Many difficult computational problems involve the simultaneous satisfaction of multiple constraints that are individually easy to satisfy. These constraints might be derived from measurements (as in tomography or diffractive imaging), interparticle interactions (as in spin glasses), or a combination of sources (as in protein folding). We present a simple geometric framework to express and solve such problems and apply it to two benchmarks. In the first application (3SAT, a Boolean satisfaction problem), the resulting method exhibits similar performance scaling as a leading context-specific algorithm (WALKSAT). In the second application (sphere packing), the method allowed us to find improved solutions to some old and well-studied optimization problems. Based upon its simplicity and observed efficiency, we argue that this framework provides a competitive alternative to stochastic methods such as simulated annealing. © 2008 The American Physical Society.