Search

In this paper, we significantly extend the applicability of state-of-the-art ELDP (equations for linearizing discrete product terms) method by providing a new linearization to handle more complicated non-linear terms involving both of discrete and bounded continuous variables. A general class of “representable programming problems” is formally proposed for a much wider range of engineering applications. Moreover, by exploiting the logarithmic feature embedded in the discrete structure, we present an enhanced linear reformulation model which requires half an order fewer equations than the original ELDP. Computational experiments on various engineering design problems support the superior computational efficiency of the proposed linearization reformulation in solving engineering optimization problems with discrete and bounded continuous variables. [ABSTRACT FROM AUTHOR]

This paper introduces a new disjunctive formulation for the simultaneous optimization and heat integration of systems with variable inlet and outlet temperatures in process streams as well as the possibility of selecting and using different utilities. The starting point is the original compact formulation of the Pinch Location Method, however, instead of approximating the “maximum” operator with smooth, but non-convex functions, these operators are modeled by means of a disjunction. The new formulation has shown to have equal or lower relaxation gap than the best alternative reformulation, thus reducing computational time and numerical problems related to non-convex approximations. [ABSTRACT FROM AUTHOR]

Abstract: This paper discusses stabilization of an optimization procedure to determine an appropriate semivariogram parameter using the empirical semivariogram for Kriging-based approximation. From a viewpoint of a computational cost for constructing a surrogate model, a semivariogram fitting approach using the empirical semivariogram can be usable for the parameter estimation of a semivariogram function. However, instability of the optimization procedure for determination of the semivariogram parameter may be observed in some cases and it causes to generate an invalid surrogate model. For this problem, a simple technique for stabilization of the optimization for the parameter determination is proposed in this paper. The proposed approach employs a normalization technique of input data with respect to values of variables and an objective function. The proposed method is applied to some numerical examples, and the numerical results illustrate validity and effectiveness of the proposed method. [Copyright &y& Elsevier]

A soft computing approach based on artificial neural network (ANN) and optimization is presented for the numerical solution of the unsteady one-dimensional advection–dispersion equation (ADE) arising in contaminant transport through porous media. A length factor ANN method, based on automatic satisfaction of arbitrary boundary conditions (BCs) was chosen for the numerical solution of ADE. The strength of ANN is exploited to construct a trial approximate solution (TAS) for ADE in a way that it satisfies the initial or BCs exactly. An unsupervised error is constructed in approximating the solution of ADE which is minimized by training ANN using gradient descent algorithm (GDA). Two challenging test problems of ADE are considered in this paper, in which, the first problem has steep boundary layers near x = 1 and many numerical methods create non-physical oscillation near steep boundaries. Also for the second problem many numerical schemes suffer from computational noise and instability issues. The proposed method is advantageous as it does not require temporal discretization for the solution of the ADEs as well as it does not suffer from numerical instability. The reliability and effectiveness of the presented algorithm is investigated by sufficient large number of independent runs and comparison of results with other existing numerical methods. The results show that the present method removes the difficulties arising in the solution of the ADEs and provides solution with good accuracy. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the low rank approximation of the symmetric positive semidefinite matrix, which arises in machine learning, quantum chemistry and inverse problem. We first characterize the feasible set by , and then transform low rank approximation into an unconstrained optimization problem. Finally, we use the nonlinear conjugate gradient method with exact line search to compute the optimal low rank symmetric positive semidefinite approximation of the given matrix. Numerical examples show that the new method is feasible and effective. [Copyright &y& Elsevier]

Abstract: In this paper, the solution of fuzzy linear system (FLS) is investigated based on a -level expansion. To this end, -level of FLS is solved for calculating the core of fuzzy solution and then its spreads are obtained by solving an optimization problem with a special objective function. Using our proposed method, if the computed solution satisfies the FLS, it can be as exact or approximated solution. Finally, the existence of solution of FLS is proved in details and some numerical examples are solved to illustrate the accuracy and capability of the method. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to solve the problem of identifying an unknown heat source depending simultaneously on both space and time variables. This problem is transformed into an optimization problem and the uniqueness of minimum element is proved rigorously. Then a variational formulation for solving the optimization problem is given. A conjugate gradient method and a finite difference method are used to solve the variational problem. Some numerical examples are also provided to show the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: Metamodelling is an increasingly more popular approach for alleviating the computational burden associated with computationally intensive optimization/management problems in environmental and water resources systems. Some studies refer to the metamodelling approach as function approximation, surrogate modelling, response surface methodology or model emulation. A metamodel-enabled optimizer approximates the objective (or constraint) function in a way that eliminates the need to always evaluate this function via a computationally expensive simulation model. There is a sizeable body of literature developing and applying a variety of metamodelling strategies to various environmental and water resources related problems including environmental model calibration, water resources systems analysis and management, and water distribution network design and optimization. Overall, this literature generally implies metamodelling yields enhanced solution efficiency and (almost always) effectiveness of computationally intensive optimization problems. This paper initially develops a comparative assessment framework which presents a clear computational budget dependent definition for the success/failure of the metamodelling strategies, and then critically evaluates metamodelling strategies, through numerical experiments, against other common optimization strategies not involving metamodels. Three different metamodel-enabled optimizers involving radial basis functions, kriging, and neural networks are employed. A robust numerical assessment within different computational budget availability scenarios is conducted over four test functions commonly used in optimization as well as two real-world computationally intensive optimization problems in environmental and water resources systems. Numerical results show that metamodelling is not always an efficient and reliable approach to optimizing computationally intensive problems. For simpler response surfaces, metamodelling can be very efficient and effective. However, in some cases, and in particular for complex response surfaces when computational budget is not very limited, metamodelling can be misleading and a hindrance, and better solutions are achieved with optimizers not involving metamodels. Results also demonstrate that neural networks are not appropriate metamodelling tools for limited computational budgets while metamodels employing kriging and radial basis functions show comparable overall performance when the available computational budget is very limited. [Copyright &y& Elsevier]

Abstract: Using principles from finite-time thermodynamics, this paper introduces a class of bilinear models suitable for financial optimization and risk management of commodity energy conversion processes. This class of high-level models is intended to predict the efficiency of energy conversion in campus scale utilities with complex energy requirements, multiple fuel sources, and significant operational flexibility. The models incorporate first and second law thermodynamic considerations. The bilinear character of these models derives from the multiplicative coupling between entropy flux and temperature driving forces that are intrinsic to the second law. Economic optimization of these models yields non-convex bilinear optimization problems. Special cases for optimal operation reduce to generalized eigenvalue problems, or more generally complex rank constraints that can be solved using numerical algebraic geometry. For general use, we propose a computational strategy based on a linear outer approximation coupled with a branch and bound methodology to reduce the search region. We demonstrate modeling and optimization using several example problems drawn from the existing literature on finite-time thermodynamics. [Copyright &y& Elsevier]

Abstract: Expensive optimization aims to find the global minimum of a given function within a very limited number of function evaluations. It has drawn much attention in recent years. The present expensive optimization algorithms focus their attention on metamodeling techniques, and call existing global optimization algorithms as subroutines. So it is difficult for them to keep a good balance between model approximation and global search due to their two-part property. To overcome this difficulty, we try to embed a metamodel mechanism into an efficient evolutionary algorithm, low dimensional simplex evolution (LDSE), in this paper. The proposed algorithm is referred to as the low dimensional simplex evolution extension (LDSEE). It is inherently parallel and self-contained. This renders it very easy to use. Numerical results show that our proposed algorithm is a competitive alternative for expensive optimization problems. [Copyright &y& Elsevier]

Abstract: This paper proposes a new ranking method for fuzzy numbers, which uses a defuzzification of fuzzy numbers and a weighting function. Following Saeidifar and Pasha (2008), first, we define a weighted distance measure on fuzzy numbers, and then, by minimizing this distance, the weighted interval and point approximations of fuzzy numbers are obtained. These indices are applied to rank the fuzzy numbers. This method is new and interesting for ranking fuzzy numbers, and it can be applied for solving and optimizing engineering and economics problems in a fuzzy environment. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new symmetric rank-one (SR1) method for the solution of unconstrained optimization problems. The proposed method involves an algorithm in which the usual SR1 Hessian is updated a number of times in a way to be specified in some iterations, to improve the performance of the Hessian approximation. In particular, we discuss how to consider a criterion for indicating at each iteration whether it is necessary to employ extra updates. However it is well known that there are some theoretical difficulties when applying the SR1 update. Even for a current positive definite Hessian approximation, it is possible that the SR1 update may not be defined or the SR1 update may not preserve positive definiteness at some iterations. We then employ a restarting procedure that guarantees that updated matrices will be well-defined while preserving positive definiteness of updates. Numerical results support these theoretical considerations. They show that the implementation of the SR1 method using extra updating techniques improves the performance of the SR1 method substantially for a number of test problems from the literature. [Copyright &y& Elsevier]

Abstract: Recently, Hager and Zhang (2005) proposed a new conjugate gradient method which generates sufficient descent direction , this property is independent of the line search used. In this paper, we take a modification of this method, such that the sufficient descent direction satisfies , this property is also independent of the line search used. Under appropriate conditions, we prove that the proposed method is globally convergent. Moreover, we give a sufficient condition for the global convergence of the proposed general method. The numerical results show that the proposed method is efficient. [Copyright &y& Elsevier]

Abstract: Modeling strategies often result in dynamical systems of very high dimension. It is then desirable to find systems of the same form but of lower complexity, whose input–output behavior approximates the behavior of the original system. Here we consider linear time-invariant discrete-time dynamical systems. The cornerstone of this paper is a relation between optimal model reduction in the -norm and (tangential) rational Hermite interpolation. First order necessary conditions for -optimal model reduction are presented for discrete Multiple-Input–Multiple-Output (MIMO) systems. These conditions suggest a specific choice of interpolation data and a novel algorithm aiming for an-optimal model reduction for MIMO systems. It is also shown that the conditions are equivalent to two known gramian-based first order necessary conditions. Numerical experiments demonstrate the approximation quality of the method. [Copyright &y& Elsevier]

Abstract: In this paper we propose a fundamentally different conjugate gradient method, in which the well-known parameter is computed by an approximation of the Hessian/vector product through finite differences. For search direction computation, the method uses a forward difference approximation to the Hessian/vector product in combination with a careful choice of the finite difference interval. For the step length computation we suggest an acceleration scheme able to improve the efficiency of the algorithm. Under common assumptions, the method is proved to be globally convergent. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with conjugate gradient algorithms including CONMIN by Shanno and Phua [D.F. Shanno, K.H. Phua, Algorithm 500, minimization of unconstrained multivariate functions, ACM Trans. Math. Softw. 2 (1976) 87–94], SCALCG by Andrei [N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl. 38 (2007) 401–416; N. Andrei, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Methods Softw. 22 (2007) 561–571; N. Andrei, A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett. 20 (2007) 645–650], and new conjugacy condition and related new conjugate gradient by Li, Tang and Wei [G. Li, C. Tang, Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math. 202 (2007) 523–539] or truncated Newton TN by Nash [S.G. Nash, Preconditioning of truncated-Newton methods, SIAM J. on Scientific and Statistical Computing 6 (1985) 599–616] using a set of 750 unconstrained optimization test problems show that the suggested algorithm outperforms these conjugate gradient algorithms as well as TN. [Copyright &y& Elsevier]

Abstract: In this paper we consider a conic trust-region method for unconstrained optimization problems and analyze its convergence properties. We propose a convenient curvilinear search method to approximately solve the arising conic trust-region subproblem. Note that this approximate method preserves the strong convergence properties of the exact solution methods and is easy to implement. Both the linear and superlinear convergence of the method are established under commonly used conditions. Numerical experiments are conducted to show the efficiency of the new method. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a family of derivative-free conjugate gradient methods for large-scale nonlinear systems of equations. They come from two modified conjugate gradient methods [W.Y. Cheng, A two term PRP based descent Method, Numer. Funct. Anal. Optim. 28 (2007) 1217–1230; L. Zhang, W.J. Zhou, D.H. Li, A descent modified Polak–Ribiére–Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal. 26 (2006) 629–640] recently proposed for unconstrained optimization problems. Under appropriate conditions, the global convergence of the proposed method is established. Preliminary numerical results show that the proposed method is promising. [Copyright &y& Elsevier]