The highly non-convex optimization landscape of modern lens design necessitates extensive human expertise, resulting in inefficiency and constrained design diversity. While automated methods are desirable, existing approaches remain limited to simple tasks or produce complex lenses with suboptimal image quality. Drawing inspiration from the synaptic pruning mechanism in mammalian neural development, this study proposes OptiNeuro--a novel automated lens design framework that first generates diverse initial structures and then progressively eliminates low-performance lenses while refining remaining candidates through gradient-based optimization. By fully automating the design of complex aspheric imaging lenses, OptiNeuro demonstrates quasi-human-level performance, identifying multiple viable candidates with minimal human intervention. This advancement not only enhances the automation level and efficiency of lens design but also facilitates the exploration of previously uncharted lens architectures.

The rise of parallel computing hardware has made it increasingly important to understand which nonlinear state space models can be efficiently parallelized. Recent advances like DEER (arXiv:2309.12252) or DeepPCR (arXiv:2309.16318) have shown that evaluating a state space model can be recast as solving a parallelizable optimization problem, and sometimes this approach can yield dramatic speed-ups in evaluation time. However, the factors that govern the difficulty of these optimization problems remain unclear, limiting the larger adoption of the technique. In this work, we establish a precise relationship between the dynamics of a nonlinear system and the conditioning of its corresponding optimization formulation. We show that the predictability of a system, defined as the degree to which small perturbations in state influence future behavior, impacts the number of optimization steps required for evaluation. In predictable systems, the state trajectory can be computed in $O((\log T)^2)$ time, where $T$ is the sequence length, a major improvement over the conventional sequential approach. In contrast, chaotic or unpredictable systems exhibit poor conditioning, with the consequence that parallel evaluation converges too slowly to be useful. Importantly, our theoretical analysis demonstrates that for predictable systems, the optimization problem is always well-conditioned, whereas for unpredictable systems, the conditioning degrades exponentially as a function of the sequence length. We validate our claims through extensive experiments, providing practical guidance on when nonlinear dynamical systems can be efficiently parallelized, and highlighting predictability as a key design principle for parallelizable models.

Contact-implicit motion planning-embedding contact sequencing as implicit complementarity constraints-holds the promise of leveraging continuous optimization to discover new contact patterns online. Nevertheless, the resulting optimization, being an instance of Mathematical Programming with Complementary Constraints, fails the classical constraint qualifications that are crucial for the convergence of popular numerical solvers. We present robust contact-implicit motion planning with sequential convex programming (CRISP), a solver that departs from the usual primal-dual algorithmic framework but instead only focuses on the primal problem. CRISP solves a convex quadratic program with an adaptive trust region radius at each iteration, and its convergence is evaluated by a merit function using weighted penalty. We (i) provide sufficient conditions on CRISP's convergence to first-order stationary points of the merit function; (ii) release a high-performance C++ implementation of CRISP with a generic nonlinear programming interface; and (iii) demonstrate CRISP's surprising robustness in solving contact-implicit planning with naive initialization. In fact, CRISP solves several contact-implicit problems with all-zero initialization.

Many classical and modern machine learning algorithms require solving optimization tasks under orthogonality constraints. Solving these tasks with feasible methods requires a gradient descent update followed by a retraction operation on the Stiefel manifold, which can be computationally expensive. Recently, an infeasible retraction-free approach, termed the landing algorithm, was proposed as an efficient alternative. Motivated by the common occurrence of orthogonality constraints in tasks such as principle component analysis and training of deep neural networks, this paper studies the landing algorithm and establishes a novel linear convergence rate for smooth non-convex functions using only a local Riemannian P{\L} condition. Numerical experiments demonstrate that the landing algorithm performs on par with the state-of-the-art retraction-based methods with substantially reduced computational overhead.

Trajectory generation for fully autonomous flights of tail-sitter unmanned aerial vehicles (UAVs) presents substantial challenges due to their highly nonlinear aerodynamics. In this paper, we introduce, to the best of our knowledge, the world's first fully autonomous tail-sitter UAV capable of high-speed navigation in unknown, cluttered environments. The UAV autonomy is enabled by cutting-edge technologies including LiDAR-based sensing, differential-flatness-based trajectory planning and control with purely onboard computation. In particular, we propose an optimization-based tail-sitter trajectory planning framework that generates high-speed, collision-free, and dynamically-feasible trajectories. To efficiently and reliably solve this nonlinear, constrained \textcolor{black}{problem}, we develop an efficient feasibility-assured solver, EFOPT, tailored for the online planning of tail-sitter UAVs. We conduct extensive simulation studies to benchmark EFOPT's superiority in planning tasks against conventional NLP solvers. We also demonstrate exhaustive experiments of aggressive autonomous flights with speeds up to 15m/s in various real-world environments, including indoor laboratories, underground parking lots, and outdoor parks. A video demonstration is available at https://youtu.be/OvqhlB2h3k8, and the EFOPT solver is open-sourced at https://github.com/hku-mars/EFOPT.

This paper introduces a modified Byrd-Omojokun (BO) trust region algorithm to address the challenges posed by noisy function and gradient evaluations. The original BO method was designed to solve equality constrained problems and it forms the backbone of some interior point methods for general large-scale constrained optimization. A key strength of the BO method is its robustness in handling problems with rank-deficient constraint Jacobians. The algorithm proposed in this paper introduces a new criterion for accepting a step and for updating the trust region that makes use of an estimate in the noise in the problem. The analysis presented here gives conditions under which the iterates converge to regions of stationary points of the problem, determined by the level of noise. This analysis is more complex than for line search methods because the trust region carries (noisy) information from previous iterates. Numerical tests illustrate the practical performance of the algorithm.

Dynamic games can be an effective approach for modeling interactive behavior between multiple competitive agents in autonomous racing and they provide a theoretical framework for simultaneous prediction and control in such scenarios. In this work, we propose DG-SQP, a numerical method for the solution of local generalized Nash equilibria (GNE) for open-loop general-sum dynamic games for agents with nonlinear dynamics and constraints. In particular, we formulate a sequential quadratic programming (SQP) approach which requires only the solution of a single convex quadratic program at each iteration. The three key elements of the method are a non-monotonic line search for solving the associated KKT equations, a merit function to handle zero sum costs, and a decaying regularization scheme for SQP step selection. We show that our method achieves linear convergence in the neighborhood of local GNE and demonstrate the effectiveness of the approach in the context of head-to-head car racing, where we show significant improvement in solver success rate when comparing against the state-of-the-art PATH solver for dynamic games. An implementation of our solver can be found at https://github.com/zhu-edward/DGSQP.

Existing approaches to fairness in stochastic multi-armed bandits (MAB) primarily focus on exposure guarantee to individual arms. When arms are naturally grouped by certain attribute(s), we propose Bi-Level Fairness, which considers two levels of fairness. At the first level, Bi-Level Fairness guarantees a certain minimum exposure to each group. To address the unbalanced allocation of pulls to individual arms within a group, we consider meritocratic fairness at the second level, which ensures that each arm is pulled according to its merit within the group. Our work shows that we can adapt a UCB-based algorithm to achieve a Bi-Level Fairness by providing (i) anytime Group Exposure Fairness guarantees and (ii) ensuring individual-level Meritocratic Fairness within each group. We first show that one can decompose regret bounds into two components: (a) regret due to anytime group exposure fairness and (b) regret due to meritocratic fairness within each group. Our proposed algorithm BF-UCB balances these two regrets optimally to achieve the upper bound of $O(\sqrt{T})$ on regret; $T$ being the stopping time. With the help of simulated experiments, we further show that BF-UCB achieves sub-linear regret; provides better group and individual exposure guarantees compared to existing algorithms; and does not result in a significant drop in reward with respect to UCB algorithm, which does not impose any fairness constraint.