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该子问题可通过模拟退火算法来解决。The subproblem is solved by simulated annealing algorithm.

其中信赖域子问题的求解是关键的一部分。And how to solve the subproblem is the critical part of this method.

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也就是说，每个单元格包含原始问题的一个字问题的解。That is, each cell will contain a solution to a subproblem of the original problem.

本文提出“后向混合整数规划法”求解动态决策问题。This paper presents the "Backward Mixed-Integer programming Method" addressing the latter subproblem.

基于子问题的精确求解，该文证明了算法的收敛性。The convergence theorem of the proposed method is proved based on the exact solution of the subproblem.

进一步分析了这两个问题的组合复杂性，提出了分层求解方案。Then with the analysis on the combination of each subproblem , a logic process for solving them is presented.

在锥模型信赖域方法中，解锥模型信赖域子问题是关键，因此本文主要研究锥模型信赖域子问题及求解方法。This paper mainly discusses the conic trust region subproblem which is a key problem in conic trust region method.

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然后在ANSYS上建立了有限元模型，采用子问题近似方法进行了动力响应优化求解，获得了合理的优化结果。A finite element model is built using ANSYS, and the reasonable results are got by subproblem approximation method.

由于每个子问题可能仍然十分复杂，需要进一步的分解，这个过程就将不断的循环往复。The process will often be self-repeating since each subproblem may still be complex enough to require further decomposition.

基于广义消去法，本文中原问题可转化为等式约束矩阵的零空间中的一个无约束优化问题。By using the generalized elimination method, the subproblem is equivalent to an unconstrained optimization problem in the null space.

在很多具体问题上可证明子问题平衡对时问复杂性的常系数可以有改进。In many practical problems it is proved that the coefficient of time complexity can be improved by balancing the size of the subproblem.

利用最优性函数构造了松弛子问题的优化算法，并证明了算法的收敛性。The optimization algorithm for the relaxed subproblem is constructed by virtue of the optimality function and the convergence is proved.

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进一步，又提出了一类非精确交替方向法，每步迭代计算只需非精确求解子问题。Furthermore, the authors develop the proposed alternating direction method as an inexact method, which only needs to solve the subproblem inexactly.

既避免了重复求解信赖域子问题，又减少了线搜索方法计算函数值的次数。The method not only avoids resolving the subproblem repeatedly but also reduces the numbers of computing function values in the line search algorithm.

该算法通过构造子问题来寻找优于当前局部最优解的可行解。In this algorithm, a subproblem is set up to search for a new feasible point at which the value of the objective function is lower than the current local minimum.

作者利用有效集策略，在每次迭代中只需求解一个低维的信赖域子问题，该信赖域子问题是通过截断共轭梯度法来近似求解的。By using active set strategy, the authors need only to solve a reduced trust region subproblem which is solved inexactly by the truncated conjugate gradient method.

将该算法应用于二维布局优化子问题，数值实验表明该遗传神经网络进行布局逼近是有效的。Thereafter, the algorithm is used to solve 2D layout optimal subproblem. Numerical experiments show that approximating the layout by genetic neural network is effective.

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在每次迭代中，主方向是通过求解一个总有解的寻找方向子问题产生，而辅助方向是由一显式公式计算产生。At each iteration, a master direction is obtained by solving one direction finding subproblem which always possesses a solution, and an auxiliary direction is yielded by an explicit formula.

当试探步不成功时，并不重解子问题，而是利用非单调线搜索得到试探步及下一迭代点，有效地减少了计算量。It takes line search instead of resolving the subproblem to get the new trial step and the next iteration point when the trial step is not successful. It allows a considerable computational saving.