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.     

解非线性优化问题的锥模型信赖域方法

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.     

解无约束最优化的基于锥模型的过滤集- 信赖域方法

锥模型优化方法是一类非二次模型优化方法, 它在每次迭代中比标准的二次模型方法含有更丰富的插值信息. Di 和Sun (1996) 提出了解无约束优化问题的锥模型信赖域方法. 本文根据Fletcher 和Leyffer (2002) 的过滤集技术的思想, 在Di 和Sun (1996) 工作的基础上, 提出了解无约束优化问题的基于锥模型的过滤集信赖域算法. 在适当的条件下, 我们证明了新算法的收敛性. 有限的数值试验结果表明新算法是有效的.     

解新锥模型信赖域子问题的折线法

本文以新锥模型信赖域子问题的最优性条件为理论基础,认真讨论了新子问题的锥函数性质,分析了此函数在梯度方向及与牛顿方向连线上的单调性.在此基础上本文提出了一个求解新锥模型信赖域子问题折线法,并证明了这一子算法保证解无约束优化问题信赖域法全局收敛性要满足的下降条件.本文获得的数值实验表明该算法是有效的.     

Globally convergent DC trust-region methods

In this paper, we investigate the use of DC (Difference of Convex functions) models and algorithms in the application of trust-region methods to the solution of a class of nonlinear optimization problems where the constrained set is closed and convex (and, from a practical point of view, where projecting onto the feasible region is computationally affordable). We consider DC local models for the quadratic model of the objective function used to compute the trust-region step, and apply a primal-dual subgradient method to the solution of the corresponding trust-region subproblems. One is able to prove that the resulting scheme is globally convergent to first-order stationary points. The theory requires the use of exact second-order derivatives but, in turn, the computation of the trust-region step asks only for one projection onto the feasible region (in comparison to the calculation of the generalized Cauchy point which may require more). The numerical efficiency and robustness of the proposed new scheme when applied to bound-constrained problems is measured by comparing its performance against some of the current state-of-the-art nonlinear programming solvers on a vast collection of test problems.     

Adaptive Algorithm for Constrained Least-Squares Problems

This paper is concerned with the implementation and testing of an algorithm for solving constrained least-squares problems. The algorithm is an adaptation to the least-squares case of sequential quadratic programming (SQP) trust-region methods for solving general constrained optimization problems. At each iteration, our local quadratic subproblem includes the use of the Gauss–Newton approximation but also encompasses a structured secant approximation along with tests of when to use this approximation. This method has been tested on a selection of standard problems. The results indicate that, for least-squares problems, the approach taken here is a viable alternative to standard general optimization methods such as the Byrd–Omojokun trust-region method and the Powell damped BFGS line search method.     

A trust-region method with a conic model for unconstrained optimization

In this paper, we propose and analyze a new conic trust-region algorithm for solving the unconstrained optimization problems. A new strategy is proposed to construct the conic model and the relevant conic trust-region subproblems are solved by an approximate solution method. This approximate solution method is not only easy to implement but also preserves the strong convergence properties of the exact solution methods. Under reasonable conditions, the locally linear and superlinear convergence of the proposed algorithm is established. The numerical experiments show that this algorithm is both feasible and efficient. Copyright © 2008 John Wiley & Sons, Ltd.     

On the Global Convergence of a Projective Trust Region Algorithm for Nonlinear Equality Constrained Optimization

A trust-region sequential quadratic programming (SQP) method is developed and analyzed for the solution of smooth equality constrained optimization problems. The trust-region SQP algorithm is based on filter line search technique and a composite-step approach, which decomposes the overall step as sum of a vertical step and a horizontal step. The algorithm includes critical modifications of horizontal step computation. One orthogonal projective matrix of the Jacobian of constraint functions is employed in trust-region subproblems. The orthogonal projection gives the null space of the transposition of the Jacobian of the constraint function. Theoretical analysis shows that the new algorithm retains the global convergence to the first-order critical points under rather general conditions. The preliminary numerical results are reported.     

A nonmonotone trust-region method of conic model for unconstrained optimization

In this paper, we present a nonmonotone trust-region method of conic model for unconstrained optimization. The new method combines a new trust-region subproblem of conic model proposed in [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231] with a nonmonotone technique for solving unconstrained optimization. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the method of [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231].     

Projected Hessian algorithm with backtracking interior point technique for linear constrained optimization

In this paper, we propose a new trust-region-projected Hessian algorithm with nonmonotonic backtracking interior point technique for linear constrained optimization. By performing the QR decomposition of an affine scaling equality constraint matrix, the conducted subproblem in the algorithm is changed into the general trust-region subproblem defined by minimizing a quadratic function subject only to an ellipsoidal constraint. By using both the trust-region strategy and the line-search technique, each iteration switches to a backtracking interior point step generated by the trustregion subproblem. The global convergence and fast local convergence rates for the proposed algorithm are established under some reasonable assumptions. A nonmonotonic criterion is used to speed up the convergence in some ill-conditioned cases. Selected from Journal of Shanghai Normal University (Natural Science), 2003, 32(4): 7–13