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Self-optimizing efficiency of vapor compression cycles (VCCs) involves assigning multiple decision variables simultaneously in order to minimize power consumption while maintaining safe operating conditions. Due to the modeling complexity associated with cycle dynamics (and other smart building energy systems), online self-optimization requires algorithms that can safely and efficiently explore the search space in a derivative-free and model-agnostic manner. This makes Bayesian optimization (BO) a strong candidate for self-optimization. Unfortunately, classical BO algorithms ignore the relationship between consecutive optimizer candidates, resulting in jumps in the search space that can lead to fail-safe mechanisms being triggered, or undesired transient dynamics that violate operational constraints. To this end, we propose safe local search region (LSR)-BO, a global optimization methodology that builds on the BO framework while enforcing two types of safety constraints including black-box constraints on the output and LSR constraints on the input. We provide theoretical guarantees that under standard assumptions on the performance and constraint functions, LSR-BO guarantees constraints will be satisfied at all iterations with high probability. Furthermore, in the presence of only input LSR constraints, we show the method will converge to the true (unknown) globally optimal solution. We demonstrate the potential of our proposed LSR-BO method on a high-fidelity simulation model of a commercial vapor compression system with both LSR constraints on expansion valve positions and fan speeds, in addition to other safety constraints on discharge and evaporator temperatures.