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This article presents a two-stage stochastic equilibrium problem with equilibrium constraints (SEPEC) model. Some source problems which motivate the model are discussed. Monte Carlo sampling method is applied to solve the SEPEC. Convergence analysis on the statistical estimators of Nash equilibria and Nash stationary points are presented. [ABSTRACT FROM AUTHOR]

This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainty and nonsmoothness. The well-known sample average approximation method is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke generalized gradients. Under some moderate conditions, it is shown that with probability one, a statistical estimator (a Nash equilibrium or a Nash-C-stationary point) obtained from sample average approximate equilibrium problem converges to its true counterpart. Moreover, under some calmness conditions of the Clarke generalized derivatives, it is shown that with probability approaching one exponentially fast by increasing sample size, the Nash-C-stationary point converges to a weak Nash-C-stationary point of the true problem. Finally, the model is applied to stochastic Nash equilibrium problem in the wholesale electricity market. [ABSTRACT FROM AUTHOR]

Stochastic programming is a well-known instrument to model many risk management problems in finance. In this paper we consider a stochastic programming model where the objective function is the variance of a random function and the constraint function is the expected value of the random function. Instead of using popular scenario tree methods, we apply the well-known sample average approximation (SAA) method to solve it. An advantage of SAA is that it can be implemented without knowing the distribution of the random data. We investigate the asymptotic properties of statistical estimators obtained from the SAA problem including examining the rate of convergence of optimal solutions of the SAA problem as sample size increases. By using the classical penalty function technique and recent results on uniform exponential convergence of sample average random functions, we show that under some mild conditions the statistical estimator of the optimal solution converges to its true counterpart at an exponential rate. We apply the proposed model and the numerical method to a portfolio management problem and present some numerical results. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Padé approximation method introduced in to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies. The problem of reconstruction of the Stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite, is used to characterize its geometric structure. We present two different proofs of the Stieltjes analytical representation for the effective complex permittivity in the 3D Bruggeman effective medium model: one proof is based on direct calculation, the other one is the derivation of the representation using Stieltjes inversion formula. We show that the continuous spectral density in the integral representation for the Bruggeman EMA model can be efficiently approximated by a rational function. A rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition. We show results of numerical rational approximation of Bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium. The volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values. [Copyright &y& Elsevier]

Abstract: The paper deals with the problem of reconstruction of microstructural information from known effective complex permittivity of a composite material. A numerical method for recovering geometric information from measurements of frequency dependent effective complex permittivity is developed based on Stieltjes analytic representation of the effective permittivity tensor of a two-component mixture. We derive the Stieltjes representation for the effective permittivity of the medium using the eigenfunction expansion of the solution of a boundary-value problem. The spectral function in this representation contains all information about the microgeometry of the mixture. A discrete approximation of the spectral measure is derived from a rational (Padé) approximation followed by its partial fractions decomposition. The approach is based on the least squares minimization with regularization constraints provided by the spectral properties of the operator. The method is applied to calculation of volume fractions of the components in a mixture of two materials in a Bruggeman effective medium analytic model which has a continuous spectral density and to analytical models of two-phase composites with coated cylindrical and ellipsoidal inclusions. The numerical results of reconstruction of spectral measure for a mixture of silver and silicon dioxide and a composite of magnesium and magnesium fluoride show good agreement between theoretical and predicted values. The approach is applicable to geological materials, biocomposites, porous media, etc. [Copyright &y& Elsevier]

Inspired by a recent work by Alexander et al. (J Bank Finance 30:583–605, 2006) which proposes a smoothing method to deal with nonsmoothness in a conditional value-at-risk problem, we consider a smoothing scheme for a general class of nonsmooth stochastic problems. Assuming that a smoothed problem is solved by a sample average approximation method, we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w.p.1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem. Moreover, under some metric regularity conditions, we obtain an error bound on approximate stationary points. The convergence result is applied to a conditional value-at-risk problem and an inventory control problem. [ABSTRACT FROM AUTHOR]

The article presents a numerical method developed for identification of information about structural parameters of a two-component mixture from effective complex permittivity measurements. The identification is based on the reconstruction of the spectral function in the analytic Stieltjes representation of the effective permittivity using Pade approximation. The spectral function contains all information about the microgeometry of the mixture, it is used to calculate volume fractions of the components in the mixture. Pade approximation is derived from a constrained minimization problem. Numerical results of recovering volume fraction of air in mixtures of air prolate and oblate spheroidal inclusions in water and in ethanol show good agreement of theoretical and predicted values. The proposed method can be used for estimating volume fractions and other structural parameters using the effective complex permittivity of two-component composite materials. [ABSTRACT FROM AUTHOR]