Lagrangian relaxation is a versatile mathematical technique employed to relax constraints in an optimization problem, enabling the generation of dual bounds to prove the optimality of feasible solutions and the design of efficient propagators in constraint programming (such as the weighted circuit constraint). However, the conventional process of deriving Lagrangian multipliers (e.g., using subgradient methods) is often computationally intensive, limiting its practicality for large-scale or time-sensitive problems. To address this challenge, we propose an innovative unsupervised learning approach that harnesses the capabilities of graph neural networks to exploit the problem structure, aiming to generate accurate Lagrangian multipliers efficiently. We apply this technique to the well-known Held-Karp Lagrangian relaxation for the travelling salesman problem. The core idea is to predict accurate Lagrangian multipliers and to employ them as a warm start for generating Held-Karp relaxation bounds. These bounds are subsequently utilized to enhance the filtering process carried out by branch-and-bound algorithms. In contrast to much of the existing literature, which primarily focuses on finding feasible solutions, our approach operates on the dual side, demonstrating that learning can also accelerate the proof of optimality. We conduct experiments across various distributions of the metric travelling salesman problem, considering instances with up to 200 cities. The results illustrate that our approach can improve the filtering level of the weighted circuit global constraint, reduce the optimality gap by a factor two for unsolved instances up to a timeout, and reduce the execution time for solved instances by 10%.

Automated machine learning (AutoML) systems propose an end-to-end solution to a given machine learning problem, creating either fixed or flexible pipelines. Fixed pipelines are task independent constructs: their general composition remains the same, regardless of the data. In contrast, the structure of flexible pipelines varies depending on the input, making them finely tailored to individual tasks. However, flexible pipelines can be structurally overcomplicated and have poor explainability. We propose the EVOSA approach that compensates for the negative points of flexible pipelines by incorporating a sensitivity analysis which increases the robustness and interpretability of the flexible solutions. EVOSA quantitatively estimates positive and negative impact of an edge or a node on a pipeline graph, and feeds this information to the evolutionary AutoML optimizer. The correctness and efficiency of EVOSA was validated in tabular, multimodal and computer vision tasks, suggesting generalizability of the proposed approach across domains.

Latent multi-view subspace clustering has been demonstrated to have desirable clustering performance. However, the original latent representation method vertically concatenates the data matrices from multiple views into a single matrix along the direction of dimensionality to recover the latent representation matrix, which may result in an incomplete information recovery. To fully recover the latent space representation, we in this paper propose an Enhanced Latent Multi-view Subspace Clustering (ELMSC) method. The ELMSC method involves constructing an augmented data matrix that enhances the representation of multi-view data. Specifically, we stack the data matrices from various views into the block-diagonal locations of the augmented matrix to exploit the complementary information. Meanwhile, the non-block-diagonal entries are composed based on the similarity between different views to capture the consistent information. In addition, we enforce a sparse regularization for the non-diagonal blocks of the augmented self-representation matrix to avoid redundant calculations of consistency information. Finally, a novel iterative algorithm based on the framework of Alternating Direction Method of Multipliers (ADMM) is developed to solve the optimization problem for ELMSC. Extensive experiments on real-world datasets demonstrate that our proposed ELMSC is able to achieve higher clustering performance than some state-of-art multi-view clustering methods.

Designing online algorithms with machine learning predictions is a recent technique beyond the worst-case paradigm for various practically relevant online problems (scheduling, caching, clustering, ski rental, etc.). While most previous learning-augmented algorithm approaches focus on integrating the predictions of a single oracle, we study the design of online algorithms with \emph{multiple} experts. To go beyond the popular benchmark of a static best expert in hindsight, we propose a new \emph{dynamic} benchmark (linear combinations of predictions that change over time). We present a competitive algorithm in the new dynamic benchmark with a performance guarantee of $O(\log K)$, where $K$ is the number of experts, for $0-1$ online optimization problems. Furthermore, our multiple-expert approach provides a new perspective on how to combine in an online manner several online algorithms - a long-standing central subject in the online algorithm research community.

Recently, many carefully crafted graph representation learning methods have achieved impressive performance on either strong heterophilic or homophilic graphs, but not both. Therefore, they are incapable of generalizing well across real-world graphs with different levels of homophily. This is attributed to their neglect of homophily in heterophilic graphs, and vice versa. In this paper, we propose a two-fold filtering mechanism to extract homophily in heterophilic graphs and vice versa. In particular, we extend the graph heat equation to perform heterophilic aggregation of global information from a long distance. The resultant filter can be exactly approximated by the Possion-Charlier (PC) polynomials. To further exploit information at multiple orders, we introduce a powerful graph convolution PC-Conv and its instantiation PCNet for the node classification task. Compared with state-of-the-art GNNs, PCNet shows competitive performance on well-known homophilic and heterophilic graphs. Our implementation is available at https://github.com/uestclbh/PC-Conv.

Molecular design based on generative models, such as variational autoencoders (VAEs), has become increasingly popular in recent years due to its efficiency for exploring high-dimensional molecular space to identify molecules with desired properties. While the efficacy of the initial model strongly depends on the training data, the sampling efficiency of the model for suggesting novel molecules with enhanced properties can be further enhanced via latent space optimization. In this paper, we propose a multi-objective latent space optimization (LSO) method that can significantly enhance the performance of generative molecular design (GMD). The proposed method adopts an iterative weighted retraining approach, where the respective weights of the molecules in the training data are determined by their Pareto efficiency. We demonstrate that our multi-objective GMD LSO method can significantly improve the performance of GMD for jointly optimizing multiple molecular properties.

We consider the problem of finding the smallest or largest entry of a tensor of order N that is specified via its rank decomposition. Stated in a different way, we are given N sets of R-dimensional vectors and we wish to select one vector from each set such that the sum of the Hadamard product of the selected vectors is minimized or maximized. We show that this fundamental tensor problem is NP-hard for any tensor rank higher than one, and polynomial-time solvable in the rank-one case. We also propose a continuous relaxation and prove that it is tight for any rank. For low-enough ranks, the proposed continuous reformulation is amenable to low-complexity gradient-based optimization, and we propose a suite of gradient-based optimization algorithms drawing from projected gradient descent, Frank-Wolfe, or explicit parametrization of the relaxed constraints. We also show that our core results remain valid no matter what kind of polyadic tensor model is used to represent the tensor of interest, including Tucker, HOSVD/MLSVD, tensor train, or tensor ring. Next, we consider the class of problems that can be posed as special instances of the problem of interest. We show that this class includes the partition problem (and thus all NP-complete problems via polynomial-time transformation), integer least squares, integer linear programming, integer quadratic programming, sign retrieval (a special kind of mixed integer programming / restricted version of phase retrieval), and maximum likelihood decoding of parity check codes. We demonstrate promising experimental results on a number of hard problems, including state-of-art performance in decoding low density parity check codes and general parity check codes.

Multi-modal optimization is often encountered in engineering problems, especially when different and alternative solutions are sought. Evolutionary algorithms can efficiently tackle multi-modal optimization thanks to their features such as the concept of population, exploration/exploitation, and being suitable for parallel computation. This paper introduces a multi-modal optimization version of the Big Bang-Big Crunch algorithm based on clustering, namely, k-BBBC. This algorithm guarantees a complete convergence of the entire population, retrieving on average the 99\% of local optima for a specific problem. Additionally, we introduce two post-processing methods to (i) identify the local optima in a set of retrieved solutions (i.e., a population), and (ii) quantify the number of correctly retrieved optima against the expected ones (i.e., success rate). Our results show that k-BBBC performs well even with problems having a large number of optima (tested on 379 optima) and high dimensionality (tested on 32 decision variables). When compared to other multi-modal optimization methods, it outperforms them in terms of accuracy (in both search and objective space) and success rate (number of correctly retrieved optima) -- especially when elitism is applied. Lastly, we validated our proposed post-processing methods by comparing their success rate to the actual one. Results suggest that these methods can be used to evaluate the performance of a multi-modal optimization algorithm by correctly identifying optima and providing an indication of success -- without the need to know where the optima are located in the search space.

Advancements in mathematical programming have made it possible to efficiently tackle large-scale real-world problems that were deemed intractable just a few decades ago. However, provably optimal solutions may not be accepted due to the perception of optimization software as a black box. Although well understood by scientists, this lacks easy accessibility for practitioners. Hence, we advocate for introducing the explainability of a solution as another evaluation criterion, next to its objective value, which enables us to find trade-off solutions between these two criteria. Explainability is attained by comparing against (not necessarily optimal) solutions that were implemented in similar situations in the past. Thus, solutions are preferred that exhibit similar features. Although we prove that already in simple cases the explainable model is NP-hard, we characterize relevant polynomially solvable cases such as the explainable shortest path problem. Our numerical experiments on both artificial as well as real-world road networks show the resulting Pareto front. It turns out that the cost of enforcing explainability can be very small.

Future spacecraft and surface robotic missions require increasingly capable autonomy stacks for exploring challenging and unstructured domains and trajectory optimization will be a cornerstone of such autonomy stacks. However, the nonlinear optimization solvers required remain too slow for use on relatively resource constrained flight-grade computers. In this work, we turn towards amortized optimization, a learning-based technique for accelerating optimization run times, and present TOAST: Trajectory Optimization with Merit Function Warm Starts. Offline, using data collected from a simulation, we train a neural network to learn a mapping to the full primal and dual solutions given the problem parameters. Crucially, we build upon recent results from decision-focused learning and present a set of decision-focused loss functions using the notion of merit functions for optimization problems. We show that training networks with such constraint-informed losses can better encode the structure of the trajectory optimization problem and jointly learn to reconstruct the primal-dual solution while also yielding improved constraint satisfaction. Through numerical experiments on a Lunar rover problem, we demonstrate that TOAST outperforms benchmark approaches in terms of both computation times and network prediction constraint satisfaction.