通过编码进行优化:重整化组的视角

Konstantin Klemm, Anita Mehta, Peter F Stadler
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摘要

难点优化问题,特别是np完全优化问题,传统上是用搜索启发式近似求解的。由于局部极小值阻碍了搜索过程,因此遇到的崎岖地形通常会减慢搜索速度。覆盖编码地图的设计是为了规避这个问题,通过将原始景观转变为一个没有局部最小值和丰富的近最优解决方案的景观。根据定义,这些涉及到将原始(较大)搜索空间映射到较小的子空间,通过通常相当于某种形式的粗粒度处理。在本文中,我们使用形式参数以及覆盖编码映射的具体示例来探索这种粗粒度的细节,这些示例通过分析和计算进行了研究。我们的结果强烈地表明,覆盖编码映射中的粗粒度与重整化群方案中遇到的粗粒度具有很强的相似性。考虑到这两种形式明显不同的本质,这些强烈的相似性是相当惊人的,并暗示了等待进一步探索的深层数学基础。
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Optimisation via encodings: a renormalisation group perspective
Abstract Difficult, in particular NP-complete, optimization problems are traditionally solved approximately using search heuristics. These are usually slowed down by the rugged landscapes encountered, because local minima arrest the search process. Cover-encoding maps were devised to circumvent this problem by transforming the original landscape to one that is free of local minima and enriched in near-optimal solutions. By definition, these involve the mapping of the original (larger) search space into smaller subspaces, by processes that typically amount to a form of coarse-graining. In this paper, we explore the details of this coarse-graining using formal arguments, as well as concrete examples of cover-encoding maps, that are investigated analytically as well as computationally. Our results strongly suggest that the coarse-graining involved in cover-encoding maps bears a strong resemblance to that encountered in renormalisation group schemes. Given the apparently disparate nature of these two formalisms, these strong similarities are rather startling, and suggest deep mathematical underpinnings that await further exploration.
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