This paper presents a trust-region method for multiobjective heterogeneous optimization problems. One of the objective functions is an expensive black-box function, given, for example, by a time-consuming simulation. ...
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This paper presents a trust-region method for multiobjective heterogeneous optimization problems. One of the objective functions is an expensive black-box function, given, for example, by a time-consuming simulation. For this function, derivative information cannot be used, and the computation of function values involves high computational effort. The other objective functions are given analytically, and derivatives can easily be computed. The method uses the basic trust-region approach by restricting the computations in every iteration to a local area and replacing the objective functions by suitable models. The search direction is generated in the image space by using local ideal points. The algorithm generates a sequence of iterates. It is proved that any limit point is Pareto critical. Numerical results are presented and compared to two other algorithms.
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 accu...
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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.
A two-step derivative-free iterative algorithm is presented for solving nonlinear equations. Error analysis shows that the algorithm is fourth-order with efficiency index equal to 1.5874. A lot of numerical results sh...
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A two-step derivative-free iterative algorithm is presented for solving nonlinear equations. Error analysis shows that the algorithm is fourth-order with efficiency index equal to 1.5874. A lot of numerical results show that the algorithm is effective and is preferable to some existing derivative-free methods in terms of computation cost. (C) 2010 Elsevier B.V. All rights reserved.
We introduce a new class of mappings, called duplomonotone, which is strictly broader than the class of monotone mappings. We study some of the main properties of duplomonotone functions and provide various examples, ...
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We introduce a new class of mappings, called duplomonotone, which is strictly broader than the class of monotone mappings. We study some of the main properties of duplomonotone functions and provide various examples, including nonlinear duplomonotone functions arising from the study of systems of biochemical reactions. Finally, we present three variations of a derivative-free line search algorithm for finding zeros of systems of duplomonotone equations, and we prove their linear convergence to a zero of the function.
We consider unconstrained black-box biobjective optimization problems in which analytic forms of the objective functions are not available and function values can be obtained only through computationally expensive sim...
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We consider unconstrained black-box biobjective optimization problems in which analytic forms of the objective functions are not available and function values can be obtained only through computationally expensive simulations. We propose a new algorithm to approximate the Pareto optimal solutions of such problems based on a trust-region approach. At every iteration, we identify a trust region, then sample and evaluate points from it. To determine nondominated solutions in the trust region, we employ a scalarization method to convert the two objective functions into one. We construct and optimize quadratic regression models for the two original objectives and the converted single objective. We then remove dominated points from the current Pareto approximation and construct a new trust region around the most isolated point in order to explore areas that have not been visited. We prove convergence of the method under general regularity conditions and present numerical results suggesting that the method efficiently generates well-distributed Pareto optimal solutions.
In this paper, we propose two new derivative-free algorithms for nonlinear equations. The first is based on quasi-Newton method and is globally and superlinearly convergent under some mild assumptions. The second comb...
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In this paper, we propose two new derivative-free algorithms for nonlinear equations. The first is based on quasi-Newton method and is globally and superlinearly convergent under some mild assumptions. The second combines the ideas of the first with the filter strategy, which helps to reduce the backtracking steps in calculating the stepsizes, for evaluating candidate points. We show its convergence under the same assumptions. The resulting algorithms show some attractive features. Some encouraging preliminary computational results for both algorithms are reported.
In this work, we propose a new globally convergent derivative-free algorithm for the minimization of a continuously differentiable function in the case that some of (or all) the variables are bounded. This algorithm i...
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In this work, we propose a new globally convergent derivative-free algorithm for the minimization of a continuously differentiable function in the case that some of (or all) the variables are bounded. This algorithm investigates the local behaviour of the objective function on the feasible set by sampling it along the coordinate directions. Whenever a "suitable" descent feasible coordinate direction is detected a new point is produced by performing a linesearch along this direction. The information progressively obtained during the iterates of the algorithm can be used to build an approximation model of the objective function. The minimum of such a model is accepted if it produces an improvement of the objective function value. We also derive a bound for the limit accuracy of the algorithm in the minimization of noisy functions. Finally, we report the results of a preliminary numerical experience.
This paper presents sequential and parallel derivative-free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest. It is proved that, under mild assumptions, a sufficient decre...
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This paper presents sequential and parallel derivative-free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest. It is proved that, under mild assumptions, a sufficient decrease condition holds for a nonsmooth function. Based on this property, the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value. If the function is strictly differentiable at its limit points, a ( sub) sequence of points generated by the algorithm converges to a first-order stationary point (delf(x) = 0). If the function is convex around its limit points, convergence ( of a subsequence) to a point with nonnegative directional derivatives on a set of search directions is ensured. Preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods.
Estimation of variance components by REML via derivative-free algorithms requires the determinant of the coefficient matrix of mixed model equations. A Monte Carlo-based method is proposed to approximate the increment...
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Estimation of variance components by REML via derivative-free algorithms requires the determinant of the coefficient matrix of mixed model equations. A Monte Carlo-based method is proposed to approximate the increment of the logarithm of the determinant of this coefficient matrix that corresponds to increments of the ratio between residual and additive-genetic variance. The computing cost of this method is linear with the order of the coefficient matrix. Results of approximate and exact methods were compared. A bias, detected when the difference of the ratio between residual and additive genetic variance is large, vanishes as convergence is reached. The proposed procedure is accurate enough to estimate the maximum of the likelihood function, The REML estimates of variance components using the Monte Carlo approximation are also presented.
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