Bayesian optimization over the latent spaces of deep autoencoder models (DAEs) has recently emerged as a promising new approach for optimizing challenging black-box functions over structured, discrete, hard-to-enumerate search spaces (e.g., molecules). Here the DAE dramatically simplifies the search space by mapping inputs into a continuous latent space where familiar Bayesian optimization tools can be more readily applied. Despite this simplification, the latent space typically remains high-dimensional. Thus, even with a well-suited latent space, these approaches do not necessarily provide a complete solution, but may rather shift the structured optimization problem to a high-dimensional one. In this paper, we propose LOL-BO, which adapts the notion of trust regions explored in recent work on high-dimensional Bayesian optimization to the structured setting.

We study two of the most popular performance metrics in medical image segmentation, Accuracy and Dice, when the target labels are noisy. For both metrics, several statements related to characterization and volume properties of the set of optimal segmentations are proved, and associated experiments are provided. Our main insights are: (i) the volume of the solutions to both metrics may deviate significantly from the expected volume of the target, (ii) the volume of a solution to Accuracy is always less than or equal to the volume of a solution to Dice and (iii) the optimal solutions to both of these metrics coincide when the set of feasible segmentations is constrained to the set of segmentations with the volume equal to the expected volume of the target.

The incorporation of cutting planes within the branch-and-bound algorithm, known as branch-and-cut, forms the backbone of modern integer programming solvers. These solvers are the foremost method for solving discrete optimization problems and thus have a vast array of applications in machine learning, operations research, and many other fields. Choosing cutting planes effectively is a major research topic in the theory and practice of integer programming. We conduct a novel structural analysis of branch-and-cut that pins down how every step of the algorithm is affected by changes in the parameters defining the cutting planes added to the input integer program. Our main application of this analysis is to derive sample complexity guarantees for using machine learning to determine which cutting planes to apply during branch-and-cut.

This paper addresses the problem of Unbalanced Optimal Transport (UOT) in which the marginal conditions are relaxed (using weighted penalties in lieu of equality) and no additional regularization is enforced on the OT plan. In this context, we show that the corresponding optimization problem can be reformulated as a non-negative penalized linear regression problem. This reformulation allows us to propose novel algorithms inspired from inverse problems and nonnegative matrix factorization. In particular, we consider majorization-minimization which leads in our setting to efficient multiplicative updates for a variety of penalties. Furthermore, we derive for the first time an efficient algorithm to compute the regularization path of UOT with quadratic penalties.

Recent years have witnessed a surge of approaches to use neural networks to help tackle combinatorial optimization problems, including graph optimization problems. However, theoretical understanding of such approaches remains limited. In this paper, we consider the geometric setting, where graphs are induced by points in a fixed dimensional Euclidean space. We show that several graph optimization problems can be approximated by an algorithm that is polynomial in graph size n via a framework we propose, call the Baker-paradigm. More importantly, a key advantage of the Baker-paradigm is that it decomposes the input problem into (at most linear number of) small sub-problems of fixed sizes (independent of the size of the input).

In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods. Our unified theorem only requires to verify several representative conditions and is not tailored to any specific algorithm. As a direct application, we recover expected and almost sure convergence results of the stochastic gradient method (SGD) and random reshuffling (RR) under more general settings. Moreover, we establish new expected and almost sure convergence results for the stochastic proximal gradient method (prox-SGD) and stochastic model-based methods for nonsmooth nonconvex optimization problems. These applications reveal that our unified theorem provides a plugin-type convergence analysis and strong convergence guarantees for a wide class of stochastic optimization methods.

Generative Adversarial Networks (GANs) have gained prominence in refining model fitting tasks in computer vision, particularly in domains involving deformable models like Active Appearance Models (AAMs). This paper explores the integration of GANs to enhance the AAM fitting process, addressing challenges in optimizing nonlinear parameters associated with appearance and shape variations. By leveraging GANs' adversarial training framework, the aim is to minimize fitting errors and improve convergence rates. Achieving robust performance even in cases with high appearance variability and occlusions. Our approach demonstrates significant improvements in accuracy and computational efficiency compared to traditional optimization techniques, thus establishing GANs as a potent tool for advanced image model fitting.

Efficient and socially equitable restoration of transportation networks post disasters is crucial for community resilience and access to essential services. The ability to rapidly recover critical infrastructure can significantly mitigate the impacts of disasters, particularly in underserved communities where prolonged isolation exacerbates vulnerabilities. Traditional restoration methods prioritize functionality over computational efficiency and equity, leaving low-income communities at a disadvantage during recovery. To address this gap, this research introduces a novel framework that combines quantum computing technology with an equity-focused approach to network restoration. Optimization of road link recovery within budget constraints is achieved by leveraging D Wave's hybrid quantum solver, which targets the connectivity needs of low, average, and high income communities. This framework combines computational speed with equity, ensuring priority support for underserved populations. Findings demonstrate that this hybrid quantum solver achieves near instantaneous computation times of approximately 8.7 seconds across various budget scenarios, significantly outperforming the widely used genetic algorithm. It offers targeted restoration by first aiding low-income communities and expanding aid as budgets increase, aligning with equity goals. This work showcases quantum computing's potential in disaster recovery planning, providing a rapid and equitable solution that elevates urban resilience and social sustainability by aiding vulnerable populations in disasters.

Combinatorial optimization (CO) problems are pivotal across various industrial applications, where the speed of solving these problems is crucial. Improving the performance of CO solvers across diverse input instances requires fine-tuning solver parameters for each instance. However, this tuning process is time-consuming, and the time required increases with the number of instances. To address this, a method called instance-specific algorithm configuration (ISAC) has been devised. This approach involves two main steps: training and execution. During the training step, features are extracted from various instances and then grouped into clusters. For each cluster, parameters are fine-tuned. This cluster-specific tuning process results in a set of generalized parameters for instances belonging to each class. In the execution step, features are extracted from an unknown instance to determine its cluster, and the corresponding pre-tuned parameters are applied. Generally, the running time of a solver is evaluated by the time to solution ($TTS$). However, methods like ISAC require preprocessing. Therefore, the total execution time is $T_{tot}=TTS+T_{tune}$, where $T_{tune}$ represents the tuning time. While the goal is to minimize $T_{tot}$, it is important to note that extracting features in the ISAC method requires a certain amount of computational time. The extracting features include summary statistics of the solver execution logs, which takes several 10 seconds. This research presents a method to significantly reduce the time of the ISAC execution step by streamlining feature extraction and class determination with a graph neural network. Experimental results show that $T_{tune}$ in the execution step, which take several 10 seconds in the original ISAC manner, could be reduced to sub-seconds.

Direct policy search has been widely applied in modern reinforcement learning and continuous control. However, the theoretical properties of direct policy search on nonsmooth robust control synthesis have not been fully understood. The optimal \mathcal{H}_\infty control framework aims at designing a policy to minimize the closed-loop \mathcal{H}_\infty norm, and is arguably the most fundamental robust control paradigm. In this work, we show that direct policy search is guaranteed to find the global solution of the robust \mathcal{H}_\infty state-feedback control design problem. Notice that policy search for optimal \mathcal{H}_\infty control leads to a constrained nonconvex nonsmooth optimization problem, where the nonconvex feasible set consists of all the policies stabilizing the closed-loop dynamics.