Non-derivative methods for optimization have had a sometimes rocky relationship for more than 50 years with applied mathematicians who specialize in optimization.
Although practitioners have never wavered in their fondness for non-derivative methods, their mathematical foundations were mostly lacking until the late 1980s. Since then, significant progress has been made concerning theoretical underpinnings, but several perplexing mysteries remain.
In addition, there has been continuing and lively controversy about which methods are ``most effective''
on real-world applications, with disagreements about both the selection of test problems and the choice of criteria for assessing computational results. This talk will briefly survey the current state of the art, trying along the way to highlight a few of the interesting open questions.