Populations unable to evolve to selectively favored states are constrained. Genetic constraints occur when additive genetic variance in selectively favored directions is absent (absolute constraints) or present but small (quantitative constraints). Quantitative--unlike absolute--constraints are presumed surmountable given time. This ignores that a population might become extinct before reaching the favored state, in which case demography effectively converts a quantitative into an absolute constraint. Here, we derive criteria for predicting when such conversions occur. We model the demography and evolution of populations subject to optimizing selection that experience either a single shift or a constant change in the optimum. In the single-shift case, we consider whether a population can evolve significantly without declining or else declines temporarily while avoiding low sizes consistent with high extinction risk. We analyze when populations in constantly changing environments evolve sufficiently to ensure long-term growth. From these, we derive formulas for critical levels of genetic variability that define demography-caused absolute constraints. The formulas depend on estimable properties of fitness, population size, or environmental change rates. Each extends to selection on multivariate traits. Our criteria define the nearly null space of a population's G matrix, the set of multivariate directions effectively inaccessible to it via adaptive evolution.