The convergence of many numerical optimization techniques is highly dependent on the initial guess given to the solver. To address this issue, we propose a novel approach that utilizes tensor methods to initialize existing optimization solvers near global optima. Our method does not require access to a database of good solutions. We first transform the cost function, which depends on both task parameters and optimization variables, into a probability density function. Unlike existing approaches, the joint probability distribution of the task parameters and optimization variables is approximated using the Tensor Train model, which enables efficient conditioning and sampling. We treat the task parameters as random variables, and for a given task, we generate samples for decision variables from the conditional distribution to initialize the optimization solver. Our method can produce multiple solutions (when they exist) faster than existing methods. We first evaluate the approach on benchmark functions for numerical optimization that are hard to solve using gradient-based optimization solvers with a naive initialization. The results show that the proposed method can generate samples close to global optima and from multiple modes. We then demonstrate the generality and relevance of our framework to robotics by applying it to inverse kinematics with obstacles and motion planning problems with a 7-DoF manipulator.

This optimization paradigm can naturally be used to formulate fundamental problems in interpretable supervised learning (e.g., sparse regression and decision trees), in unsupervised learning (e.g., clustering), and beyond; BackboneLearn solves the aforementioned problems faster than exact methods and with higher accuracy than commonly used heuristics. The package is built in Python and is user-friendly and easily extensible: users can directly implement a backbone algorithm for their MIO problem at hand.

Engineering Design is undergoing a transformative shift with the advent of AI, marking a new era in how we approach product, system, and service planning. Large language models have demonstrated impressive capabilities in enabling this shift. Yet, with text as their only input modality, they cannot leverage the large body of visual artifacts that engineers have used for centuries and are accustomed to. This gap is addressed with the release of multimodal vision language models, such as GPT-4V, enabling AI to impact many more types of tasks. In light of these advancements, this paper presents a comprehensive evaluation of GPT-4V, a vision language model, across a wide spectrum of engineering design tasks, categorized into four main areas: Conceptual Design, System-Level and Detailed Design, Manufacturing and Inspection, and Engineering Education Tasks. Our study assesses GPT-4V's capabilities in design tasks such as sketch similarity analysis, concept selection using Pugh Charts, material selection, engineering drawing analysis, CAD generation, topology optimization, design for additive and subtractive manufacturing, spatial reasoning challenges, and textbook problems. Through this structured evaluation, we not only explore GPT-4V's proficiency in handling complex design and manufacturing challenges but also identify its limitations in complex engineering design applications. Our research establishes a foundation for future assessments of vision language models, emphasizing their immense potential for innovating and enhancing the engineering design and manufacturing landscape. It also contributes a set of benchmark testing datasets, with more than 1000 queries, for ongoing advancements and applications in this field.

Control barrier functions (CBFs) have been widely applied to safety-critical robotic applications. However, the construction of control barrier functions for robotic systems remains a challenging task. Recently, collision detection using differentiable optimization has provided a way to compute the minimum uniform scaling factor that results in an intersection between two convex shapes and to also compute the Jacobian of the scaling factor. In this letter, we propose a framework that uses this scaling factor, with an offset, to systematically define a CBF for obstacle avoidance tasks. We provide theoretical analyses of the continuity and continuous differentiability of the proposed CBF. We empirically evaluate the proposed CBF's behavior and show that the resulting optimal control problem is computationally efficient, which makes it applicable for real-time robotic control. We validate our approach, first using a 2D mobile robot example, then on the Franka-Emika Research 3 (FR3) robot manipulator both in simulation and experiment.

Traditional smart meter measurements lack the granularity needed for real-time decision-making. To address this practical problem, we create a generative adversarial networks (GAN) model that enforces temporal consistency on its high-resolution outputs via hard inequality constraints using a convex optimization layer. A unique feature of our GAN model is that it is trained solely on slow timescale aggregated power information obtained from historical smart meter data. The results demonstrate that the model can successfully create minutely interval temporally-correlated instantaneous power injection profiles from 15-minute average power consumption information. This innovative approach, emphasizing inter-neuron constraints, offers a promising avenue for improved high-speed state estimation in distribution systems and enhances the applicability of data-driven solutions for monitoring such systems.

We introduce a new Langevin dynamics based algorithm, called e-TH$\varepsilon$O POULA, to solve optimization problems with discontinuous stochastic gradients which naturally appear in real-world applications such as quantile estimation, vector quantization, CVaR minimization, and regularized optimization problems involving ReLU neural networks. We demonstrate both theoretically and numerically the applicability of the e-TH$\varepsilon$O POULA algorithm. More precisely, under the conditions that the stochastic gradient is locally Lipschitz in average and satisfies a certain convexity at infinity condition, we establish non-asymptotic error bounds for e-TH$\varepsilon$O POULA in Wasserstein distances and provide a non-asymptotic estimate for the expected excess risk, which can be controlled to be arbitrarily small. Three key applications in finance and insurance are provided, namely, multi-period portfolio optimization, transfer learning in multi-period portfolio optimization, and insurance claim prediction, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world datasets illustrate the superior empirical performance of e-TH$\varepsilon$O POULA compared to SGLD, TUSLA, ADAM, and AMSGrad in terms of model accuracy.

Resided at the intersection of multi-fidelity optimization (MFO) and Bayesian optimization (BO), MF BO has found a niche in solving expensive engineering design optimization problems, thanks to its advantages in incorporating physical and mathematical understandings of the problems, saving resources, addressing exploitation-exploration trade-off, considering uncertainty, and processing parallel computing. The increasing number of works dedicated to MF BO suggests the need for a comprehensive review of this advanced optimization technique. In this paper, we survey recent developments of two essential ingredients of MF BO: Gaussian process (GP) based MF surrogates and acquisition functions. We first categorize the existing MF modeling methods and MFO strategies to locate MF BO in a large family of surrogate-based optimization and MFO algorithms. We then exploit the common properties shared between the methods from each ingredient of MF BO to describe important GP-based MF surrogate models and review various acquisition functions. By doing so, we expect to provide a structured understanding of MF BO. Finally, we attempt to reveal important aspects that require further research for applications of MF BO in solving intricate yet important design optimization problems, including constrained optimization, high-dimensional optimization, optimization under uncertainty, and multi-objective optimization.

Multi-objective optimization (MOO) aims to optimize multiple, possibly conflicting objectives with widespread applications. We introduce a novel interacting particle method for MOO inspired by molecular dynamics simulations. Our approach combines overdamped Langevin and birth-death dynamics, incorporating a "dominance potential" to steer particles toward global Pareto optimality. In contrast to previous methods, our method is able to relocate dominated particles, making it particularly adept at managing Pareto fronts of complicated geometries. Our method is also theoretically grounded as a Wasserstein-Fisher-Rao gradient flow with convergence guarantees. Extensive experiments confirm that our approach outperforms state-of-the-art methods on challenging synthetic and real-world datasets.

The goal of multi-objective optimization is to identify a collection of points which describe the best possible trade-offs between the multiple objectives. In order to solve this vector-valued optimization problem, practitioners often appeal to the use of scalarization functions in order to transform the multi-objective problem into a collection of single-objective problems. This set of scalarized problems can then be solved using traditional single-objective optimization techniques. In this work, we formalise this convention into a general mathematical framework. We show how this strategy effectively recasts the original multi-objective optimization problem into a single-objective optimization problem defined over sets. An appropriate class of objective functions for this new problem is the R2 utility function, which is defined as a weighted integral over the scalarized optimization problems. We show that this utility function is a monotone and submodular set function, which can be optimised effectively using greedy optimization algorithms. We analyse the performance of these greedy algorithms both theoretically and empirically. Our analysis largely focusses on Bayesian optimization, which is a popular probabilistic framework for black-box optimization.

We propose novel fast algorithms for optimal transport (OT) utilizing a cyclic symmetry structure of input data. Such OT with cyclic symmetry appears universally in various real-world examples: image processing, urban planning, and graph processing. Our main idea is to reduce OT to a small optimization problem that has significantly fewer variables by utilizing cyclic symmetry and various optimization techniques. On the basis of this reduction, our algorithms solve the small optimization problem instead of the original OT. As a result, our algorithms obtain the optimal solution and the objective function value of the original OT faster than solving the original OT directly. In this paper, our focus is on two crucial OT formulations: the linear programming OT (LOT) and the strongly convex-regularized OT, which includes the well-known entropy-regularized OT (EROT). Experiments show the effectiveness of our algorithms for LOT and EROT in synthetic/real-world data that has a strict/approximate cyclic symmetry structure. Through theoretical and experimental results, this paper successfully introduces the concept of symmetry into the OT research field for the first time.