Unconstrained binary integer programming (UBIP) is a challenging optimization problem due to the presence of binary variables. To address the challenge, we introduce a novel class of functions named sharp-peak functions (SPFs), which equivalently reformulate the binary constraints as equality constraints, giving rise to an SPF-constrained optimization. Rather than solving this constrained reformulation directly, we focus on its associated penalty model. The established exact penalty theory shows that the global minimizers of UBIP and the penalty model coincide when the penalty parameter exceeds a threshold, a constant independent of the solution set of UBIP. To analyze the penalty model, we introduce Karush-Kuhn-Tucker (KKT) points and a new type of stationarity, referred to as P-stationarity, and provide a comprehensive characterization of its optimality conditions. We then develop an efficient algorithm called ShaPeak based on the inexact alternating direction methods of multipliers, which is guaranteed to converge to a P-stationary point at a linear rate under appropriate parameter choices and a single mild assumption, namely, the local Lipschitz continuity of the gradient over a bounded box. Finally, numerical experiments demonstrate the high performance of ShaPeak in comparison to several established solvers.