Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear ... more Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear equations. They use quasi-Newton updates for approximating the Jacobian matrix. Owing to their ability to more completely utilize the information about the Jacobian matrix gathered at the previous iterations, these methods are especially efficient in the case of expensive functions. They are known to be local and superlinearly convergent. We combine these methods with the nonmonotone line search proposed by Li and Fukushima (2000), and study global and superlinear convergence of this combination. Results of numerical experiments are presented. They indicate that the multipoint secant and interpolation methods tend to be more robust and efficient than Broyden's method globalized in the same way.
In this paper, we study the problem of finding a common element of the set of solutions of a syst... more In this paper, we study the problem of finding a common element of the set of solutions of a system of monotone inclusion problems and the set of common fixed points of a finite family of generalized demimetric mappings in Hilbert spaces. We propose a new and efficient algorithm for solving this problem. Our method relies on the inertial algorithm, Tseng's splitting algorithm and the viscosity algorithm. Strong convergence analysis of the proposed method is established under standard and mild conditions. As applications we use our algorithm for finding the common solutions to variational inequality problems, the constrained multiple-set split convex feasibility problem, the convex minimization problem and the common minimizer problem. Finally, we give some numerical results to show that our proposed algorithm is efficient and implementable from the numerical point of view.
Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear ... more Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear equations. They use quasi-Newton updates for approximating the Jacobian matrix. Owing to their ability to more completely utilize the information about the Jacobian matrix gathered at the previous iterations, these methods are especially efficient in the case of expensive functions. They are known to be local and superlinearly convergent. We combine these methods with the nonmonotone line search proposed by Li and Fukushima (2000), and study global and superlinear convergence of this combination. Results of numerical experiments are presented. They indicate that the multipoint secant and interpolation methods tend to be more robust and efficient than Broyden's method globalized in the same way.
We propose a new and efficient nonmonotone adaptive trust region algorithm to solve unconstrained... more We propose a new and efficient nonmonotone adaptive trust region algorithm to solve unconstrained optimization problems. This algorithm incorporates two novelties: it benefits from a radius dependent shrinkage parameter for adjusting the trust region radius that avoids undesirable directions and exploits a new strategy to prevent sudden increments of objective function values in nonmonotone trust region techniques. Global convergence of this algorithm is investigated under some mild conditions. Numerical experiments demonstrate the efficiency and robustness of the proposed algorithm in solving a collection of unconstrained optimization problems from the CUTEst package.
In this paper, we propose a new and efficient nonmonotone adaptive trust region algorithm to solv... more In this paper, we propose a new and efficient nonmonotone adaptive trust region algorithm to solve unconstrained optimization problems. This algorithm incorporates two novelties: it benefits from a radius dependent shrinkage parameter for adjusting the trust region radius that avoids undesirable directions; and it exploits a strategy to prevent sudden increments of objective function values in nonmonotone trust region techniques. Global convergence of this algorithm is investigated under mild conditions. Numerical experiments demonstrate the efficiency and robustness of the proposed algorithm in solving a collection of unconstrained optimization problems from CUTEst package.
In this paper, a new conjugate gradient-like algorithm is proposed to solve unconstrained optimiz... more In this paper, a new conjugate gradient-like algorithm is proposed to solve unconstrained optimization problems. The step directions generated by the new algorithm satisfy sufficient descent condition independent of the line search. The global convergence of the new algorithm, with the Armijo backtracking line search, is proved. Numerical experiments indicate the efficiency and robustness of the new algorithm.
In this paper, we propose a new conjugate gradient-like algorithm. The step directions generated ... more In this paper, we propose a new conjugate gradient-like algorithm. The step directions generated by the new algorithm satisfy a sufficient descent condition independent of the line search. The global convergence of the new algorithm, with the Armijo backtracking line search, is proved. Numerical experiments indicate the efficiency and robustness of the new algorithm in solving a collection of unconstrained optimization problems from CUTEst package.
This paper gives a variant trust-region method, where its radius is automatically adjusted by usi... more This paper gives a variant trust-region method, where its radius is automatically adjusted by using the model information gathered at the current and preceding iterations. The primary aim is to decrease the number of function evaluations and solving subproblems, which increases the efficiency of the trust-region method. The next aim is to update the new radius for large-scale problems without imposing too much computational cost to the scheme. Global convergence to first-order stationary points is proved under classical assumptions. Preliminary numerical experiments on a set of test problems from the CUTEst collection show that the presented method is promising for solving unconstrained optimization problems.
Exergy efficiency investigation and optimization of an Al2O3–water nanofluid based Flat-plate solar collector
Energy and Buildings, 2015
ABSTRACT The present study aims to investigate exergy efficiency of a Flat-plate solar collector ... more ABSTRACT The present study aims to investigate exergy efficiency of a Flat-plate solar collector containing Al2O3–water nanofluid as base fluid. The effect of various parameters like mass flow rate of fluid, nanoparticle volume concentration, collector inlet fluid temperature, solar radiation, and ambient temperature on the collector exergy efficiency is investigated. Also, the procedure to determine optimum values of nanoparticle volume concentration, mass flow rate of fluid, and collector inlet fluid temperature for maximum exergy efficiency delivery has been developed by means of interior-point method for constrained optimization under the given conditions. According to the results, each of these parameters can differently affect the collector exergy by changing the value of the other parameters. The optimization results indicate that under the actual constraints, in both pure water and nanofluid cases the optimized exergy efficiency is increased with increasing solar radiation value. By suspending Al2O3 nanoparticles in the base fluid (water) the maximum collector exergy efficiency is increased about 1% and also the corresponding optimum values of mass flow rate of fluid and collector inlet fluid temperature are decreased about 68% and 2%, respectively.
Projection based methods are a family of efficient and applicable derivative free methods for sol... more Projection based methods are a family of efficient and applicable derivative free methods for solving systems of nonlinear monotone equations. These methods, at each iteration, use a backtracking line search to generate a hyperplane which strictly separates the current approximation from the solution set of the problem. Numerical experiments indicate that choosing an appropriate line search highly affects the efficiency of projection based methods. In this paper we introduce a new line search procedure for generating the separating hyperplane. The convergence properties of the new procedure is established in a simple and short way. Numerical results show that the new line search is very effective and increases the efficiency of projection based methods.
A double-projection-based algorithm for large-scale nonlinear systems of monotone equations
Numerical Algorithms, 2014
ABSTRACT In this paper, we propose a derivative free algorithm for solving large-scale nonlinear ... more ABSTRACT In this paper, we propose a derivative free algorithm for solving large-scale nonlinear systems of monotone equations which combines a new idea of projection methodology with a line search strategy, while an improvement in the projection step is exerted. At each iteration, the algorithm constructs two appropriate hyperplanes which strictly separate the current approximation from the solution set of the problem. Then the new approximation is determined by projecting the current point onto the intersection of two halfspaces that are constructed by these hyperplanes and contain the solution set of the problem. Under some mild conditions, the global convergence of the algorithm is established. Preliminary numerical results indicate that the proposed algorithm is promising.
ORBIT is a derivative-free trust-region framework that employs a radial basis function (RBF) inte... more ORBIT is a derivative-free trust-region framework that employs a radial basis function (RBF) interpolation to solve the computationally expensive optimization problems. The accuracy and stability of RBFs depend on a so-called shape parameter and number of data points. So, it is more appropriate to determine these parameters properly. In this paper, we evaluate the performance of ORBIT algorithm by different types of RBFs, different numbers of data points and different shape parameter values. We utilize Dolan-Moré performance profile and Moré-Wild data profile to investigate the performance of algorithms. Finally, based on this numerical study we propose some recommendations for the type of RBF, the number of data points and the shape parameter value.
Optimization using radial basis functions as an interpolation tool in trust region (ORBIT) is a d... more Optimization using radial basis functions as an interpolation tool in trust region (ORBIT) is a derivative-free framework based on fully linear radial basis function (RBF) models. In this paper, an improved version of ORBIT algorithm based on two novel ideas is proposed. The accuracy and stability of RBFs depend on a so-called shape parameter, so it is more appropriate to determine the shape parameter according to the optimization problem. While ORBIT in all problems uses a fixed value as a shape parameter, our new version, Hybrid-ORBIT, uses a statistical technique to select an appropriate shape parameter. In addition, ORBIT uses some stored points to build a fully linear RBF model without considering their function values, while in the Hybrid-ORBIT algorithm the stored points are sorted based on their function values and the RBF model is built using the points with lower function values, and the best point in the sense of function value is defined as the trust-region center. Numerical results indicate the efficiency of the improved version compared with the original version.
In this paper, a novel hybrid trust-region algorithm using radial basis function (RBF) interpolat... more In this paper, a novel hybrid trust-region algorithm using radial basis function (RBF) interpolations is proposed. The new algorithm is an improved version of ORBIT algorithm based on two novel ideas. Because the accuracy and stability of RBF interpolation depends on a shape parameter, so it is more appropriate to select this parameter according to the optimization problem. In the new algorithm, the appropriate shape parameter value is determined according to the optimization problem based on an effective statistical approach, while the ORBIT algorithm in all problems uses a fixed shape parameter value. In addition, the new algorithm is equipped with a new intelligent nonmonotone strategy which improves the speed of convergence, while the monotonicity of the sequence of objective function values in the ORBIT may decrease the rate of convergence, especially when an iteration is trapped near a narrow curved valley. The global convergence of the new hybrid algorithm is analyzed under some mild assumptions. The numerical results significantly indicate the superiority of the new algorithm compared with the original version.
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Papers by Ahmad Kamandi