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Designing novel proteins with desired characteristics remains a significant challenge due to the large sequence space and the complexity of sequence-function relationships. Efficient exploration of this space to identify sequences that meet specific design criteria is crucial for advancing therapeutics and biotechnology. Here, we present BoGA (Bayesian Optimization Genetic Algorithm), a framework that combines evolutionary search with Bayesian optimization to efficiently navigate the sequence space. By integrating a genetic algorithm as a stochastic proposal generator within a surrogate modeling loop, BoGA prioritizes candidates based on prior evaluations and surrogate model predictions, enabling data-efficient optimization. We demonstrate the utility of BoGA through benchmarking on sequence and structure design tasks, followed by its application in designing peptide binders against pneumolysin, a key virulence factor of \textit{Streptococcus pneumoniae}. BoGA accelerates the discovery of high-confidence binders, demonstrating the potential for efficient protein design across diverse objectives. The algorithm is implemented within the BoPep suite and is available under an MIT license at \href{this https URL}{GitHub}.