders are provided.     

Semidilute solution structure of cellulose in an ionic liquid and its mixture with a polar organic co‐solvent studied by small‐angle X‐ray scattering

Structural and thermodynamic properties of cellulose solutions in the ionic liquid 1‐ethyl‐3‐methylimidazolium acetate (EMIMAc) and its binary mixtures with N,N‐dimethyl formamide (DMF) are studied by small‐angle X‐ray scattering (SAXS). These measurements indicate molecular dissolution of the cellulose chains without any significant aggregation. The power–law relationships of the evaluated correlation length and osmotic modulus to concentration exhibit exponents of ?0.76 and 2.06 for EMIMAc and ?0.80 and 2.14 for DMF/EMIMAc solvent mixture, respectively. Thus, these solvents can be considered to be good solvents for cellulose. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55 , 888–894     

The Linear-Array Conjecture in Communication Complexity Is False

linear array network consists of k+1 processors with links only between and (0≤i<k). It is required to compute some boolean function f(x,y) in this network, where initially x is stored at and y is stored at . Let be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, , where D(f) is the standard two-party communication complexity of f. Tiwari proved that for almost all functions and conjectured that this is true for all functions. In this paper we disprove Tiwari's conjecture, by exhibiting an infinite family of functions for which is essentially at most . Our construction also leads to progress on another major problem in this area: It is easy to bound the two-party communication complexity of any function, given the least number of monochromatic rectangles in any partition of the input space. How tight are such bounds? We exhibit certain functions, for which the (two-party) communication complexity is twice as large as the best lower bound obtainable this way. Received: March 1, 1996     

Turbulent Energy Spectrum via an Interaction Potential

For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier–Stokes equations, because they are destroyed by the Bogoliubov–Born–Green–Kirkwood–Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas–Fermi interatomic potential (as well as its variations, such as the Ziegler–Biersack–Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas–Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas–Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling.

     

Composite particles of polyethylene @ silica

Polyethylene (PE) and silica are perhaps the simplest and most common organic and inorganic polymers, respectively. We describe, for the first time, a physically interpenetrating nanocomposite between these two elementary polymers. While polymer-silica composites are well known, the nanometric physical blending of PE and silica has remained a challenge. A method for the preparation of such materials, which is based on the entrapment of dissolved PE in a polymerizing tetraethoxysilane (TEOS) system, has been developed. Specifically, the preparation of submicron particles of low-density PE@silica and high-density PE@silica is detailed, which is based on carrying out a silica sol-gel polycondensation process within emulsion droplets of TEOS dissolved PE, at elevated temperatures. The key to the successful preparation of this new composite has been the identification of a surfactant, PE-b-PEG, that is capable of stabilizing the emulsion and promoting the dissolution of the PE. A mechanism for the formation of the particles as well as their inner structure are proposed, based on a large battery of analyses, including transmission electron microscopy (TEM) and scanning electron microscopies (SEM), surface area and porosity analyses, various thermal analyses including thermal gravimetric analysis (TGA/DTA) and differential scanning calorimetry (DSC) measurements, small-angle X-ray scattering (SAXS) measurements and solid-state NMR spectroscopy.     

Statistical approach to some ill-posed problems for linear partial differential equations

For linear partial differential equations, some inverse source problems are treated statistically based on nonparametric estimation ideas. By observing the solution in a small Gaussian white noise, the kernel type of estimators is used to estimate the unknown source function and its partial derivatives.. It is proved that such estimators are consistent as the noise intensity tends to zero. Depending on the principal part of the differential operator, the optimal asymptotic rate of convergence is ascertained within a wide class of risk functions in a minimax sense. Received: 5 May 1997 / Revised version: 18 June 1998