Constrained optimization in high-dimensional black-box settings is difficult due to expensive evaluations, the lack of gradient information, and complex feasibility regions. In this work, we propose a Bayesian optimization method that combines a penalty formulation, a surrogate model, and a trust region strategy. The constrained problem is converted to an unconstrained form by penalizing constraint violations, which provides a unified modeling framework. A trust region restricts the search to a local region around the current best solution, which improves stability and efficiency in high dimensions. Within this region, we use the Expected Improvement acquisition function to select evaluation points by balancing improvement and uncertainty. The proposed Trust Region method integrates penalty-based constraint handling with local surrogate modeling. This combination enables efficient exploration of feasible regions while maintaining sample efficiency. We compare the proposed method with state-of-the-art methods on synthetic and real-world high-dimensional constrained optimization problems. The results show that the method identifies high-quality feasible solutions with fewer evaluations and maintains stable performance across different settings.

In this work, we present a new deterministic partition-based global optimization algorithm, HALO (Hybrid Adaptive Lipschitzian Optimization), which uses estimates of the local Lipschitz constants associated with different sub-regions of the objective function's domain to compute lower bounds and guide the search toward global minimizers. These estimates are obtained by adaptively balancing the global and local information collected from the algorithm, based on absolute slopes. HALO is hyperparameter-free, eliminating the need for manual tuning, and it highlights the most important variables to help interpret the optimization problem. We also introduce a coupling strategy with local optimization algorithms, both gradient-based and derivative-free, to accelerate convergence. We compare HALO with popular global optimization algorithms on hundreds of test functions. The numerical results are very promising and demonstrate that HALO can expand our arsenal of efficient procedures of efficient procedures for challenging real-world black-box optimization problems. The Python code of HALO is publicly available on GitHub. https://github.com/dannyzx/HALO

We present a derivation and theoretical investigation of the Adams-Bashforth and Adams-Moulton family of linear multistep methods for solving ordinary differential equations, starting from a Gaussian process (GP) framework. In the limit, this formulation coincides with the classical deterministic methods, which have been used as higher-order initial value problem solvers for over a century. Furthermore, the natural probabilistic framework provided by the GP formulation allows us to derive probabilistic versions of these methods, in the spirit of a number of other probabilistic ODE solvers presented in the recent literature. In contrast to higher-order Runge-Kutta methods, which require multiple intermediate function evaluations per step, Adams family methods make use of previous function evaluations, so that increased accuracy arising from a higher-order multistep approach comes at very little additional computational cost. We show that through a careful choice of covariance function for the GP, the posterior mean and standard deviation over the numerical solution can be made to exactly coincide with the value given by the deterministic method and its local truncation error respectively. We provide a rigorous proof of the convergence of these new methods, as well as an empirical investigation (up to fifth order) demonstrating their convergence rates in practice.

Bayesian optimization (BO) is a model-based approach for gradient-free black-box function optimization, such as hyperparameter optimization. Typically, BO relies on conventional Gaussian process (GP) regression, whose algorithmic complexity is cubic in the number of evaluations. As a result, GP-based BO cannot leverage large numbers of past function evaluations, for example, to warm-start related BO runs. We propose a multi-task adaptive Bayesian linear regression model for transfer learning in BO, whose complexity is linear in the function evaluations: one Bayesian linear regression model is associated to each black-box function optimization problem (or task), while transfer learning is achieved by coupling the models through a shared deep neural net. Experiments show that the neural net learns a representation suitable for warm-starting the black-box optimization problems and that BO runs can be accelerated when the target black-box function (e.g., validation loss) is learned together with other related signals (e.g., training loss). The proposed method was found to be at least one order of magnitude faster that methods recently published in the literature.

Global optimization of expensive, potentially gradient-free functions has long been a critical component of many complex problems in science and engineering.

A.1 Proof of Proposition 4.1 Proposition 4.1 For any bounded measurable function τ(t): [0, T ] R, the following Reverse SDEs [ (1 + τ Eq. (20) is a reverse-time SDE running[ from T to 0, thus (there)are two additional minus ] signs in Eq. (21) before term A.2 Two Reparameterizations and Exact Solution under Exponential Integrator In this subsection, we will show the exact solution of SDE in both data prediction reparameterization and noise prediction reparameterization. The noise term in data prediction has smaller variance than noise prediction ones, implying the necessity of adopting data prediction reparameterization for the SDE sampler. The computation of variance uses the Itô Isometry, which is a crucial fact of Itô integral. Similar with Proposition 4.2, Eq. (37) can be solved analytically, which is shown in the following propositon: Following the derivation in Proposition 4.2, the mean of the Itô integral term is: [ A.2.4 Comparison between Data and Noise Reparameterizations In Table 1 we perform an ablation study on data and noise reparameterizations, the experiment results show that under the same magnitude of stochasticity, the proposed SA-Solver in data reparameterization has a better convergence which leads to better FID results under the same NFEs. In this subsection, we provide a theoretical view of this phenomenon.

Diffusion Probabilistic Models (DPMs) have achieved considerable success in generation tasks. As sampling from DPMs is equivalent to solving diffusion SDE or ODE which is time-consuming, numerous fast sampling methods built upon improved differential equation solvers are proposed.

Diffusion-based generative models, as first introduced by Sohl-Dickstein et al. (2015), have shown