Many critical business and societal decisions in areas such as supply chain and healthcare involve numerous potential actions, complex constraints, and goals that can be modeled as objective functions. Mathematical optimization, a core area in Operations Research (OR), provides robust, mathematically grounded methodologies to address such decisions and has shown tremendous benefits in many applications. However, its application requires the creation of accurate and efficient optimization models, necessitating rare expertise and considerable time, creating a barrier to widespread adoption in decision-making. Thus, it is a long-standing goal to make these capabilities widely accessible. The advent of Large Language Models (LLMs) has made advanced Artificial Intelligence (AI) capabilities widely accessible through natural language. LLMs can accelerate expert work in creating formal models like computer programs, and emerging research indicates they can also speed up the development of optimization models by OR experts. We, therefore, propose integrating and advancing LLM and optimization modeling to empower organizational decision-makers to model and solve such complex problems without requiring deep expertise in optimization. In this work, we present our vision for democratizing optimization modeling for organizational decision-making by such a combination of LLMs and optimization modeling. We identify a set of fundamental requirements for the vision's implementation and describe the state of the art through a literature survey and some experimentation. We show that a) LLMs already provide substantial novel capabilities relevant to realizing this vision, but that b) major research challenges remain to be addressed. We also propose possible research directions to overcome these gaps. We would like this work to serve as a call to action to bring together the LLM and OR optimization modeling communities to pursue this vision, thereby enabling much more widespread improved decision-making and increasing by orders of magnitude the benefits AI and OR can bring to enterprises and society.

Graph convolutional networks (GCNs) are *discriminative models* that directly model the class posterior $p(y|\mathbf{x})$ for semi-supervised classification of graph data. While being effective, as a representation learning approach, the node representations extracted from a GCN often miss useful information for effective clustering, because the objectives are different. In this work, we design normalizing flows that replace GCN layers, leading to a *generative model* that models both the class conditional likelihood $p(\mathbf{x}|y)$ and the class prior $p(y)$. The resulting neural network, GC-Flow, retains the graph convolution operations while being equipped with a Gaussian mixture representation space. It enjoys two benefits: it not only maintains the predictive power of GCN, but also produces well-separated clusters, due to the structuring of the representation space. We demonstrate these benefits on a variety of benchmark data sets. Moreover, we show that additional parameterization, such as that on the adjacency matrix used for graph convolutions, yields additional improvement in clustering.

Label switching is a phenomenon arising in mixture model posterior inference that prevents one from meaningfully assessing posterior statistics using standard Monte Carlo procedures. This issue arises due to invariance of the posterior under actions of a group; for example, permuting the ordering of mixture components has no effect on the likelihood. We propose a resolution to label switching that leverages machinery from optimal transport. Our algorithm efficiently computes posterior statistics in the quotient space of the symmetry group. We give conditions under which there is a meaningful solution to label switching and demonstrate advantages over alternative approaches on simulated and real data.

The ability to measure similarity between documents enables intelligent summarization and analysis of large corpora. Past distances between documents suffer from either an inability to incorporate semantic similarities between words or from scalability issues. As an alternative, we introduce hierarchical optimal transport as a meta-distance between documents, where documents are modeled as distributions over topics, which themselves are modeled as distributions over words. We then solve an optimal transport problem on the smaller topic space to compute a similarity score. We give conditions on the topics under which this construction defines a distance, and we relate it to the word mover's distance. We evaluate our technique for k-NN classification and show better interpretability and scalability with comparable performance to current methods at a fraction of the cost.

A key bottleneck in structured output prediction is the need for inference during training and testing, usually requiring some form of dynamic programming. Rather than using approximate inference or tailoring a specialized inference method for a particular structure---standard responses to the scaling challenge---we propose to embed prediction constraints directly into the learned representation. By eliminating the need for explicit inference a more scalable approach to structured output prediction can be achieved, particularly at test time. We demonstrate the idea for multi-label prediction under subsumption and mutual exclusion constraints, where a relationship to maximum margin structured output prediction can be established. Experiments demonstrate that the benefits of structured output training can still be realized even after inference has been eliminated.

We present a general framework for association learning, where entities are embedded in a common latent space to express relatedness by geometry -- an approach that underlies the state of the art for link prediction, relation learning, multi-label tagging, relevance retrieval and ranking. Although current approaches rely on local training applied to non-convex formulations, we demonstrate how general convex formulations can be achieved for entity embedding, both for standard multi-linear and prototype-distance models. We investigate an efficient optimization strategy that allows scaling. An experimental evaluation reveals the advantages of global training in different case studies.