For prohibitively large-scale Travelling Salesman Problems (TSPs), existing algorithms face big challenges in terms of both computational efficiency and solution quality. To address this issue, we propose a hierarchical destroy-and-repair (HDR) approach, which attempts to improve an initial solution by applying a series of carefully designed destroy-and-repair operations. A key innovative concept is the hierarchical search framework, which recursively fixes partial edges and compresses the input instance into a small-scale TSP under some equivalence guarantee. This neat search framework is able to deliver highly competitive solutions within a reasonable time. Fair comparisons based on nineteen famous large-scale instances (with 10,000 to 10,000,000 cities) show that HDR is highly competitive against existing state-of-the-art TSP algorithms, in terms of both efficiency and solution quality. Notably, on two large instances with 3,162,278 and 10,000,000 cities, HDR breaks the world records (i.e., best-known results regardless of computation time), which were previously achieved by LKH and its variants, while HDR is completely independent of LKH. Finally, ablation studies are performed to certify the importance and validity of the hierarchical search framework.

Scaling to arbitrarily large bundle adjustment problems requires data and compute to be distributed across multiple devices. Centralized methods in prior works are only able to solve small or medium size problems due to overhead in computation and communication. In this paper, we present a fully decentralized method that alleviates computation and communication bottlenecks to solve arbitrarily large bundle adjustment problems. We achieve this by reformulating the reprojection error and deriving a novel surrogate function that decouples optimization variables from different devices. This function makes it possible to use majorization minimization techniques and reduces bundle adjustment to independent optimization subproblems that can be solved in parallel. We further apply Nesterov's acceleration and adaptive restart to improve convergence while maintaining its theoretical guarantees. Despite limited peer-to-peer communication, our method has provable convergence to first-order critical points under mild conditions. On extensive benchmarks with public datasets, our method converges much faster than decentralized baselines with similar memory usage and communication load. Compared to centralized baselines using a single device, our method, while being decentralized, yields more accurate solutions with significant speedups of up to 953.7x over Ceres and 174.6x over DeepLM. Code: https://joeaortiz.github.io/daba.

Federated optimization, an emerging paradigm which finds wide real-world applications such as federated learning, enables multiple clients (e.g., edge devices) to collaboratively optimize a global function. The clients do not share their local datasets and typically only share their local gradients. However, the gradient information is not available in many applications of federated optimization, which hence gives rise to the paradigm of federated zeroth-order optimization (ZOO). Existing federated ZOO algorithms suffer from the limitations of query and communication inefficiency, which can be attributed to (a) their reliance on a substantial number of function queries for gradient estimation and (b) the significant disparity between their realized local updates and the intended global updates. To this end, we (a) introduce trajectory-informed gradient surrogates which is able to use the history of function queries during optimization for accurate and query-efficient gradient estimation, and (b) develop the technique of adaptive gradient correction using these gradient surrogates to mitigate the aforementioned disparity. Based on these, we propose the federated zeroth-order optimization using trajectory-informed surrogate gradients (FZooS) algorithm for query- and communication-efficient federated ZOO. Our FZooS achieves theoretical improvements over the existing approaches, which is supported by our real-world experiments such as federated black-box adversarial attack and federated non-differentiable metric optimization.

This paper presents a framework for smooth optimization of objectives with $\ell_q$ and $\ell_{p,q}$ regularization for (structured) sparsity. Finding solutions to these non-smooth and possibly non-convex problems typically relies on specialized optimization routines. In contrast, the method studied here is compatible with off-the-shelf (stochastic) gradient descent that is ubiquitous in deep learning, thereby enabling differentiable sparse regularization without approximations. The proposed optimization transfer comprises an overparametrization of selected model parameters followed by a change of penalties. In the overparametrized problem, smooth and convex $\ell_2$ regularization induces non-smooth and non-convex regularization in the original parametrization. We show that the resulting surrogate problem not only has an identical global optimum but also exactly preserves the local minima. This is particularly useful in non-convex regularization, where finding global solutions is NP-hard and local minima often generalize well. We provide an integrative overview that consolidates various literature strands on sparsity-inducing parametrizations in a general setting and meaningfully extend existing approaches. The feasibility of our approach is evaluated through numerical experiments, demonstrating its effectiveness by matching or outperforming common implementations of convex and non-convex regularizers.

Monoclonal antibodies (mAbs) have emerged as indispensable assets in medicine, and are currently at the forefront of biopharmaceutical product development. However, the growing market demand and the substantial doses required for mAb clinical treatments necessitate significant progress in its large-scale production. Most of the processes for industrial mAb production rely on batch operations, which result in significant downtime. The shift towards a fully continuous and integrated manufacturing process holds the potential to boost product yield and quality, while eliminating the extra expenses associated with storing intermediate products. The integrated continuous mAb production process can be divided into the upstream and downstream processes. One crucial aspect that ensures the continuity of the integrated process is the switching of the capture columns, which are typically chromatography columns operated in a fed-batch manner downstream. Due to the discrete nature of the switching operation, advanced process control algorithms such as economic MPC (EMPC) are computationally difficult to implement. This is because an integer nonlinear program (INLP) needs to be solved online at each sampling time. This paper introduces two computationally-efficient approaches for EMPC implementation, namely, a sigmoid function approximation approach and a rectified linear unit (ReLU) approximation approach. It also explores the application of deep reinforcement learning (DRL). These three methods are compared to the traditional switching approach which is based on a 1% product breakthrough rule and which involves no optimization.

Stochastic sequential quadratic optimization (SQP) methods for solving continuous optimization problems with nonlinear equality constraints have attracted attention recently, such as for solving large-scale data-fitting problems subject to nonconvex constraints. However, for a recently proposed subclass of such methods that is built on the popular stochastic-gradient methodology from the unconstrained setting, convergence guarantees have been limited to the asymptotic convergence of the expected value of a stationarity measure to zero. This is in contrast to the unconstrained setting in which almost-sure convergence guarantees (of the gradient of the objective to zero) can be proved for stochastic-gradient-based methods. In this paper, new almost-sure convergence guarantees for the primal iterates, Lagrange multipliers, and stationarity measures generated by a stochastic SQP algorithm in this subclass of methods are proved. It is shown that the error in the Lagrange multipliers can be bounded by the distance of the primal iterate to a primal stationary point plus the error in the latest stochastic gradient estimate. It is further shown that, subject to certain assumptions, this latter error can be made to vanish by employing a running average of the Lagrange multipliers that are computed during the run of the algorithm. The results of numerical experiments are provided to demonstrate the proved theoretical guarantees.

SecureBoost is a tree-boosting algorithm leveraging homomorphic encryption to protect data privacy in vertical federated learning setting. It is widely used in fields such as finance and healthcare due to its interpretability, effectiveness, and privacy-preserving capability. However, SecureBoost suffers from high computational complexity and risk of label leakage. To harness the full potential of SecureBoost, hyperparameters of SecureBoost should be carefully chosen to strike an optimal balance between utility, efficiency, and privacy. Existing methods either set hyperparameters empirically or heuristically, which are far from optimal. To fill this gap, we propose a Constrained Multi-Objective SecureBoost (CMOSB) algorithm to find Pareto optimal solutions that each solution is a set of hyperparameters achieving optimal tradeoff between utility loss, training cost, and privacy leakage. We design measurements of the three objectives. In particular, the privacy leakage is measured using our proposed instance clustering attack. Experimental results demonstrate that the CMOSB yields not only hyperparameters superior to the baseline but also optimal sets of hyperparameters that can support the flexible requirements of FL participants.

In many engineering applications it is useful to reason about "negative information". For example, in planning problems, providing an optimal solution is the same as giving a feasible solution (the "positive" information) together with a proof of the fact that there cannot be feasible solutions better than the one given (the "negative" information). We model negative information by introducing the concept of "norphisms", as opposed to the positive information of morphisms. A "nategory" is a category that has "nom"-sets in addition to hom-sets, and specifies the interaction between norphisms and morphisms. In particular, we have composition rules of the form morphism + norphism $\to$ norphism. Norphisms do not compose by themselves; rather, they use morphisms as catalysts. After providing several applied examples, we connect nategories to enriched category theory. Specifically, we prove that categories enriched in de Paiva's dialectica categories GC, in the case C = Set and equipped with a modified monoidal product, define nategories which satisfy additional regularity properties. This formalizes negative information categorically in a way that makes negative and positive morphisms equal citizens.

Abstract: Balance assessment during physical rehabilitation often relies on rubricoriented battery tests to score a patient's physical capabilities, leading to subjectivity. While some objective balance assessments exist, they are often limited to tracking the center of pressure (COP), which does not fully capture the whole-body postural stability. This study explores the use of the center of mass (COM) state space and presents a promising avenue for monitoring the balance capabilities in humans. We employ a musculoskeletal model integrated with a balance controller, trained through reinforcement learning (RL), to investigate balancing capabilities. The RL framework consists of two interconnected neural networks governing balance recovery and muscle coordination respectively, trained using Proximal Policy Optimization (PPO) with reference state initialization, early termination, and multiple training strategies. By exploring recovery from random initial COM states (position and velocity) space for a trained controller, we obtain the final BR enclosing successful balance recovery trajectories. Comparing the BRs with analytical postural stability limits from a linear inverted pendulum model, we observe a similar trend in successful COM states but more limited ranges in the recoverable areas. We further investigate the effect of muscle weakness and neural excitation delay on the BRs, revealing reduced balancing capability in different regions. Overall, our approach of learning muscular balance controllers presents a promising new method for establishing balance recovery limits and objectively assessing balance capability in bipedal systems, particularly in humans. Keywords: Balance, Reinforcement Learning, Musculoskeletal Modeling, Bipedal Systems, Motor Disorders 1. Introduction Falls and subsequent injuries pose a significant health risk for the elderly and mobility-impaired populations. Poor balancing capabilities are the leading cause of falls in the elderly population, which reduces the overall quality of life of aging patients [1-3]. The injuries sustained by these patients can range from lower-body fractures, particularly in the hip, to head injuries, with falls being the leading cause of traumatic brain injuries [4]. Therefore, effective balance assessment and rehabilitation are critical components not only to health monitoring and injury prevention in mobility-impaired individuals, but also to the diagnoses of other serious underlying medical conditions. Since balance is maintained through a complicated network of physiological systems in the body, it is difficult to pinpoint a single origin causing deficiencies in patients and to assess balance through simple isolated measures. In most clinical environments, balance assessment is performed as a battery of balance exercises designed to evaluate the patient's ability to perform selected tasks.

For example, a grid-based search is a classical approach for spacecraft preliminary trajectory design. However, this technique is more suitable for impulsive trajectory since the search space is much smaller. Due to the curse of dimensionality, low-thrust trajectory design often needs a more intelligent global search algorithm. Evolutionary algorithms, including Differential Evolution (DE) [4], Genetic algorithm (GA) [5], Particle swarm optimization (PSO) [6], etc., have been widely used in global optimization problems in spacecraft trajectory design [7, 8, 9, 10]. These algorithms iteratively generate new solutions by introducing randomness to previously obtained solutions and downselecting the solutions based on specific quality metrics. In addition, researchers also combine stochastic search algorithms with local gradient-based optimizers to attempt to find the globally optimal solution. The multistart method samples the search space with a fixed distribution and feeds the samples into a local optimizer as starting points for local search [10]. Inspired by energy minimization principles in computational chemistry, Monotonic Basin Hopping (MBH) [11, 12] adds random perturbations during the local search to uncover multiple local optima solutions that are close to each other. MBH rapidly became popular in the sphere of spacecraft trajectory design [1, 13, 14] and has been established as the state-of-the-art algorithm in terms of efficiency and solution quality through various benchmarks [15, 9, 10].