In hyperspectral target detection, one must contend with variability in both target materials and background clutter. While most algorithms focus on the background clutter, there are some materials for which there is substantial variability in the signatures of the target. When multiple signatures can be used to describe a target material, subspace detectors are often the detection algorithm of choice. However, as the number of variable target spectra increases, so does the size of the target subspace spanned by these spectra, which in turn increases the number of false alarms. Here, we propose a modification to this approach, wherein the target subspace is instead a constrained subspace, or a simplex without the sum-to-one constraint. We derive the simplex adaptive matched filter (simplex AMF) and the simplex adaptive cosine estimator (simplex ACE), which are constrained basis adaptations of the traditional subspace AMF and subspace ACE detectors. We present results using simplex AMF and simplex ACE for variable targets, and compare their performances against their subspace counterparts. Our primary interest is in the simplex ACE detector, and as such, the experiments herein seek to evaluate the robustness of simplex ACE, with simplex AMF included for comparison. Results are shown on hyperspectral images using both implanted and ground-truthed targets, and demonstrate the robustness of simplex ACE to target variability.