Many approaches for addressing Global Optimization problems typically rely on
relaxations of nonlinear constraints over specific mathematical primitives.
This is restricting in applications with constraints that are black-box,
implicit or consist of more general primitives. Trying to address such
limitations, Bertsimas and Ozturk (2023) proposed OCTHaGOn as a way of solving
black-box global optimization problems by approximating the nonlinear
constraints using hyperplane-based Decision-Trees and then using those trees to
construct a unified mixed integer optimization (MIO) approximation of the
original problem. We provide extensions to this approach, by (i) approximating
the original problem using other MIO-representable ML models besides Decision
Trees, such as Gradient Boosted Trees, Multi Layer Perceptrons and Suport
Vector Machines, (ii) proposing adaptive sampling procedures for more accurate
machine learning-based constraint approximations, (iii) utilizing robust
optimization to account for the uncertainty of the sample-dependent training of
the ML models, and (iv) leveraging a family of relaxations to address the
infeasibilities of the final MIO approximation. We then test the enhanced
framework in 81 Global Optimization instances. We show improvements in solution
feasibility and optimality in the majority of instances. We also compare
against BARON, showing improved optimality gaps or solution times in 11
instances.

通过使用基于超平面的决策树将非线性约束进行建模，并利用这些树构建原问题的统一混合整数优化（MIO）近似，本研究提出了扩展解决方案，包括使用其他可转化为 MIO 的机器学习模型（如梯度提升树、多层感知器和支持向量机）逼近原问题，提出自适应采样方法以获得更准确的基于机器学习的约束近似，利用鲁棒优化来考虑样本依赖性训练的不确定性，并利用一组松弛方法来处理最终的 MIO 近似的不可行性。然后我们在 81 个全局优化实例中测试了增强的框架，并展示了大多数实例中解的可行性和最优性的改进。我们还与 BARON 进行了比较，在 11 个实例中显示了更好的最优间隙或解决时间。

全局优化：机器学习方法

Global Optimization: A Machine Learning Approach

This paper investigates the transmission power control in over-the-air
federated edge learning (Air-FEEL) system. Different from conventional power
control designs (e.g., to minimize the individual mean squared error (MSE) of
the over-the-air aggregation at each round), we consider a new power control
design aiming at directly maximizing the convergence speed. Towards this end,
we first analyze the convergence behavior of Air-FEEL (in terms of the
optimality gap) subject to aggregation errors at different communication
rounds. It is revealed that if the aggregation estimates are unbiased, then the
training algorithm would converge exactly to the optimal point with mild
conditions; while if they are biased, then the algorithm would converge with an
error floor determined by the accumulated estimate bias over communication
rounds. Next, building upon the convergence results, we optimize the power
control to directly minimize the derived optimality gaps under both biased and
unbiased aggregations, subject to a set of average and maximum power
constraints at individual edge devices. We transform both problems into convex
forms, and obtain their structured optimal solutions, both appearing in a form
of regularized channel inversion, by using the Lagrangian duality method.
Finally, numerical results show that the proposed power control policies
achieve significantly faster convergence for Air-FEEL, as compared with
benchmark policies with fixed power transmission or conventional MSE
minimization.

研究了 Air-FEEL 系统中的传输功率控制，通过最大化收敛速度的方法优化功率控制策略，并使用拉格朗日对偶方法得到了优化解，结果表明该策略比传统策略实现了更快的收敛速度。