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We present a homotopic approach to solving challenging, optimization-based motion planning problems. The approach uses Homotopy Optimization, which, unlike standard continuation methods for solving homotopy problems, solves a sequence of constrained optimization problems rather than a sequence of nonlinear systems of equations. The insight behind our proposed algorithm is formulating the discovery of this sequence of optimization problems as a search problem in a multidimensional homotopy parameter space. Our proposed algorithm, the Probabilistic Homotopy Optimization algorithm, switches between solve and sample phases, using solutions to easy problems as initial guesses to more challenging problems. We analyze how our algorithm performs in the presence of common challenges to homotopy methods, such as bifurcation, folding, and disconnectedness of the homotopy solution manifold. Finally, we demonstrate its utility via a case study on two dynamic motion planning problems: the cart-pole and the MIT Humanoid.

Facial expression recognition (FER) is vital for human-computer interaction and emotion analysis, yet recognizing expressions in low-resolution images remains challenging. This paper introduces a practical method called Dynamic Resolution Guidance for Facial Expression Recognition (DRGFER) to effectively recognize facial expressions in images with varying resolutions without compromising FER model accuracy. Our framework comprises two main components: the Resolution Recognition Network (RRN) and the Multi-Resolution Adaptation Facial Expression Recognition Network (MRAFER). The RRN determines image resolution, outputs a binary vector, and the MRAFER assigns images to suitable facial expression recognition networks based on resolution. We evaluated DRGFER on widely-used datasets RAFDB and FERPlus, demonstrating that our method retains optimal model performance at each resolution and outperforms alternative resolution approaches. The proposed framework exhibits robustness against resolution variations and facial expressions, offering a promising solution for real-world applications.

Obtaining annotated table structure data for complex tables is a challenging task due to the inherent diversity and complexity of real-world document layouts. The scarcity of publicly available datasets with comprehensive annotations for intricate table structures hinders the development and evaluation of models designed for such scenarios. This research paper introduces a novel approach for generating annotated images for table structure by leveraging conditioned mask images of rows and columns through the application of latent diffusion models. The proposed method aims to enhance the quality of synthetic data used for training object detection models. Specifically, the study employs a conditioning mechanism to guide the generation of complex document table images, ensuring a realistic representation of table layouts. To evaluate the effectiveness of the generated data, we employ the popular YOLOv5 object detection model for training. The generated table images serve as valuable training samples, enriching the dataset with diverse table structures. The model is subsequently tested on the challenging pubtables-1m testset, a benchmark for table structure recognition in complex document layouts. Experimental results demonstrate that the introduced approach significantly improves the quality of synthetic data for training, leading to YOLOv5 models with enhanced performance. The mean Average Precision (mAP) values obtained on the pubtables-1m testset showcase results closely aligned with state-of-the-art methods. Furthermore, low FID results obtained on the synthetic data further validate the efficacy of the proposed methodology in generating annotated images for table structure.

When optimizing problems with uncertain parameter values in a linear objective, decision-focused learning enables end-to-end learning of these values. We are interested in a stochastic scheduling problem, in which processing times are uncertain, which brings uncertain values in the constraints, and thus repair of an initial schedule may be needed. Historical realizations of the stochastic processing times are available. We show how existing decision-focused learning techniques based on stochastic smoothing can be adapted to this scheduling problem. We include an extensive experimental evaluation to investigate in which situations decision-focused learning outperforms the state of the art for such situations: scenario-based stochastic optimization.

Scalarization is a general, parallizable technique that can be deployed in any multiobjective setting to reduce multiple objectives into one, yet some have dismissed this versatile approach because linear scalarizations cannot explore concave regions of the Pareto frontier. To that end, we aim to find simple non-linear scalarizations that provably explore a diverse set of $k$ objectives on the Pareto frontier, as measured by the dominated hypervolume. We show that hypervolume scalarizations with uniformly random weights achieves an optimal sublinear hypervolume regret bound of $O(T^{-1/k})$, with matching lower bounds that preclude any algorithm from doing better asymptotically. For the setting of multiobjective stochastic linear bandits, we utilize properties of hypervolume scalarizations to derive a novel non-Euclidean analysis to get regret bounds of $\tilde{O}( d T^{-1/2} + T^{-1/k})$, removing unnecessary $\text{poly}(k)$ dependencies. We support our theory with strong empirical performance of using non-linear scalarizations that outperforms both their linear counterparts and other standard multiobjective algorithms in a variety of natural settings.

We propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract Hilbert spaces under the Guignard constraint qualification. The method is equivalent to performing projected backward Euler timestepping on a projected gradient/antigradient flow of the augmented Lagrangian. The projected backward Euler equations can be interpreted as the necessary optimality conditions of a primal-dual proximal regularization of the original problem. The regularized problems are always feasible, satisfy a strong constraint qualification guaranteeing uniqueness of Lagrange multipliers, yield unique primal solutions provided that the stepsize is sufficiently small, and can be solved by a continuation in the stepsize. We show that equilibria of the projected gradient/antigradient flow and critical points of the optimization problem are identical, provide sufficient conditions for the existence of global flow solutions, and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient/antigradient flow, practically eradicating convergence to saddle points and maxima. The sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework. We demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth Newton approach for the regularized subproblems. In contrast to the published article, this version contains a correction that the associate editor considers as too insignificant to justify publication in the journal.

In combinatorial optimization, matroids provide one of the most elegant structures for algorithm design. This is perhaps best identified by the Edmonds-Rado theorem relating the success of the simple greedy algorithm to the anatomy of the optimal basis of a matroid [Edm71; Rad57]. As a response, much energy has been devoted to understanding a matroid's favorable computational properties. Yet surprisingly, not much is understood where parallel algorithm design is concerned. Specifically, while prior work has investigated the task of finding an arbitrary basis in parallel computing settings [KUW88], the more complex task of finding the optimal basis remains unexplored. We initiate this study by reexamining Bor\r{u}vka's minimum weight spanning tree algorithm in the language of matroid theory, identifying a new characterization of the optimal basis by way of a matroid's cocircuits as a result. Furthermore, we then combine such insights with special properties of binary matroids to reduce optimization in a binary matroid to the simpler task of search for an arbitrary basis, with only logarithmic asymptotic overhead. Consequentially, we are able to compose our reduction with a known basis search method of [KUW88] to obtain a novel algorithm for finding the optimal basis of a binary matroid with only sublinearly many adaptive rounds of queries to an independence oracle. To the authors' knowledge, this is the first parallel algorithm for matroid optimization to outperform the greedy algorithm in terms of adaptive complexity, for any class of matroid not represented by a graph.

The similarity between the question and indexed documents is a crucial factor in document retrieval for retrieval-augmented question answering. Although this is typically the only method for obtaining the relevant documents, it is not the sole approach when dealing with entity-centric questions. In this study, we propose Entity Retrieval, a novel retrieval method which rather than relying on question-document similarity, depends on the salient entities within the question to identify the retrieval documents. We conduct an in-depth analysis of the performance of both dense and sparse retrieval methods in comparison to Entity Retrieval. Our findings reveal that our method not only leads to more accurate answers to entity-centric questions but also operates more efficiently.

Solving partial differential equations (PDEs) is a fundamental problem in engineering and science. While neural PDE solvers can be more efficient than established numerical solvers, they often require large amounts of training data that is costly to obtain. Active Learning (AL) could help surrogate models reach the same accuracy with smaller training sets by querying classical solvers with more informative initial conditions and PDE parameters. While AL is more common in other domains, it has yet to be studied extensively for neural PDE solvers. To bridge this gap, we introduce AL4PDE, a modular and extensible active learning benchmark. It provides multiple parametric PDEs and state-of-the-art surrogate models for the solver-in-the-loop setting, enabling the evaluation of existing and the development of new AL methods for PDE solving. We use the benchmark to evaluate batch active learning algorithms such as uncertainty- and feature-based methods. We show that AL reduces the average error by up to 71% compared to random sampling and significantly reduces worst-case errors. Moreover, AL generates similar datasets across repeated runs, with consistent distributions over the PDE parameters and initial conditions. The acquired datasets are reusable, providing benefits for surrogate models not involved in the data generation.

We develop a generic computational model that can be used effectively for establishing the existence of winning strategies for concrete finite combinatorial games. Our modelling is (equational) logic-based involving advanced techniques from algebraic specification, and it can be executed by equational programming systems such as those from the OBJ-family. We show how this provides a form of experimental mathematics for strategy problems involving combinatorial games.