Quadratic Approximation Method Research Articles

The optimal evaporation temperature and working fluids for subcritical organic Rankine cycle

A theoretical formula is proposed to calculate the OET (optimal evaporation temperature) of subcritical ORC (organic Rankine cycle) based on thermodynamic theory when the net power output is selected as the objective function. The OETs of 22 working fluids including wet, isentropic and dry fluids are determined under the given conditions. In order to compare the accuracy of these results, the quadratic approximation method in EES (Engineering Equation Solver) is used to optimize the net power output and the OETs are obtained by numerical simulation. The results show that the OETs calculated by the theoretical formula are consistent with the numerical simulation results. In addition, the average computational accuracy of OETs from the theoretical formula is higher than that from the simplified formula recommended by the related literature. The larger net power output will be produced when the critical temperature of working fluid approaches to the temperature of the waste heat source. According to the maximum net power output, suitable working pressure, total heat transfer capacity and expander SP (size parameter), R114, R245fa, R123, R601a, n-pentane, R141b and R113 are suited as working fluids for subcritical ORC under the given conditions in this paper.

Uncertainty quantifications of Pareto optima in multiobjective problems

Design is a multi-objective decision-making process considering manufacturing, cost, aesthetics, usability among many other product attributes. The set of optimal solutions, the Pareto set, indicates the trade-offs between objectives. Decision-makers generally select their own optima from the Pareto set based on personal preferences or other judgements. However, uncertainties from manufacturing processes and from operating conditions will change the performances of the Pareto optima. Evaluating the impacts of uncertainties on Pareto optima requires a large amount of data and resources. Comparing multiple Pareto solutions under uncertainty are also very costly. In this work, local Pareto set approximation is integrated with uncertainty propagation technique to quantify design variations in the objective space. An optimality influence range is proposed using linear combinations of objective functions that creates a more accurate polygon objective variation subspace. A set of `virtual samples' is then generated to form two quantifications of the objective variation subspace, namely an influence noise to indicate how a design remains optimal, and an influence range that quantifies the overall variations of a design. In most engineering practices, a Pareto optimum with a smaller influence noise and a smaller influence range is preferred. We also extend the influence noise/range concept to nonlinear Pareto set with the second-order approximation. The quadratic local Pareto approximation method in the literature is also extended in this work to solve multi-objective engineering problems with black-box functions. The usefulness of the proposed quantification method is demonstrated using a numerical example as well as using an engineering problem in structural design.

A serial supply chain of newsvendor problem with safety stocks under complete and partial information sharing

Abstract We consider a supply chain where multiple members are serially connected. The decision is to determine the ordering quantity of a member to the next upstream member in the supply chain. The basic cost model is similar to the newsvendor problem with additional consideration to safety stock. This paper presents optimal approaches for coordination of the supply chain under both complete and partial information sharing in order to maximize the total expected benefit. For complete information sharing we develop an optimal coordination algorithm. For partial information sharing, we propose an optimal coordination algorithm based on the Alternating Direction Method and the Diagonal Quadratic Approximation Method. A numerical example is discussed to show the optimal convergence of ordering quantities and discuss the properties of the proposed algorithms.

Efficient and accurate quadratic approximation methods for pricing Asian strike options

This study is on valuing Asian strike options and presents efficient and accurate quadratic approximation methods that work extremely well, both with regard to the volatility of a wide range of underlying assets, and longer average time windows. We demonstrate that most of the well-known quadratic approximation methods used in the literature for pricing Asian strike options are special cases of our model, with the numerical results demonstrating that our method significantly outperforms the other quadratic approximation methods examined here. Using our method for the calculation of hundreds of Asian strike options, the pricing errors (in terms of the root mean square errors) are reasonably small. Compared with the Monte Carlo benchmark method, our method is shown to be rapid and accurate. We further extend our method to the valuing of quanto forward-starting Asian strike options, with the pricing accuracy of these options being largely the same as the pricing of plain vanilla Asian strike options.

Examination of Shape Optimization for the Nonlinear Buckling Phenomenon of Suspension Parts

This paper presents a numerical solution to a non-parametric shape optimization problem for design of suspension arm in which strength of suspension arm is evaluated by reaction force to plastic buckling load due to compulsory displacement. To deal with buckling phenomena, the geometrical non-linearity and material non-linearity are considered. Hyper-elastic theory is applied to calculate the deformation of suspension arm, under assumption of monotonous loading. Mass and the reaction force integral to the buckling phenomena are chosen as an objective function and a constraint function, respectively. The shape derivatives of these functions are evaluated by the shape optimization theory. A numerical scheme based on a sequential quadratic approximation method is applied to reshape by using the shape gradients. In this scheme, the traction method is used to find the decent directions of the cost functions. The scheme is implemented by using a commercial shape optimization program. In this program, the shape gradients are calculated by a user sub-program which is developed by using the result of non-linear FEM analysis of a commercial solver. The numerical example for a suspension arm model shows 12% of mass reduction while keeping the reaction force integral constant.

Constructing composite search directions with parameters in quadratic interpolation models

In this paper, we introduce the weighted composite search directions to develop the quadratic approximation methods. The purpose is to make fully use of the information disclosed by the former steps to construct possibly more promising directions. Firstly, we obtain these composite directions based on the properties of simplex methods and use them to construct trust region subproblems. Then, these subproblems are solved in the algorithm to find solutions of some benchmark optimization problems. The computation results show that for most tested problems, the improved quadratic approximation methods can obviously reduce the number of function evaluations compared with the existing ones. Finally, we conclude that the algorithm will perform better if the composite directions approach the previous steepest descent direction of the sub-simplex so far. We also point out the potential applications of this improved quadratic interpolation method in business intelligence systems.

Quadratic approximation on SCAD penalized estimation

In this paper, we propose a method of quadratic approximation that unifies various types of smoothly clipped absolute deviation (SCAD) penalized estimations. For convenience, we call it the quadratically approximated SCAD penalized estimation (Q-SCAD). We prove that the proposed Q-SCAD estimator achieves the oracle property and requires only the least angle regression (LARS) algorithm for computation. Numerical studies including simulations and real data analysis confirm that the Q-SCAD estimator performs as efficient as the original SCAD estimator.

Equivalence of Three Quadratic Approximation Methods of Stability Boundaries of Nonlinear Dynamic Systems

Approximation of stable manifolds of unstable equilibrium points is a key problem in stability region estimation of nonlinear dynamic systems. Different quadratic approximation methods have been proposed in past literature. Then what is their relationship and which has better performance are of concern. The mathematical equivalence of three quadratic approximation methods is proved in this note. Their computation burdens of numerical implementation are compared, and the one with the least computation time is recommended for application.

A distributed coordination for a single warehouse-multiple retailer problem under private information

In this paper, we consider a distributed coordination for single warehouse and multiple retailer problems. The objective is to minimize the sum of the setup cost and inventory holding cost incurring among warehouse and retailers in a distributed manner. We propose two algorithms based on the alternating direction method (ADM) and diagonal quadratic approximation method (DQAM) that utilize Augmented Lagrangian function and multiplier updating scheme. The ADM-based algorithm solves the problem by communicating partial information between a coordinator, the warehouse and retailers. The DQAM-based algorithm does not require a coordinator. The cooperation to minimize the system-wide cost is performed by the direct communication of the warehouse and retailers. The proposed algorithms guarantee 6% deviation from optimal solution in the worst case. The computational experiments and the comparison with existing decentralized approaches are presented.

WELFARE-MAXIMIZING OPERATIONAL MONETARY AND TAX POLICY RULES

This paper computes welfare-maximizing monetary and tax policy feedback rules in a calibrated dynamic general equilibrium model with sticky prices. The government makes exogenous final good purchases, levies a proportional income tax, and issues nominal one-period bonds. A quadratic approximation method is used to solve the model and to compute household welfare. Optimized policy has a strong anti-inflation stance and implies persistent fluctuations of the tax rate and of public debt. Very simple optimized policy rules, under which the interest rate just responds to inflation and the tax rate just responds to public debt, yield a welfare level very close to that generated by richer rules.

Enhanced two-point diagonal quadratic approximation methods for design optimization

Based on two-point diagonal quadratic approximation (referred to as TDQA), developed by Kim et al. [Min-Soo Kim, Jong-Rip Kim, Jae-Young Jeon, Dong-Hoon Choi, Efficient mechanical system optimization using two-point diagonal approximation in the nonlinear intervening variable space, J. KSME 15 (2001) 1257–126], enhanced two-point approximation methods are proposed in this paper. The suggested methods reinforce TDQA with new quadratic correction terms using the concept of TANA-3. These methods overcome the disadvantage of TDQA when the derivatives at two design points have the same sign. In addition, the values and derivatives of the proposed approximation functions are completely equal to those of an original function at the two design points whether the derivatives at those points have the same sign or not. Several examples show the numerical performance and accuracy of the proposed methods compared to previous work, and optimization examples show that these methods can successfully reach the optimum via sequential approximation optimization.

Variable-Number Sample-Path Optimization

The sample-path method is one of the most important tools in simulation-based optimization. The basic idea of the method is to approximate the expected simulation output by the average of sample observations with a common random number sequence. In this paper, we describe a new variant of Powell’s unconstrained optimization by quadratic approximation (UOBYQA) method, which integrates a Bayesian variable-number sample-path (VNSP) scheme to choose appropriate number of samples at each iteration. The statistically accurate scheme determines the number of simulation runs, and guarantees the global convergence of the algorithm. The VNSP scheme saves a significant amount of simulation operations compared to general purpose ‘fixed-number’ sample-path methods. We present numerical results based on the new algorithm.

On the use of simplex methods in constructing quadratic models

In this paper, we investigate the quadratic approximation methods. After studying the basic idea of simplex methods, we construct several new search directions by combining the local information progressively obtained during the iterates of the algorithm to form new subspaces. And the quadratic model is solved in the new subspaces. The motivation is to use the information disclosed by the former steps to construct more promising directions. For most tested problems, the number of function evaluations have been reduced obviously through our algorithms.

An efficient approximation method for American exotic options

AbstractThe authors suggest a modified quadratic approximation scheme, and apply this scheme to American barrier (knock‐out) and floating‐strike lookback options. This modified scheme introduces an additional parameter into the quadratic approximation method, originally suggested by G. Barone‐Adesi and R. Whaley (1987), to reduce pricing errors. When the barrier is close to the underlying asset's current price, the approximation formula is more accurate than lattice methods because the optimal exercise boundary is independent of the underlying asset's current price. That is, the proposed method overcomes the “near‐barrier” problem that occurs in lattice methods. In addition, the pricing error decreases when the underlying asset's volatility is high. This approximation scheme is more efficient than B. Gao, J. Huang, and M. Subrahmanyam's (2000) method. As a second application of the modified approximation scheme, the authors provide an approximation formula for American floating‐strike lookback options which is the first approximation formula ever suggested in the literature. Compared to S. Babbs' (2000) binomial approach, our approximation method is more efficient after controlling for pricing errors, and is more accurate after controlling for computing time. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:29–59, 2007

개선된 이점 대각 이차 근사화를 이용한 함수 근사화

Function approximation is one of the most important and active research fields in design optimization. Accurate function approximations can reduce the repetitive computational effort for system analysis. So this study presents an enhanced two-point diagonal quadratic approximation method. The proposed method is based on the Two-point Diagonal Quadratic Approximation method. But unlike TDQA, the suggested method has two quadratic terms, the diagonal term and the correction term. Therefore this method overcomes the disadvantage of TDQA when the derivatives of two design points are same signed values. And in the proposed method, both the approximate function and derivative values at two design points are equal to the exact counterparts whether the signs of derivatives at two design points are the same or not. Several numerical examples are presented to show the merits of the proposed method compared to the other forms used in the literature.

Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function

A calculation of the proton structure function is reported with an approximation method that relates the reduced cross section derivative and the scaling violation at low x by using quadratic form for the structure function. This quadratic form approximation method can be used to determine the structure function from the HERA reduced cross section data taken at low x. This new approach can determine the structure functions with reasonable precision even for low x values which have not been investigated. We observe that the Q2 dependence is quadratic over the full kinematic covered range. To test the validity of our new determined structure functions, we find the gluon distribution function in the leading order approximation with our new calculation for the structure functions and compare them with the QCD parton distribution functions.

Method of quadratic approximation: A new approach to identification of analysers and channels in human vision

Models comprising, in sequence, linear analysers, Ψ 1, …, Ψ n , non-linear transducer functions, and the Minkowski decision rule, are widely used to fit detection and discrimination data, especially when necessary to take into account the effect of probability summation. However, the analysers’ characteristics cannot be derived from detection/discrimination data because of an unavoidable trade-off between these characteristics and the decision rule. Here we show how to overcome this problem, i.e. how to identify the analysers Ψ i= ∑ m aijφ j=1 despite the probability summation between them. The observer's performance is assumed to be quantitatively defined in terms of an equi-detection (discrimination) surface. Each analyser Ψ i is expressed as a weighted sum of linear (coordinate) functionals φ j :Ψ 1,…,Ψ n , so that an identification of the analysers ϕ i is then reduced to evaluating the weight matrix A={ a ij }. It is proved that A can be uniquely recovered from a quadratic approximation of the equi-detection (discrimination) surface at the neighbourhood of at least two points. More specifically, the following equation holds true: A T D( A T ) #= H 1 H 2 #, where A # is the generalised inverse of the matrix A, D is an unknown diagonal matrix, H 1 and H 2 are the matrices of the quadratic forms determining the quadratic surfaces approximating the equi-detection (discrimination) surface at two different points. Thus, the matrix H 1 H 2 # known from experiment is a similarity transform of the diagonal matrix, the rows of A being the eigenvectors of H 1 H 2 #. Hence, any eigensystem routine can be used to derive A from H 1 H 2 #.

이점 대각 이차 근사화 기법과 통계적 제한조건을 적용한 강건 최적설계 기법

This study presents an efficient method for robust optimal design. In order to avoid the excessive evaluations of the exact performance functions, two-point diagonal quadratic approximation method is employed for approximating them during optimization process. This approximation method is one of the two point approximation methods. Therefore, the second order sensitivity information of the approximated performance functions are calculated by an analytical method. As a result, this enables one to avoid the expensive evaluations of the exact derivatives of the performance functions unlike the conventional robust optimal design methods based on the gradient information. Finally, in order to show the numerical performance of the proposed method, one mathematical problem and two mechanical design problems are solved and their results are compared with those of the conventional methods.

REFINING THE QUADRATIC APPROXIMATION FORMULA FOR AN AMERICAN OPTION

The aim of this paper is to provide a more refined approximation for the valuation of an American option, based on the quadratic approximation method. We not only show that the result using the old method is the special case of our method, but also investigate the qualitative behavior of American options with respect to a certain new parameter.

Efficient approximation method for constructing quadratic response surface model

For a large scaled optimization based on response surface methods, an efficient quadratic approximation method is presented in the context of the trust region model management strategy. If the number of design variables inn, the proposed method requires only 2n+1 design points for one approximation, which are a center point and two additional axial points within a systematically adjusted trust region. These design points are used to uniquely determine the main effect terms such as the linear and quadratic regression coefficients. A quasi-Newton formula then uses these linear and quadratic coefficients to progressively update the two-factor interaction effect terms as the sequential approximate optimization progresses. In order to show the numerical performance of the proposed method, a typical unconstrained optimization problem and two dynamic response optimization problems with multiple objective are solved. Finally, their optimization results compared with those of the central composite designs (CCD) or the over-determined D-optimality criterion show that the proposed method gives more efficient results than others.