An Efficient Algorithm for Finding the Unique Best Approximation Pair Between Two Disjoint Intersections of Convex Sets

핵심 개념

The authors propose an efficient iterative algorithm based on projections onto the individual sets that generate the two disjoint intersections, without the need to project directly onto the intersections themselves, which can be computationally demanding. The algorithm is proven to converge to the unique best approximation pair under certain conditions.

초록

The paper addresses the problem of finding the best approximation pair between two nonempty and disjoint intersections of closed and convex sets in a Euclidean space. The authors propose an iterative algorithm inspired by the Halpern-Lions-Wittmann-Bauschke (HLWB) algorithm and the classical alternating process of Cheney and Goldstein.
The key highlights are:

The algorithm eliminates the need to project directly onto the intersection sets, which can be computationally demanding. Instead, it uses a weighted sum of projections onto the individual sets that generate the intersections.
The algorithm alternates between the two intersections, performing successive weighted sums of projections, with the number of projections increasing from one sweep to the next.
Under certain conditions, such as the sets being strictly convex, the authors prove that the algorithm converges to the unique best approximation pair.
The proof of convergence involves a generalization of Dini's theorem for uniform convergence of operators, as well as properties of fixed points of compositions of averaged nonexpansive operators.
The result extends the work of Aharoni et al. who considered the case of finite-dimensional polyhedra, by allowing for more general convex sets.

통계

None.

인용구

None.

핵심 통찰 요약

The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets

by Yair Censor,... 게시일 arxiv.org 04-25-2024

https://arxiv.org/pdf/2304.09600.pdf

더 깊은 질문

How can the proposed algorithm be extended to handle the case where the best approximation pair is not unique

To extend the proposed algorithm to handle cases where the best approximation pair is not unique, we can introduce a refinement step in the algorithm. After obtaining a candidate best approximation pair, we can evaluate the distance between this pair and the disjoint intersections of convex sets. If the distance is below a certain threshold, we can consider this pair as the best approximation pair. However, if the distance exceeds the threshold, we can iterate the algorithm with different initial conditions or adjust the weights in the weighted sums to explore alternative solutions. By incorporating this refinement step, the algorithm can adapt to situations where the best approximation pair may not be unique and explore multiple potential solutions.

Are there other types of convex sets, beyond strict convexity, for which the existence and uniqueness of the best approximation pair can be guaranteed

Beyond strict convexity, there are other types of convex sets for which the existence and uniqueness of the best approximation pair can be guaranteed. One such type is strongly convex sets. Strongly convex sets have a more pronounced curvature than strictly convex sets, leading to a unique best approximation pair. Additionally, polyhedral sets, which are defined by a finite number of linear inequalities, can also ensure the existence and uniqueness of the best approximation pair. By considering these types of convex sets in the algorithm, we can broaden its applicability to a wider range of scenarios while still guaranteeing the convergence to a unique best approximation pair.

What are the potential applications of this algorithm in real-world scenarios involving disjoint constraint sets

The proposed algorithm has various potential applications in real-world scenarios involving disjoint constraint sets. One application is in optimization problems where there are multiple constraints that need to be satisfied simultaneously. By finding the best approximation pair relative to two disjoint intersections of convex sets, the algorithm can help in identifying solutions that balance competing constraints effectively. This can be useful in engineering design, resource allocation, and decision-making processes where conflicting objectives need to be considered.
Another application is in signal processing and data analysis, where the algorithm can be used to find optimal solutions that adhere to different constraints or criteria. For example, in image processing, the algorithm can be applied to find the best approximation pair for image reconstruction from incomplete or noisy data, ensuring that the reconstructed image satisfies both fidelity to the original data and smoothness constraints. Overall, the algorithm's ability to handle disjoint constraint sets makes it versatile for various optimization and modeling tasks in diverse fields.