收敛离散仿射系统可达值集上的二次极大化：可对角化情形
Quadratic Maximization over the Reachable Values Set of a Convergent Discrete-time Affine System : The diagonalizable case
摘要：
在本文中，我们解决了一个最大化问题，其中目标函数是二次和凸或凹的，约束集是收敛的离散仿射系统的可达值集。此外，我们假设定义系统的矩阵是可对角化的。该问题的难点在于约束集中需要处理的无穷序列。等价地，该问题需要求解无穷多个二次规划。因此，主要思想是提取有限的，并保证提取的问题的解决方案提供了初始问题的最优值和最大化。要解决的二次规划的数量必须尽可能小。实际上，我们使用线性代数的基本思想构造了一个整数族，它过近似于要求解的二次规划的精确数目。这个整数族用于最终算法。一个新的计算的一个整数的家庭内的算法，确保减少循环迭代的次数。文中所提出的方法是说明小的学术例子。最后，该算法进行了实验，随机生成的问题的实例。
英文摘要：
In this paper, we solve a maximization problem where the objective function is quadratic and convex or concave and the constraints set is the reachable value set of a convergent discrete-time affine system. Moreover, we assume that the matrix defining the system is diagonalizable. The difficulty of the problem lies in the infinite sequence to handle in the constraint set. Equivalently, the problem requires to solve an infinite number of quadratic programs. Therefore, the main idea is to extract a finite of them and to guarantee that the resolution of the extracted problems provides the optimal value and a maximizer for the initial problem. The number of quadratic programs to solve has to be the smallest possible. Actually, we construct a family of integers that over-approximate the exact number of quadratic programs to solve using basic ideas of linear algebra. This family of integers is used in the final algorithm. A new computation of an integer of the family within the algorithm ensures a reduction of the number of loop iterations. The method proposed in the paper is illustrated on small academic examples. Finally, the algorithm is experimented on randomly generated instances of the problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
