A Retrospective Filter Trust Region Algorithm for Unconstrained Optimization

In this paper, we propose a retrospective filter trust region algorithm for unconstrained optimization, which is based on the framework of the retrospective trust region method and associated with the technique of the multi-dimensional filter. The new algorithm gives a good estimation of trust region radius, relaxes the condition of accepting a trial step for the usual trust region methods. Under reasonable assumptions, we analyze the global convergence of the new method and report the preliminary results of numerical tests. We compare the results with those of the basic trust region algorithm, the filter trust region algorithm and the retrospective trust region algorithm, which shows the effectiveness of the new algorithm.

A Non-Monotone Trust Region Method with Non-Monotone Wolfe-Type Line Search Strategy for Unconstrained Optimization

In this paper, we propose and analyze a non-monotone trust region method with non-monotone line search strategy for unconstrained optimization problems. Unlike the traditional non-monotone trust region method, our algorithm utilizes non-monotone Wolfe line search to get the next point if a trial step is not adopted. Thus, it can reduce the number of solving sub-problems. Theoretical analysis shows that the new proposed method has a global convergence under some mild conditions.

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.

Nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization

We propose a nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization. Unlike traditional trust region methods, the subproblem in our method is a simple conic model, where the Hessian of the objective function is approximated by a scalar matrix. The trust region radius is adjusted with a new self-adaptive adjustment strategy which makes use of the information of the previous iteration and current iteration. The new method needs less memory and computational efforts. The global convergence and Q-superlinear convergence of the algorithm are established under the mild conditions. Numerical results on a series of standard test problems are reported to show that the new method is effective and attractive for large scale unconstrained optimization problems.

A NEW NONMONOTONE TRUST REGION ALGORITHM FOR SOLVING UNCONSTRAINED OPTIMIZATION PROBLEMS

Based on the nonmonotone line search technique proposed by Gu and Mo （Appl. Math. Comput. 55, （2008） pp. 2158-2172）, a new nonmonotone trust region algorithm is proposed for solving unconstrained optimization problems in this paper. The new algorithm is developed by resetting the ratio ρk for evaluating the trial step dk whenever acceptable. The global and superlinear convergence of the algorithm are proved under suitable conditions. Numerical results show that the new algorithm is effective for solving unconstrained optimization problems.

CURVILINEAR PATHS AND TRUST REGION METHODS WITH NONMONOTONIC BACK TRACKING TECHNIQUE FOR UNCONSTRAINED OPTIMIZATION

Focuses on a study which examined the modification of type approximate trust region methods via two curvilinear paths for unconstrained optimization. Properties of the curvilinear paths; Description of a method which combines line search technique with an approximate trust region algorithm; Information on the convergence analysis; Details on the numerical experiments.

A NEW FAMILY OF TRUST REGION ALGORITHMS FOR UNCONSTRAINED OPTIMIZATION

Trust region (TR) algorithms are a class of recently developed algorithms for nonlinear optimization. A new family of TR algorithms for unconstrained optimization, which is the extension of the usual TR method, is presented in this paper. When the objective function is bounded below and continuously, differentiable, and the norm of the Hesse approximations increases at most linearly with the iteration number, we prove the global convergence of the algorithms. Limited numerical results are reported, which indicate that our new TR algorithm is competitive.

An improved trust region method for unconstrained optimization

In this paper,we propose an improved trust region method for solving unconstrained optimization problems.Different with traditional trust region methods,our algorithm does not resolve the subproblem within the trust region centered at the current iteration point,but within an improved one centered at some point located in the direction of the negative gradient,while the current iteration point is on the boundary set.We prove the global convergence properties of the new improved trust region algorithm and give the computational results which demonstrate the effectiveness of our algorithm.

A randomized nonmonotone adaptive trust region method based on the simulated annealing strategy for unconstrained optimization

Purpose–The purpose of this paper is to employ stochastic techniques to increase efficiency of the classical algorithms for solving nonlinear optimization problems.Design/methodology/approach–The well-known simulated annealing strategy is employed to search successive neighborhoods of the classical trust region(TR)algorithm.Findings–An adaptive formula for computing the TR radius is suggested based on an eigenvalue analysis conducted on the memoryless Broyden-Fletcher-Goldfarb-Shanno updating formula.Also,a(heuristic)randomized adaptive TR algorithm is developed for solving unconstrained optimization problems.Results of computational experiments on a set of CUTEr test problems show that the proposed randomization scheme can enhance efficiency of the TR methods.Practical implications–The algorithm can be effectively used for solving the optimization problems which appear in engineering,economics,management,industry and other areas.Originality/value–The proposed randomization scheme improves computational costs of the classical TR algorithm.Especially,the suggested algorithm avoids resolving the TR subproblems for many times.

A class of nonmonotone trust region algorithms for unconstrained optimization problems

A class of nonmonotone trust region algorithms is presented for unconstrained optimizations. Under suitable conditions, the global and Q quadratic convergences of the algorithm are proved. Several rules of choosing trial steps and trust region radii are also discussed.

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.

An Improved Line Search and Trust Region Algorithm

In this paper, we present a new line search and trust region algorithm for unconstrained optimization problems. The trust region center locates at somewhere in the negative gradient direction with the current best iterative point being on the boundary. By doing these, the trust region subproblems are constructed at a new way different with the traditional ones. Then, we test the efficiency of the new line search and trust region algorithm on some standard benchmarking. The computational results reveal that, for most test problems, the number of function and gradient calculations are reduced significantly.

Adaptive optimization methodology based on Kriging modeling and a trust region method

Surrogate-Based Optimization(SBO) is becoming increasingly popular since it can remarkably reduce the computational cost for design optimizations based on high-fidelity and expensive numerical analyses. However, for complicated optimization problems with a large design space, many design variables, and strong nonlinearity, SBO converges slowly and shows imperfection in local exploitation. This paper proposes a trust region method within the framework of an SBO process based on the Kriging model. In each refinement cycle, new samples are selected by a certain design of experiment method within a variable design space, which is sequentially updated by the trust region method. A multi-dimensional trust-region radius is proposed to improve the adaptability of the developed methodology. Further, the scale factor and the limit factor of the trust region are studied to evaluate their effects on the optimization process. Thereafter, different SBO methods using error-based exploration, prediction-based exploitation, refinement based on the expected improvement function, a hybrid refinement strategy, and the developed trust-regionbased refinement are utilized in four analytical tests. Further, the developed optimization methodology is employed in the drag minimization of an RAE2822 airfoil. Results indicate that it has better robustness and local exploitation capability in comparison with those of other SBO

A NONMONOTONE TRUST REGION TECHNIQUEFOR UNCONSTRAINED OPTIMIZAfIONPROBLEMS

Trust region methods with nonmonotone technique for unconstrained opti-mization problems are presented and analyzed. The convergence results are demonstratedfor the proposed algorithms even if the conditions are mild.

CONIC TRUST REGION METHOD FOR LINEARLY CONSTRAINED OPTIMIZATION

In this paper we present a trust region method of conic model for linearly constrained optimization problems. We discuss trust region approaches with conic model subproblems. Some equivalent variation properties and optimality conditions are given. A trust region algorithm based on conic model is constructed. Global convergence of the method is established.

A RETROSPECTIVE TRUST REGION ALGORITHM WITH TRUST REGION CONVERGING TO ZERO

We propose a retrospective trust region algorithm with the trust region converging to zero for the unconstrained optimization problem. Unlike traditional trust region algo- rithms, the algorithm updates the trust region radius according to the retrospective ratio, which uses the most recent model information. We show that the algorithm preserves the global convergence of traditional trust region algorithms. The superlinear convergence is also proved under some suitable conditions.

A SELF-ADAPTIVE TRUST REGION ALGORITHM

In this paper we propose a self-adaptive trust region algorithm. The trust region radius is updated at a variable rate according to the ratio between the actual reduction and the predicted reduction of the objective function, rather than by simply enlarging or reducing the original trust region radius at a constant rate. We show that this new algorithm preserves the strong convergence property of traditional trust region methods. Numerical results are also presented.

A NEW DERIVATIVE FREE OPTIMIZATION METHOD BASED ON CONIC INTERPOLATION MODEL

In this paper, a new derivative free trust region method is developed based on the conic interpolation model for the unconstrained optimization. The conic interpolation model is built by means of the quadratic model function, the collinear scaling formula, quadratic approximation and interpolation. All the parameters in this model are determined by objective function interpolation condition. A new derivative free method is developed based upon this model and the global convergence of this new method is proved without any information on gradient.

AN ADAPTIVE NONMONOTONIC TRUST REGION METHOD WITH CURVILINEAR SEARCHES

In this paper, an algorithm for unconstrained optimization that employs both trust region techniques and curvilinear searches is proposed. At every iteration, we solve the trust region subproblem whose radius is generated adaptively only once. Nonmonotonic backtracking curvilinear searches are performed when the solution of the subproblem is unacceptable. The global convergence and fast local convergence rate of the proposed algorithms are established under some reasonable conditions. The results of numerical ＇experiments are reported to show the effectiveness of the proposed algorithms.

An adaptive trust region method and its convergence

In this paper, a new trust region subproblem is proposed. The trust radius in the new subproblem adjusts itself adaptively. As a result, an adaptive trust region method is constructed based on the new trust region subproblem. The local and global convergence results of the adaptive trust region method are proved.Numerical results indicate that the new method is very efficient.