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1. Sequential convex programming for non-linear stochastic optimal control

   https://doi.org/10.1051/cocv/2022060  

   Bonalli, Riccardo; Lew, Thomas; Pavone, Marco (January 2022, ESAIM: Control, Optimisation and Calculus of Variations)

   This work introduces a sequential convex programming framework for non-linear, finitedimensional stochastic optimal control, where uncertainties are modeled by a multidimensional Wiener process. We prove that any accumulation point of the sequence of iterates generated by sequential convex programming is a candidate locally-optimal solution for the original problem in the sense of the stochastic Pontryagin Maximum Principle. Moreover, we provide sufficient conditions for the existence of at least one such accumulation point. We then leverage these properties to design a practical numerical method for solving non-linear stochastic optimal control problems based on a deterministic transcription of stochastic sequential convex programming. 

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2. An Efficient Way to Find Optimal Crossover Designs Using CVX for Precision Medicine: Optimal Crossover Designs via CVX

   https://doi.org/10.52933/jdssv.v4i3.83  

   Li, Yin; Wong, Weng Kee; Zhou, Hua; Chough_Carriere, Keumhee (June 2024, Journal of Data Science, Statistics, and Visualisation)

   Crossover designs play an increasingly important role in precision medicine. We show the search of an optimal crossover design can be formulated as a convex optimization problem and convex optimization tools, such as CVX, can be directly used to search for an optimal crossover design.  We first demonstrate how to transform crossover design problems into convex optimization problems and show CVX can effortlessly find optimal crossover designs that coincide with a few theoretical crossover optimal designs in the literature. The proposed approach is especially useful when it becomes problematic to construct optimal designs analytically for complicated models. We then apply CVX to find crossover designs for models with auto-correlated error structures or when the information matrices may be singular and analytical answers are unavailable. We also construct N-of-1 trials frequently used in precision medicine to estimate treatment effects on the individuals or to estimate average treatment effects, including finding dual-objective optimal crossover designs. 

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3. An uncertainty-aware deep learning framework-based robust design optimization of metamaterial units

   https://doi.org/10.1007/s00158-025-03979-z  

   Wang, Zihan; Bhaduri, Anindya; Xu, Hongyi; Wang, Liping (March 2025, Structural and Multidisciplinary Optimization)

   Mechanical metamaterials represent an innovative class of artificial structures, distinguished by their extraordinary mechanical characteristics, which are beyond the scope of traditional natural materials. The use of deep generative models has become increasingly popular in the design of metamaterial units. The effectiveness of using deep generative models lies in their capacity to compress complex input data into a simplified, lower-dimensional latent space, while also enabling the creation of novel optimal designs through sampling within this space. However, the design process does not take into account the effect of model uncertainty due to data sparsity or the effect of input data uncertainty due to inherent randomness in the data. This might lead to the generation of undesirable structures with high sensitivity to the uncertainties in the system. To address this issue, a novel uncertainty-aware deep learning framework-based robust design approach is proposed for the design of metamaterial units with optimal target properties. The proposed approach utilizes the probabilistic nature of the deep learning framework and quantifies both aleatoric and epistemic uncertainties associated with surrogate-based design optimization. We demonstrate that the proposed design approach is capable of designing high-performance metamaterial units with high reliability. To showcase the effectiveness of the proposed design approach, a single-objective design optimization problem and a multi-objective design optimization problem are presented. The optimal robust designs obtained are validated by comparing them to the designs obtained from the topology optimization method as well as the designs obtained from a deterministic deep learning framework-based design optimization where none of the uncertainties in the system are explicitly considered. 

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4. Sparse feature selection in kernel discriminant analysis via optimal scoring

   Lapanowski, Alexander F.; Gaynanova, Irina (April 2019, Proceedings of Machine Learning Research)

   We consider the two-group classification problem and propose a kernel classifier based on the optimal scoring framework. Unlike previous approaches, we provide theoretical guarantees on the expected risk consistency of the method. We also allow for feature selection by imposing structured sparsity using weighted kernels. We propose fully-automated methods for selection of all tuning parameters, and in particular adapt kernel shrinkage ideas for ridge parameter selection. Numerical studies demonstrate the superior classification performance of the proposed approach compared to existing nonparametric classifiers. 

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5. Optimal Sets and Solution Paths of ReLU Networks

   Mishkin, Aaron; Pilanci, Mert (January 2023, International Conference on Machine Learning)

   We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the non-convex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the non-convex training objective. Since all stationary points of the ReLU training problem can be represented as optima of sub-sampled convex programs, our work provides a general expression for all critical points of the non-convex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks. 

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