1. Instead of trying to solve the problem all at once there is an attempt to create sub-problems.

2. The subproblem is solved by simulated annealing algorithm.

3. And how to solve the subproblem is the critical part of this method.

4. By using the generalized elimination method, the subproblem is equivalent to an unconstrained optimization problem in the null space.

5. The optimization algorithm for the relaxed subproblem is constructed by virtue of the optimality function and the convergence is proved.

6. This paper mainly discusses the conic trust region subproblem which is a key problem in conic trust region method.

7. The process will often be self-repeating since each subproblem may still be complex enough to require further decomposition.

8. Chapter 2: A trust region subproblem model with memory is proposed.

9. In each decomposed subproblem, only parameters in projected state space related to its subgoal are reserved, and identical subproblems are integrated into one through features comparison.

10. At each iteration, the proposed subproblem consists of a strongly monotonic linear variational inequality and a well-conditioned system of nonlinear equations, which is easily to be solved.

11. Note that the subproblem() method does not copy the elements; it merely copies the array reference and offsets into an existing data structure.

12. That is, each cell will contain a solution to a subproblem of the original problem.

13. The current sequential quadratic programming ( SQP ) type algorithm may fail if the QP subproblem is infeasible.

14. Then with the analysis on the combination of each subproblem , a logic process for solving them is presented.

15. Furthermore, the authors develop the proposed alternating direction method as an inexact method, which only needs to solve the subproblem inexactly.

16. This paper presents the "Backward Mixed-Integer programming Method" addressing the latter subproblem.

17. The convergence theorem of the proposed method is proved based on the exact solution of the subproblem.

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18. I had to define the task closure, determine appropriate task granularity, split the subproblem, combine the results, and so on.

19. By virtue of this minimax problem as well as the directionally differentiable property of the functions in every subproblem, the first-order optimality conditions are obtained.

20. Thereafter, the algorithm is used to solve 2D layout optimal subproblem. Numerical experiments show that approximating the layout by genetic neural network is effective.

21. A finite element model is built using ANSYS, and the reasonable results are got by subproblem approximation method.

22. 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.

23. 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.