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 values 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 treatment of infinite sequences belonging to the constraint set. Equivalently, the problem requires to solve an infinite number of quadratic programs. Therefore, the main idea is to extract a finite number 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 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.

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凸和凹或凹且约束集是收敛的离散时间仿射系统的可达值集。此外，我们假定定义系统的矩阵是对角线的。问题的困难在于处理属于约束集的无限序列。同样，该问题需要解决无限数量的二次程序。因此，主要思想是提取有限数量的问题，并确保所提取问题的解决方案为初始问题提供最优值和最大化值。要解决的二次程序的数量必须尽可能少。其实，我们使用线性代数的基本思想构造了一个整数家族，该整数家族过度逼近二次方程式的确切数目以进行求解。此整数系列用于最终算法中。对算法中族整数的新计算可确保减少迭代次数。本文中提出的方法在较小的学术实例上得到了说明。最后，在随机生成的问题实例上对该算法进行了实验。