Reformulation of variational inequalities on a simplex and compactification of complementarity problems

Many variational inequality problems (VIPs) can be reduced, by a compactification procedure, to a VIP on the canonical simplex. Reformulations of this problem are studied, including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized Fischer-Burmeister function. It is proved that bounded level set results hold for these reformulations under quite general assumptions on the operator. Therefore, it can be guaranteed that minimization algorithms generate bounded sequences and, under monotonicity conditions, these algorithms necessarily nd solutions of the original problem. Some numerical experiments are presented.

Bound constrained smooth optimization for solving variational inequalities and related problems

Variational inequalities and related problems may be solved via smooth bound constrained optimization. A comprehensive discussion of the important features involved with this strategy is presented. Complementarity problems and mathematical programming problems with equilibrium constraints are included in this report. Numerical experiments are commented. Conclusions and directions of future research are indicated.

On the regularization of mixed complementarity problems

A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MCP). Monotonicity of the VIP implies that the MCP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable inexact solutions of the strictly monotone MCP's is bounded and every accumulation point is a solution. Under an additional condition on the precision used for solving each subproblem, the sequence converges to the minimum norm solution of the MCP. Copyright © 2000 by Marcel Dekker, Inc.

On the solution of bounded and unbounded mixed complementarity problems

A reformulation of the bounded mixed complementarity problem is introduced. It is proved that the level sets of the objective function are bounded and, under reasonable assumptions, stationary points coincide with solutions of the original variational inequality problem. Therefore, standard minimization algorithms applied to the new reformulation must succeed. This result is applied to the compactification of unbounded mixed complementarity problems. © 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint, a member of the Taylor & Francis Group.

Design of a control system using linear matrix inequalities for the active vibration control of a plate

The study of algorithms for active vibration control in smart structures is an area of interest, mainly due to the demand for better performance of mechanical systems, such as aircraft and aerospace structures. Smart structures, formed using actuators and sensors, can improve the dynamic performance with the application of several kinds of controllers. This article describes the application of a technique based on linear matrix inequalities (LMI) to design an active control system. The positioning of the actuators, the design of a robust state feedback controller and the design of an observer are all achieved using LMI. The following are considered in the controller design: limited actuator input, bounded output (energy) and robustness to parametric uncertainties. Active vibration control of a flat plate is chosen as an application example. The model is identified using experimental data by an eigensystem realization algorithm (ERA) and the placement of the two piezoelectric actuators and single sensor is determined using a finite element model (FEM) and an optimization procedure. A robust controller for active damping is designed using an LMI framework, and a reduced model with observation and control spillover effects is implemented using a computer. The simulation results demonstrate the efficacy of the approach, and show that the control system increases the damping in some of the modes.

Landau and Kolmogoroff type polynomial inequalities

Let 0inequalitiesparallel to f((f))parallel to(2) less than or equal to A parallel to f(m)parallel to + B parallel to f parallel to/ A theta(k) + B mu(k),with certain constants theta(k)e mu(k), which hold for every f is an element of pi(n) (pi(n) denotes the space of real algebraic polynomials of degree not exceeding n).For the particular case j=1 and m=2, we provide a complete characterisation of the positive constants A and B, for which the corresponding Landau type polynomial inequalities parallel to f'parallel to less than or equal toA parallel to f parallel to + B parallel to f parallel to/ A theta(k) + B mu(k)hold. In each case we determine the corresponding extremal polynomials for which equalities are attained.

Inequalities for zeros of associated polynomials and derivatives of orthogonal polynomials

It is well known and easy to see that the zeros of both the associated polynomial and the derivative of an orthogonal polynomial p(n)(x) interlace with the zeros of p(n)(x) itself. The natural question of how these zeros interlace is under discussion. We give a sufficient condition for the mutual location of kth, 1 less than or equal to k less than or equal to n - 1, zeros of the associated polynomial and the derivative of an orthogonal polynomial in terms of inequalities for the corresponding Cotes numbers. Applications to the zeros of the associated polynomials and the derivatives of the classical orthogonal polynomials are provided. Various inequalities for zeros of higher order associated polynomials and higher order derivatives of orthogonal polynomials are proved. The results involve both classical and discrete orthogonal polynomials, where, in the discrete case, the differential operator is substituted by the difference operator. (C) 2001 IMACS. Published by Elsevier B.V. B.V. All rights reserved.

Characterization of generalized Bessel polynomials in terms of polynomial inequalities

Generalized Bessel polynomials (GBPs) are characterized as the extremal polynomials in certain inequalities in L-2 norm of Markov type. (C) 1998 Academic Press.

INEQUALITIES FOR ZEROS of JACOBI POLYNOMIALS VIA OBRECHKOFF'S THEOREM

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Confined Lennard-Jones potential: a variational treatment

In this work, the energy eigenvalues for the confined Lennard-Jones potential are calculated through the Variational Method allied to the Super symmetric Quantum Mechanics. Numerical results are obtained for different energy levels, parameters of the potential and values of confinement radius. In the limit, where this radius assumes great values, the results for the non-confined case are recovered..

Variational supersymmetric approach to evaluate Fokker-Planck probability

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Convergent variational calculation of positronium-hydrogen-atom scattering lengths

We present a convergent variational basis-set calculational scheme for elastic scattering of the positronium atom by the hydrogen atom in S wave. Highly correlated trial functions with appropriate symmetry are needed to achieve convergence. We report convergent results for scattering lengths in atomic units for both singlet (= 3.49 +/-0.20) and triplet (= 2.46 +/-0.10) states.

Variational fits to the lattice energy of heavy-light mesons

We perform variational calculations of heavy-light meson masses using a fitted formula to a lattice two-quark potential. We examine the light quark mass dependence of the meson mass using the Schrodinger equation and the Dirac equation. For the Dirac equation, a saddle-point variational principle is employed, since the Dirac Hamiltonian is not bound from below.

Variational Formulation for Quaternionic Quantum Mechanics

A quaternionic version of Quantum Mechanics is constructed using the Schwinger's formulation based on measurements and a Variational Principle. Commutation relations and evolution equations are provided, and the results are compared with other formulations.

Numerical and variational solutions of the dipolar Gross-Pitaevskii equation in reduced dimensions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)