Variable Transformation Technique Research Articles

Determining The Objective Value Range For A Class Of Interval Convex Optimization Problems

This paper generalizes a method of determining the objective value range of quadratic programming problems to a general class of interval convex programming ones, where all coefficients in objective function and constraints are interval numbers. The upper bound and lower bound of the objective values of the interval quadratic program is calculated by formulating a pair of two-level mathematical programs. Based on the duality theorem and by applying the variable transformation technique, the pair of two-level mathematical programs is transformed into conventional one-level convex programming problem. Solving the pair of convex programs produces the interval of the objective values of the problem. Numerical results confirms the procedure of the presented approach.

A fuzzy modeling for fuzzy portfolio optimization

Conventional portfolio optimization models have an assumption that the future condition of stock market can be accurately predicted by historical data. However, no matter how accurate the past data is, this premise will not exist in the financial market due to the high volatility of market environment. This paper discusses the fuzzy portfolio optimization problem where the asset returns are represented by fuzzy data. A mean-absolute deviation risk function model and Zadeh’s extension principle are utilized for the solution method of portfolio optimization problem with fuzzy returns. Since the parameters are fuzzy numbers, the gain of return is a fuzzy number as well. A pair of two-level mathematical programs is formulated to calculate the upper bound and lower bound of the return of the portfolio optimization problem. Based on the duality theorem and by applying the variable transformation technique, the pair of two-level mathematical programs is transformed into a pair of ordinary one-level linear programs so they can be manipulated. It is found that the calculated results conform to an essential idea in finance and economics that the greater the amount of risk that an investor is willing to take on, the greater the potential return. An example, which utilizes the data from Taiwan stock exchange corporation, illustrates the whole idea on fuzzy portfolio optimization problem.

The mean-absolute deviation portfolio selection problem with interval-valued returns

In real-world investments, one may care more about the future earnings than the current earnings of the assets. This paper discusses the uncertain portfolio selection problem where the asset returns are represented by interval data. Since the parameters are interval valued, the gain of returns is interval valued as well. According to the concept of the mean-absolute deviation function, we construct a pair of two-level mathematical programming models to calculate the lower and upper bounds of the investment return of the portfolio selection problem. Using the duality theorem and applying the variable transformation technique, the pair of two-level mathematical programs is transformed into a conventional one-level mathematical program. Solving the pair of mathematical programs produces the interval of the portfolio return of the problem. The calculated results conform to an essential idea in finance and economics that the greater the amount of risk that an investor is willing to take on the greater the potential return.

A developed solution of the stochastic Milne problem using probabilistic transformations

This paper considers the solution of the Milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium. Properties of the medium are assumed to be continuous random functions of the spatial dimensions. The available solutions – in literature – for this stochastic integro-differential equation (SIDE) are represented only by the ensemble average of the radiant energy density. In this paper, a developed algorithm, based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties, is introduced. A complete stochastic solution represented by the probability-density function (p.d.f) of the radiant energy density is obtained. Using the closed form of the p.d.f, the nth moment of the stochastic solution is evaluated. In realization of this work, Exponential and Gaussian statistics for the medium properties are assumed. Results are physically acceptable and found to be compatible with those in the literature.

Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique

Much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation (SDE). The Random Variable Transformation technique (RVT) is a powerful technique to get the complete solution for the SDE represented by the probability-density function of the solution process. In this paper, the RVT technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one-speed transport equation for neutral particles in a semi-infinite stochastic medium with linear anisotropic scattering. The extinction function of the medium (input stochastic process) is assumed to be a continuous random function of position. The probability-density function and hence, the higher order statistical moments of the solution process are presented. Numerical results are given for different distributions of the input stochastic process.

Nationwide Freight Generation Models: A Spatial Regression Approach

This paper investigates the application of linear regression models and modeling techniques in predicting freight generation at the national level within the U.S. Specifically, the paper seeks to improve the performance and fit of linear regression models of freight generation. We provide insight into different variable transformation techniques, evaluate the use of spatial regression variables, and apply a spatial regression modeling methodology to correct for spatial autocorrelation. We conclude that the spatial regression model is the preferred specification for freight generation at the national level. The proliferation of Geographic Information Systems (GIS) within planning agencies affords more widespread use of spatial regression and our results indicate this technique would provide improvement to models that have been traditionally limited by insufficient data.

A revisit to quadratic programming with fuzzy parameters

Quadratic programming has been widely applied to solving real-world problems. Recently, Liu describes a solution method for solving a class of fuzzy quadratic programming problems, where the cost coefficients of the linear terms in objective function, constraint coefficients, and right-hand sides are fuzzy numbers [Liu ST. Quadratic programming with fuzzy parameters: a membership function approach. Chaos, Solitons & Fractals 2009;40:237–45]. In this paper, we generalize Liu’s method to a more general fuzzy quadratic programming problem, where the cost coefficients in objective function, constraint coefficients, and right-hand sides are all fuzzy numbers. A pair of two-level mathematical programs is formulated to calculate the upper bound and lower bound of the objective values of the fuzzy quadratic program. Based on the duality theorem and by applying the variable transformation technique, the pair of two-level mathematical programs is transformed into a family of conventional one-level quadratic programs. Solving the pair of quadratic programs produces the fuzzy objective values of the problem. With the ability of calculating the fuzzy objective value developed in this paper, it might help initiate wider applications.

Global solution of optimization problems with signomial parts

In this paper a new approach for the global solution of nonconvex MINLP (Mixed Integer NonLinear Programming) problems that contain signomial (generalized geometric) expressions is proposed and illustrated. By applying different variable transformation techniques and a discretization scheme a lower bounding convex MINLP problem can be derived. The convexified MINLP problem can be solved with standard methods. The key element in this approach is that all transformations are applied termwise. In this way all convex parts of the problem are left unaffected by the transformations. The method is illustrated by four example problems.

Quadratic programming with fuzzy parameters: A membership function approach

Quadratic programming has been widely applied to solving real world problems. The conventional quadratic programming model requires the parameters to be known constants. In the real world, however, the parameters are seldom known exactly and have to be estimated. This paper discusses the fuzzy quadratic programming problems where the cost coefficients, constraint coefficients, and right-hand sides are represented by convex fuzzy numbers. Since the parameters in the program are fuzzy numbers, the derived objective value is a fuzzy number as well. Using Zadeh’s extension principle, a pair of two-level mathematical programs is formulated to calculate the upper bound and lower bound of the objective values of the fuzzy quadratic program. Based on the duality theorem and by applying the variable transformation technique, the pair of two-level mathematical programs is transformed into a family of conventional one-level quadratic programs. Solving the pair of quadratic programs produces the fuzzy objective values of the problem. An example illustrates method proposed in this paper.

Exact solutions for axisymmetric bending of micro/nanoscale circular plates based onnonlocal plate theory

Axisymmetric bending of micro/nanoscale circular plates is studied using a nonlocal platetheory. The nonlocal theory allows for small scale effects. The governing equations andboundary conditions are derived for the aforementioned problem. By using a variabletransformation technique, exact nonlocal solutions for axisymmetric bending of circularplates under general loading are obtained. A detailed examination of the effects of smallscale on nonlocal solutions is carried out using uniformly loaded circular plates with eitherclamped or simply-supported edges. When compared with local plate theory, the nonlocalsolutions show larger deflections, moments and shear force and lower bendingstiffness.

Noncanonical Poisson brackets for elastic and micromorphic solids

This paper investigates the Lagrangian-to-Eulerian transformation approach to the construction of noncanonical Poisson brackets for the conservative part of elastic solids and micromorphic elastic solids. The Dirac delta function links Lagrangian canonical variables and Eulerian state variables, producing noncanonical Poisson brackets from the corresponding canonical brackets. Specifying the Hamiltonian functionals generates the evolution equations for these state variables from the Poisson brackets. Different elastic strain tensors, such as the Green deformation tensor, the Cauchy deformation tensor, and the higher-order deformation tensor, are appropriate state variables in Poisson bracket formalism since they are quantities composed of the deformation gradient. This paper also considers deformable directors to comprise the three elastic strain density measures for micromorphic solids. Furthermore, the technique of variable transformation is also discussed when a state variable is not conserved along with the motion of the body.

A numerical solution method to interval quadratic programming

Quadratic programming has been widely applied to solving real world problems. The conventional quadratic programming model requires the parameters to be known constants. In the real world, however, the parameters are seldom known exactly and have to be estimated. This paper discusses the interval quadratic programming problems where the cost coefficients, constraint coefficients, and right-hand sides, are represented by interval data. Since the parameters are interval-valued, the objective value is interval-valued as well. A pair of two-level mathematical programs is formulated to calculate the upper bound and lower bound of the objective values of the interval quadratic program. Based on the duality theorem and by applying the variable transformation technique, the pair of two-level mathematical programs is transformed into conventional one-level quadratic program. Solving the pair of quadratic programs produces the interval of the objective values of the problem. An example illustrates the whole idea and sheds some light on interval quadratic programming.

Using FEM–RVT technique for solving a randomly excited ordinary differential equation with a random operator

The technique of stochastic finite element (SFEM) which is the finite element technique FEM adapted to stochastic problems can be re-described to use random variable transformation technique RVT. A new FEM–RVT technique was successfully used in solving stochastic problems with random excitation [M. El-Tawil, W. El-Tahhan, A. Hussein, A proposed technique of SFEM on solving ordinary random differential equation, J. Appl. Math. Comput. 161 (2005) 35–47]. In this paper, the technique is adapted to solve a randomly excited differential equation with a random operator. The technique shows high accuracy when solving a case study compared with the exact solution. Finally a problem with unknown exact solution is solved using this technique.

Comparison study of parameter estimation techniques for rock failure criterion models

This paper presents a comparative study of parameter estimation of rock failure criteria using different statistical methods, such as the least squares, least median squares, and reweighted least squares. The Mises–Schleicher and Drucker–Prager unified (MSDPu) failure criterion, a nonlinear polyaxial failure criterion suitable for different rock strength data, has been considered for this study. The procedure for determining model parameters from scattered data using the least median squares method is presented, and the methods of identifying the scattered data are discussed. The use of the reweighted least squares method for improving statistical performance of model parameter estimation from scattered data is presented. Using the variable transformation technique, the parameter estimation problem has been formulated as an unconstrained minimization problem. Here, both traditional nonlinear programming techniques and recently developed evolutionary optimization algorithms have been employed, and a comparative study of their relative efficacy is presented. Of several traditional nonlinear unconstrained minimization techniques, only the Hooke–Jeeves method could be used to find the optimum value, but its robustness is much lower than that of evolutionary algorithms. Evolutionary algorithms can be applied to find the optimal solution for cases with a high degree of robustness.Key words: rock mechanics, failure criterion, parameter determination, least median squares, evolutionary optimization.

Integrating product and process design with optimal control: A case study of solvent recycling

This paper proposes the simultaneous integration of environmentally benign solvent selection (product design), solvent recycling (process design) and optimal control for the separation of azeotropic systems using batch distillation. The previous work performed by Kim et al. (2004. Entrainer selection and solvent recycling in complex batch distillation. Chemical Engineering Communications 191(12), 1606–1633) combines the chemical synthesis and process synthesis under uncertainty. For batch distillation, optimal operation is also important due to the unsteady state nature of the process and high operating costs. Optimal control allows us to optimize the column operating policy by selecting a trajectory for the reflux ratio. However, there are time-dependent uncertainties in thermodynamic models of batch distillation due to the assumption of constant relative volatility. In this paper, the uncertainties in relative volatility were modeled using Ito processes and the stochastic optimal control problem was solved by combined maximum principle and non-linear programming (NLP) techniques. Then the previous work of optimal solvent selection and recycling was coupled with optimal control. As a real world example for this integrated approach, a waste stream containing acetonitrile–water was studied. The optimal design parameters obtained by Kim et al. (2004. Entrainer selection and solvent recycling in complex batch distillation. Chemical Engineering Communications 191(12), 1606–1633), for this separation were used and the optimal control policy is computed first without considering uncertainties by variable transformation technique. The deterministic optimal control policy improves the product yield by 4.0% as compared to the base case, verified using a rigorous simulator for batch distillation. When the stochastic optimal control policy was computed representing the relative volatility as an Ito process, a similar recovery rate was obtained from simulations, but the batch time was reduced significantly, producing the most profitable operation.

A note on the extended Blasius equation

In a recent paper, the Blasius equation was extended to a nonlinear equation like a f ‴ + f f ″ = 0 with the prime denoting differentiation with respect to the similarity variable η and a being a constant parameter. The current note will show that the solution of the extended Blasius equation can be obtained from the original Blasius equation solution with a variable transformation technique. The observed phenomena in numerical solutions of previous published work are theoretically analyzed. The equation is also discussed for an arbitrary real parameter or complex parameters. It is further shown that the extended Blasius equation is a special form of the similarity equation of momentum boundary layers over a flat plate with a temperature dependent property.

Magnesium-induced vasodilation in the dorsal hand vein.

Magnesium affects blood pressure by modulating vascular tone and reactivity. In obstetric patients, magnesium is administered to prevent eclamptic seizures and as a tocolytic to treat preterm labour. Prior to studying vascular sensitivity in women with pre-eclampsia, we sought to determine the effect of magnesium on venous tone in healthy women of childbearing age. Dose-response study. Columbia-Presbyterian Medical Center, New York. Ten healthy non-pregnant women (age range 21-47 years). Vascular response to magnesium sulphate (MgSO4) was measured in a dorsal hand vein using the linear variable differential transformer (LVDT) technique. Complete dose-response curves to MgSO4 (0.0000275-0.088 mmol/min) were determined after 50% preconstriction of the vein with phenylephrine. The ED50 of MgSO4 (dose resulting in 50% of maximal effect, Emax) was determined. Blood samples from the contralateral upper extremity were obtained to assess total plasma magnesium levels at baseline and at the highest infused dose of magnesium. ED50 results are expressed as geometric mean (95% confidence interval, CI). Emax results and magnesium plasma concentrations are expressed as mean [SD]. The ED50 of MgSO4 was 0.000307 mmol/min (95% CI 0.138, 0.666) and Emax was 102% [20%]. Magnesium induces dose-dependent venodilation in healthy women in the absence of systemic haemodynamic effects. The dose resulting in vasodilation using the LVDT/hand vein model is two to three orders of magnitude less than the therapeutic doses of magnesium used for tocolysis or seizure prophylaxis. Studies of the effect of systemically administered therapeutic doses of magnesium on vascular reactivity in pre-eclampsia will be of interest.

A proposed technique of SFEM on solving ordinary random differential equation

The stochastic finite element (SFEM) is a new method to solve random differential equation using the known deterministic finite element technique (FEM) adapted to stochastic techniques for solving stochastic differential equations. In this paper, a proposed technique of FEM and random variable transformation (RVT) is used to solve a differential equation with random excitation. The technique shows high accuracy when solving a case study compared with the exact solution.

Measurement of body venous tone

The venous system contains about 70% of the blood volume, and approximately 75% of the venous volume is in the small veins and venules. Veins play an active role in the control of cardiac output (CO) and blood pressure. Drugs that interfere with venous tone have profound effects on CO and blood pressure due to the large venous capacity. Information on body venous tone cannot be obtained from studies using isolated venous preparations and perfused venous beds, which lack modulating cardiovascular reflex mechanisms. In vivo methods used for the assessment of venous function in experimental animals and humans are as follows: the mean circulatory filling pressure (MCFP) method for the determination of body venous tone, constant CO reservoir technique for measuring vascular compliance and unstressed volume, plethysmography or blood-pool scintigraphy along with venous occlusion for measuring the volume and compliance of an organ, linear variable differential transformer (LVDT) technique for estimating the diameter of a human dorsal hand vein, intravascular ultrasound (IVUS) imaging technique to monitor the cross-sectional area of a large vein, and ultrasonic crystals to estimate the dimension of an organ. These methods are described and critically evaluated to disclose their validity, merits and limitations.

Venous responsiveness to norepinephrine in healthy subjects: Effects of single doses of 325 mg aspirin

Antithrombotic doses of aspirin, which are widely used in patients with cardiovascular disease, may inhibit prostaglandin synthesis but the physiologic significance with regard to vascular tone is not well defined. We hypothesized that inhibition of vasodilator prostaglandin synthesis by aspirin would significantly increase sympathetic-mediated venoconstriction. Twelve healthy volunteers (mean age, 24.5 +/- 0.8 years; age range, 19 to 30 years) were studied on two mornings approximately 7 days apart and not less than 90 minutes after a randomized single dose of 325 mg aspirin or matching placebo. Distension of dorsal hand veins was measured with use of the linear variable differential transformer technique during local infusions of exogenous norepinephrine (0.125 to 1,024 ng/min) and during release of endogenous norepinephrine from sympathetic activation by a forehead cold pressor test. Hemodynamic parameters and venous plasma catecholamine levels were measured. Antiplatelet activity of the dose of aspirin was confirmed in three subjects. Aspirin increased venoconstriction to norepinephrine, causing a significant shift to the left (P < .03) of the norepinephrine dose-response curve and a significant decrease in logED50 (P < .03), representing a decrease in the dose of norepinephrine required to reduce vein distension by 50% from 50 to 25 ng/min. Venoconstriction to the cold pressor test was significantly increased by aspirin (10% +/- 3% versus 3% +/- 1% for placebo; P < .02). Cold pressor-induced increases in mean arterial pressure were significantly larger with aspirin compared with placebo (18 +/- 1 versus 14 +/- 1 mm Hg, respectively; P < .03). Baseline levels and stress-induced increases in plasma norepinephrine were not different between days. The results suggest that aspirin inhibits the role of vasodilator prostaglandins in modulating peripheral venoconstriction and increases vascular resistance during physiologic stress in young healthy subjects.