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Minimization methods can be modeled by iteration mappings that generate new iterate-multisets from current iterate-multisets. We introduce the notion of viability to indicate when an initial iterate-multiset generates a sequence of non-empty iterate-multisets through the repeated application of an iteration mapping. We review the minimization methods of coordinate-search, steepest-descent (with various line-searches), and Nelder–Mead, and we define the corresponding iteration mappings and discuss viability in each case. We prove an analogue to the classical Banach fixed-point theorem for iteration mappings.