Nonmonotone Adaptive Trust Region Algorithms with Indefinite Dogleg Path for Unconstrained Minimization

In this paper, we combine the nonmonotone and adaptive techniques with trust region method for unconstrained minimization problems. We set a new ratio of the actual descent and predicted descent. Then, instead of the monotone sequence, the nonmonotone sequence of function values are employed. With the adaptive technique, the radius of trust region △k can be adjusted automatically to improve the efficiency of trust region methods. By means of the Bunch-Parlett factorization, we construct a method with indefinite dogleg path for solving the trust region subproblem which can handle the indefinite approximate Hessian Bk. The convergence properties of the algorithm are established. Finally, detailed numerical results are reported to show that our algorithm is efficient.

A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC^1 CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, a new trust region algorithm for nonlinear equality constrained LC1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subprobiem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.

A Superlinerly Convergent ODE-type Trust Region Algorithm for LC^1 Optimization Problems

In this paper, a new trust region algorithm for unconstrained LC1 optimization problems is given. Compare with those existing trust regiion methods, this algorithm has a different feature: it obtains a stepsize at each iteration not by soloving a quadratic subproblem with a trust region bound, but by solving a system of linear equations. Thus it reduces computational complexity and improves computation efficiency. It is proven that this algorithm is globally convergent and locally superlinear under some conditions.

Trust Region Algorithm for a Class of Nonlinear Complementarity Problem

In this paper, an ODE-type trust region algorithm for solving a class of nonlinear complementarity problems is proposed. A feature of this algorithm is that only the solution of linear systems of equations is required at each iteration, thus avoiding the need for solving a quadratic subproblem with a trust region bound. Under some conditions, it is proven that this algorithm is globally and locally superlinear convergent. The limited numerical examples show its efficiency.

A Nonmonotone Trust Region Method for Solving Symmetric Nonlinear Equations

A trust region method combining with nonmonotone technique is proposed tor solving symmetric nonlinear equations. The global convergence of the given method will be established under suitable conditions. Numerical results show that the method is interesting for the given problems.

GLOBAL CONVERGENCE OF A TRUST REGION ALGORITHM USING INEXACT GRADIENT FOR EQUALITY-CONSTRAINED OPTIMIZATION

A trust-region algorithm is presented for a nonlinear optimization problem of equality-constraints. The characterization of the algorithm is using inexact gradient information. Global convergence results are demonstrated where the gradient values are obeyed a simple relative error condition.

A NEW ADAPTIVE TRUST REGION ALGORITHM FOR OPTIMIZATION PROBLEMS

It is well known that trust region methods are very effective for optimization problems. In this article, a new adaptive trust region method is presented for solving uncon- strained optimization problems. The proposed method combines a modified secant equation with the BFGS updated formula and an adaptive trust region radius, where the new trust region radius makes use of not only the function information but also the gradient information. Under suitable conditions, global convergence is proved, and we demonstrate the local superlinear convergence of the proposed method. The numerical results indicate that the proposed method is very efficient.

A TRUST REGION METHOD WITH A CONIC MODEL FOR NONLINEARLY CONSTRAINED OPTIMIZATION

Trust region methods are powerful and effective optimization methods. The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods. The advantages of the above two methods can be combined to form a more powerful method for constrained optimization. The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound. At the same time, the new algorithm still possesses robust global properties. The global convergence of the new algorithm under standard conditions is established.

A New Nonmonotone Adaptive Trust Region Method

The trust region method plays an important role in solving optimization problems. In this paper, we propose a new nonmonotone adaptive trust region method for solving unconstrained optimization problems. Actually, we combine a popular nonmonotone technique with an adaptive trust region algorithm. The new ratio to adjusting the next trust region radius is different from the ratio in the traditional trust region methods. Under some appropriate conditions, we show that the new algorithm has good global convergence and superlinear convergence.

A TRUST REGION ALGORITHM VIA BILEVEL LINEAR PROGRAMMING FOR SOLVING THE GENERAL MULTICOMMODITY MINIMAL COST FLOW PROBLEMS

This paper proposes a nonmonotonic backtracking trust region algorithm via bilevel linear programming for solving the general multicommodity minimal cost flow problems.Using the duality theory of the linear programming and convex theory,the generalized directional derivative of the general multicommodity minimal cost flow problems is derived.The global convergence and superlinear convergence rate of the proposed algorithm are established under some mild conditions.

An Active-Set Projected Trust Region Algorithm for Box Constrained Optimization Problems

An active-set projected trust region algorithm is proposed for box constrained optimization problems, where the given algorithm is designed by three steps. First, the projected gradient direction which normally has better numerical performance is introduced. Second, the projected trust region direction that often possesses good convergence is defined, where the matrix of trust region subproblem is updated by limited memory strategy. Third, in order to get both good numerical performance and convergence, the authors define the final search which is the convex combination of the projected gradient direction and the projected trust region direction. Under suitable conditions, the global convergence of the given algorithm is established. Numerical results show that the presented method is competitive to other similar methods.

A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS:I.ALGORITHM AND GLOBAL CONVERGENCEXIU NAIHUA

A class of trust region methods for solving linear inequality constrained problems is proposed in this paper. It is shown that the algorithm is of global convergence.The algorithm uses a version of the two-sided projection and the strategy of the unconstrained trust region methods. It keeps the good convergence properties of the unconstrained case and has the merits of the projection method. In some sense, our algorithm can be regarded as an extension and improvement of the projected type algorithm.

Nonmonotonic Trust Region Algorithm via the Conjugate Gradient Path for Unconstrained Generalized Geometric Programming

In this paper,on the basis of making full use of the characteristics of unconstrained generalized geometric programming（GGP）,we establish a nonmonotonic trust region algorithm via the conjugate path for solving unconstrained GGP problem.A new type of condensation problem is presented,then a particular conjugate path is constructed for the problem,along which we get the approximate solution of the problem by nonmonotonic trust region algorithm,and further prove that the algorithm has global convergence and quadratic convergence properties.

A New Nonmonotonic Trust Region Algorithm for A Class of Unconstrained Nonsmooth Optimization

This paper presents a new trust region algorithm for solving a class of composite nonsmooth optimizations. It is distinguished by the fact that this method does not enforce strict monotonicity of the objective function values at successive iterates and that this method extends the existing results for this type of nonlinear optimization with smooth, or piecewise smooth, or convex objective functions or their composition. It is proved that this algorithm is globally convergent under certain conditions. Finally, some numerical results for several optimization problems are reported which show that the nonmonotonic trust region method is competitive with the usual trust region method.

GLOBAL CONVERGENCE OF NONMONOTONIC TRUST REGION ALGORITHM FOR NONLINEAR OPTIMIZATION

A trust region algorithm for equality constrained optimization is given in this paper.The algorithm does not enforce strict monotonicity of the merit function for every iteration.Global convergence of the algorithm is proved under the same conditions of usual trust region method.

Elastic Full Waveform Inversion Based on the Trust Region Strategy

In this paper, we investigate the elastic wave full-waveform inversion (FWI) based on the trust region method. The FWI is an optimization problem of minimizing the misfit between the observed data and simulated data. Usually, the line search method is used to update the model parameters iteratively. The line search method generates a search direction first and then finds a suitable step length along the direction. In the trust region method, it defines a trial step length within a certain neighborhood of the current iterate point and then solves a trust region subproblem. The theoretical methods for the trust region FWI with the Newton type method are described. The algorithms for the truncated Newton method with the line search strategy and for the Gauss-Newton method with the trust region strategy are presented. Numerical computations of FWI for the Marmousi model by the L-BFGS method, the Gauss-Newton method and the truncated Newton method are completed. The comparisons between the line search strategy and the trust region strategy are given and show that the trust region method is more efficient than the line search method and both the Gauss-Newton and truncated Newton methods are more accurate than the L-BFGS method.

CONVERGENCE PROPERTIES OF IMPROVED SECANT METHODS WITH TRUST REGION MULTIPLIER

The secant methods discussed by Fontecilla (in 1988) are considerably revised through employing a trust region multiplier strategy and introducing a nondifferentiable merit function. In this paper the secant methods are also improved by adding a dogleg typed movement which allows to overcome a phenomena similar to the Maratos effect. Furthermore, these algorithms are analyzed and global convergence theorems as well as local superlinear convergence rate are proved.

Optimization Strategy Using Dynamic Metamodel Based on Trust Region and Biased Sampling Method

Combining a trust region method with a biased sampling method,a novel optimization strategy(TRBSKRG)based on a dynamic metamodel is proposed.Initial sampling points are selected by a maximin Latin hypercube design method,and the metamodel is constructed with Kriging functions.The global optimization algorithm is employed to perform the biased sampling by searching the maximum expectation improvement point or the minimum of surrogate prediction point within the trust region.And the trust region is updated according to the current known information.The iteration continues until the potential global solution of the true optimization problem satisfied the convergence conditions.Compared with the trust region method and the biased sampling method,the proposed optimization strategy can obtain the global optimal solution to the test case,in which improvements in computation efficiency are also shown.When applied to an aerodynamic design optimization problem,the aerodynamic performance of tandem UAV is improved while meeting the constraints,which verifies its engineering application.

Broken Rotor Bar Fault Detection of Induction Motors Using a Joint Algorithm of Trust Region and Modified Bare-bones Particle Swarm Optimization

A precise detection of the fault feature parameter of motor current is a new research hotspot in the broken rotor bar(BRB) fault diagnosis of induction motors. Discrete Fourier transform(DFT) is the most popular technique in this field, owing to low computation and easy realization. However, its accuracy is often limited by the data window length, spectral leakage, fence e ect, etc. Therefore, a new detection method based on a global optimization algorithm is proposed. First, a BRB fault current model and a residual error function are designed to transform the fault parameter detection problem into a nonlinear least-square problem. Because this optimization problem has a great number of local optima and needs to be resolved rapidly and accurately, a joint algorithm(called TR-MBPSO) based on a modified bare-bones particle swarm optimization(BPSO) and trust region(TR) is subsequently proposed. In the TR-MBPSO, a reinitialization strategy of inactive particle is introduced to the BPSO to enhance the swarm diversity and global search ability. Meanwhile, the TR is combined with the modified BPSO to improve convergence speed and accuracy. It also includes a global convergence analysis, whose result proves that the TR-MBPSO can converge to the global optimum with the probability of 1. Both simulations and experiments are conducted, and the results indicate that the proposed detection method not only has high accuracy of parameter estimation with short-time data window, e.g., the magnitude and frequency precision of the fault-related components reaches 10^(-4), but also overcomes the impacts of spectral leakage and non-integer-period sampling. The proposed research provides a new BRB detection method, which has enough precision to extract the parameters of the fault feature components.

A TRUST REGION METHOD FOR MICROWAVE TOMOGRAPHY

A trust region method is proposed to solve the problem of microwave tomography,which is very difficult to be solved for its ill-posedness and nonlinearity. Compared with the Levenberg-Marquardt method, this method introduces more a priori knowledge and might obtain better results, though the two methods are equal in some cases.