  The word "valley" is a popular term used in intuitively describing fitness
landscapes. What is a valley on a fitness landscape? How to identify the
direction and location of a valley if it exists? However, such questions are
seldom rigorously studied in evolutionary optimization especially when the
search space is a high dimensional continuous space. This paper presents two
methods of studying valleys on a fitness landscape. The first method is based
on the topological homeomorphism. It establishes a rigorous definition of a
valley. A valley is regarded as a one-dimensional manifold. The second method
takes a different viewpoint from statistics. It provides an algorithm of
identifying the valley direction and location using principle component
analysis.
