AI 中文总结
针对不一致情形下的通用动态串平均法，提出新择一定理，给出优化方法实现更优可行点的充分条件，解决了现有文献未完全覆盖的优化保证问题。
AI 中文摘要
我们在不一致情形下的通用动态串平均法（GDSA）语境中研究优化方法（SM），其中输入算子没有公共不动点，该方法主要旨在实现凸可行性，同时最小化目标函数。在许多建模为约束最小化任务的科学和现实问题中，追求精确的约束最优解会消耗大量时间、能量和资源，因此应用优化方法（SM）可提供实用高效的替代方案。特别地，我们为优化方法提出了一种新的“择一定理”，这有助于研究理论条件，在此条件下，GDSA算法的优化版本收敛到一个“更优”的可行点，即其目标函数值小于等于未受扰动的可行性寻求算法产生的值。尽管现有文献仅部分解决了该问题，但我们提出了新的充分条件，保证优化方法（SM）能获得这样的更优结果。
英文摘要
We study the Superiorization Methodology (SM) in the context of the General Dynamic String-Averaging (GDSA) method in the inconsistent case (that is, where the input operators don't have a common fixed point) which primarily aims at achieving convex feasibility while simultaneously reducing an objective function. In many scientific and real-world problems modeled as constrained minimization tasks, striving for the exact constrained optimum can be costly in terms of time, energy, and resources. Therefore, applying the SM can offer a practical and efficient alternative. In particular, we present a new "theorem of alternatives" for the superiorization method which leads to investigation of theoretical conditions under which the superiorized version of the GDSA algorithm converges to a "superior" feasible point, i.e., one with an objective function value that is smaller or equal to that produced by the unperturbed feasibility-seeking algorithm. While this question has only been partially addressed in the existing literature, we present new sufficient conditions that guarantee that the SM attains such a superior outcome.
CommentsAccepted for publication in Journal of Fixed Point Theory and Applications. 25 pages
