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Bayesian optimization relies on iteratively constructing and optimizing an acquisition function. The latter turns out to be a challenging, non-convex optimization problem itself. Despite the relative importance of this step, most algorithms employ sampling- or gradient-based methods, which do not provably converge to global optima. This work investigates mixed-integer programming (MIP) as a paradigm for global acquisition function optimization. Specifically, our Piecewise-linear Kernel Mixed Integer Quadratic Programming (PK-MIQP) formulation introduces a piecewise-linear approximation for Gaussian process kernels and admits a corresponding MIQP representation for acquisition functions. The proposed method is applicable to uncertainty-based acquisition functions for any stationary or dot-product kernel. We analyze the theoretical regret bounds of the proposed approximation, and empirically demonstrate the framework on synthetic functions, constrained benchmarks, and a hyperparameter tuning task. 

Abstract Primordial germ cells (PGCs) are the germline precursors that give rise to oocytes and sperm, ensuring the continuation of life. While the PGC specification is extensively studied, it remains elusive how the PGC population is sustained and expanded after they migrate to embryonic gonads before birth. This study demonstrates that NRF1, a known regulator for mitochondrial metabolism, plays critical roles in post‐migrating PGC development. We show that NRF1 protein level gradually increases in post‐migrating PGCs during embryonic development. ConditionalNrf1knockout from embryonic germ cells leads to impaired PGC proliferation and survival. In addition, NRF1 may also actively drive PGC derivation from pluripotent stem cells. Using whole genome transcriptome profiling and ChIP‐seq analyses, we further reveal that NRF1 directly regulates key signalling molecules in PGC formation, transcription factors in proliferation and cell cycle and enzymes in mitochondrial metabolism. Overall, our findings highlight an essential requirement of NRF1 in regulating a broad transcriptional network to support post‐migrating PGC development both in vitro and in vivo.