Fortified-Descent Simplicial Search Method: A General Approach

Abstract

We propose a new simplex-based direct search method for unconstrained minimization of a real-valued function f of n variables. As in other methods of this kind, the intent is to iteratively improve an n-dimensional simplex through certain reflection/expansion/contraction steps. The method has three novel features. First, a user-chosen integer $\bar m_k$ specifies the number of good vertices to be retained in constructing the initial trial simplices---reflected, then either expanded or contracted---at iteration k. Second, a trial simplex is accepted only when it satisfies the criteria of fortified descent, which are stronger than the criterion of strict descent used in most direct search methods. Third, the number of additional function evaluations needed to check a trial reflected/expanded simplex for fortified descent can be controlled. If one of the initial trial simplices satisfies the fortified-descent criteria, it is accepted as the new simplex; otherwise, the simplex is shrunk a fraction of the way toward a best vertex and the process is restarted, etc., until either a trial simplex is accepted or the simplex effectively has shrunk to a single point. We prove several theoretical properties of the new method. If f is continuously differentiable, bounded below, and uniformly continuous on its lower level set and we choose $\bar m_k$ with the same value at all iterations k, then every cluster point of the generated sequence of iterates is a stationary point. The same conclusion holds if the function is continuously differentiable, bounded below, and we choose $\bar m_k=1$ at all iterations k.