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Optimization models have been applied to solve a wide variety of decision-making problems. These models are usually developed by optimization experts but are used by practitioners without optimization expertise in various application domains. As a result, practitioners often struggle to interact with and draw useful conclusions from optimization models independently. To fill this gap, we introduce OptiChat, a natural language dialogue system designed to help practitioners interpret model formulation, diagnose infeasibility, analyze sensitivity, retrieve information, evaluate modifications, and provide counterfactual explanations. By augmenting large language models (LLMs) with functional calls and code generation tailored for optimization models, we enable seamless interaction and minimize the risk of hallucinations in OptiChat. We develop a new dataset to evaluate OptiChat's performance in explaining optimization models. Experiments demonstrate that OptiChat effectively bridges the gap between optimization models and practitioners, delivering autonomous, accurate, and instant responses.

We introduce a quadratically-constrained approximation (QCAC) of the AC optimal power flow (AC-OPF) problem. Unlike existing approximations like the DC-OPF, our model does not rely on typical assumptions such as high reactance-to-resistance ratio, near-nominal voltage magnitudes, or small angle differences, and preserves the structural sparsity of the original AC power flow equations, making it suitable for decentralized power systems optimization problems. To achieve this, we reformulate the AC-OPF problem as a quadratically constrained quadratic program. The nonconvex terms are expressed as differences of convex functions, which are then convexified around a base point derived from a warm start of the nodal voltages. If this linearization results in a non-empty constraint set, the convexified constraints form an inner convex approximation. Our experimental results, based on Power Grid Library instances of up to 30,000 buses, demonstrate the effectiveness of the QCAC approximation with respect to other well-documented conic relaxations and a linear approximation. We further showcase its potential advantages over the well-documented second-order conic relaxation of the power flow equations in two proof-of-concept case studies: optimal reactive power dispatch in transmission networks and PV hosting capacity in distribution grids., Comment: 10 pages, 5 figures, 4 tables

We show variants of the genus inequality for the irreducible components of the special fiber of an arithmetic curve over a henselian discrete valuation ring of residue characteristic zero that take into account the non-existence of rational, respectively real points on the the components. We then apply this inequality to obtain the bound $2^{ng}$ (respectively $2^{n(g+1)}$) on the totally positive sum-of-two-squares index in the function field of a curve of genus $g$ over the field of $n$-fold iterated real Laurent series with (respectively without) real points. The bound $2^{n(g+1)}$ had been previously known only for hyperelliptic curves.

Machine learning algorithms are increasingly being applied to fault detection and diagnosis (FDD) in chemical processes. However, existing data-driven FDD platforms often lack interpretability for process operators and struggle to identify root causes of previously unseen faults. This paper presents FaultExplainer, an interactive tool designed to improve fault detection, diagnosis, and explanation in the Tennessee Eastman Process (TEP). FaultExplainer integrates real-time sensor data visualization, Principal Component Analysis (PCA)-based fault detection, and identification of top contributing variables within an interactive user interface powered by large language models (LLMs). We evaluate the LLMs' reasoning capabilities in two scenarios: one where historical root causes are provided, and one where they are not to mimic the challenge of previously unseen faults. Experimental results using GPT-4o and o1-preview models demonstrate the system's strengths in generating plausible and actionable explanations, while also highlighting its limitations, including reliance on PCA-selected features and occasional hallucinations.

For a domain $\Omega$ in a finite-dimensional space $E$, we consider the space $M=(\Omega,d)$ where $d$ is the intrinsic distance in $\Omega$. We obtain an isometric representation of the space $\mathrm{Lip}_{0}(M)$ as a subspace of $L^{\infty}(\Omega;E^{*})$ and we use this representation in order to obtain the corresponding isometric representation for the Lipschitz-free space $\mathcal{F}(M)$ as a quotient of the space $L^{1}(\Omega;E)$. We compare our result with those existent in the literature for bounded domains with Lipschitz boundary, and for convex domains, which can be then deduced as a corollaries of our result.

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\hat{f}$ of a Lipschitz map $f:M\to N$ is Dunford-Pettis if and only if it is Radon-Nikod\'ym if and only if it does not fix any copy of $L_1$. We also identify and study the corresponding metric property of $f$, which is a natural extension of the curve-flatness introduced in [arXiv:2103.09370]. Further, we show that $\hat{f}$ is compact if and only if it does not fix any copy of $\ell_1$., Comment: 37 pages

This paper proposes a new data-driven methodology for predicting intervals of post-fault voltage trajectories in power systems. We begin by introducing the Quantile Attention-Fourier Deep Operator Network (QAF-DeepONet), designed to capture the complex dynamics of voltage trajectories and reliably estimate quantiles of the target trajectory without any distributional assumptions. The proposed operator regression model maps the observed portion of the voltage trajectory to its unobserved post-fault trajectory. Our methodology employs a pre-training and fine-tuning process to address the challenge of limited data availability. To ensure data privacy in learning the pre-trained model, we use merging via federated learning with data from neighboring buses, enabling the model to learn the underlying voltage dynamics from such buses without directly sharing their data. After pre-training, we fine-tune the model with data from the target bus, allowing it to adapt to unique dynamics and operating conditions. Finally, we integrate conformal prediction into the fine-tuned model to ensure coverage guarantees for the predicted intervals. We evaluated the performance of the proposed methodology using the New England 39-bus test system considering detailed models of voltage and frequency controllers. Two metrics, Prediction Interval Coverage Probability (PICP) and Prediction Interval Normalized Average Width (PINAW), are used to numerically assess the model's performance in predicting intervals. The results show that the proposed approach offers practical and reliable uncertainty quantification in predicting the interval of post-fault voltage trajectories.

This paper presents a parametric quadratic approximation of the AC optimal power flow (AC-OPF) problem for time-sensitive and market-based applications. The parametric approximation preserves the physics-based but simple representation provided by the DC-OPF model and leverages market and physics information encoded in the data-driven demand-dependent parameters. To enable the deployment of the proposed model for real-time applications, we propose a supervised learning approach to predict near-optimal parameters, given a certain metric concerning the dispatch quantities and locational marginal prices (LMPs). The training dataset is generated based on the solution of the accurate AC-OPF problem and a bilevel optimization problem, which calibrates parameters satisfying two market properties: cost recovery and revenue adequacy. We show the proposed approach's performance in various test systems in terms of cost and dispatch approximation errors, LMPs, market properties satisfaction, dispatch feasibility, and generalizability with respect to N-1 network topologies., Comment: 11 pages, 6 figures, 52nd Hawaii International Conference on System

A monomial algebra B is defined as a quotient of a polynomial ring by a monomial ideal, which is an ideal generated by a finite set of monomials. In this paper, we determine the automorphism group of a monomial algebra B, under the assumption that B is a finite-dimensional vector space over a field of characteristic zero. We achieve this by providing an explicit classification of the homogeneous locally nilpotent derivations of B. The main body of the paper addresses the more general case of semigroup algebras, with the polynomial ring being a particular case., Comment: 18 pages

Surrogate modeling is used to replace computationally expensive simulations. Neural networks have been widely applied as surrogate models that enable efficient evaluations over complex physical systems. Despite this, neural networks are data-driven models and devoid of any physics. The incorporation of physics into neural networks can improve generalization and data efficiency. The physics-informed neural network (PINN) is an approach to leverage known physical constraints present in the data, but it cannot strictly satisfy them in the predictions. This work proposes a novel physics-informed neural network, KKT-hPINN, which rigorously guarantees hard linear equality constraints through projection layers derived from KKT conditions. Numerical experiments on Aspen models of a continuous stirred-tank reactor (CSTR) unit, an extractive distillation subsystem, and a chemical plant demonstrate that this model can further enhance the prediction accuracy.

Decision-making problems can be represented as mathematical optimization models, finding wide applications in fields such as economics, engineering and manufacturing, transportation, and health care. Optimization models are mathematical abstractions of the problem of making the best decision while satisfying a set of requirements or constraints. One of the primary barriers to deploying these models in practice is the challenge of helping practitioners understand and interpret such models, particularly when they are infeasible, meaning no decision satisfies all the constraints. Existing methods for diagnosing infeasible optimization models often rely on expert systems, necessitating significant background knowledge in optimization. In this paper, we introduce OptiChat, a first-of-its-kind natural language-based system equipped with a chatbot GUI for engaging in interactive conversations about infeasible optimization models. OptiChat can provide natural language descriptions of the optimization model itself, identify potential sources of infeasibility, and offer suggestions to make the model feasible. The implementation of OptiChat is built on GPT-4, which interfaces with an optimization solver to identify the minimal subset of constraints that render the entire optimization problem infeasible, also known as the Irreducible Infeasible Subset (IIS). We utilize few-shot learning, expert chain-of-thought, key-retrieve, and sentiment prompts to enhance OptiChat's reliability. Our experiments demonstrate that OptiChat assists both expert and non-expert users in improving their understanding of the optimization models, enabling them to quickly identify the sources of infeasibility.

We prove that every sum of squares in the rational function field in two variables $K(X,Y)$ over a hereditarily pythagorean field $K$ is a sum of $8$ squares. More precisely, we show that the Pythagoras number of every finite extension of $K(X)$ is at most $5$. The main ingredients of the proof are a local-global principle for quadratic forms over function fields in one variable over a complete rank-$1$ valued field due to V. Mehmeti and a valuation theoretic characterization of hereditarily pythagorean fields due to L. Br\"ocker., Comment: 24 pages

Titanium dioxide (TiO$_2$) thin films are commonly used as photocatalytic materials. Here, we enhance the photocatalytic activity of devices based on TiO$_2$ by combining nanostructured glass substrates with metallic plasmonic nanostructures. We achieve a three-fold increase of the catalyst's surface area through nanoscale three-dimensional patterning of periodic conical grids, which creates a broadband optical absorber. The addition of aluminum and gold activates the structures plasmonically and improves the optical absorption in the TiO$_2$ films to above 70% in the visible and NIR spectral range. We demonstrate the resulting enhancement of the photocatalytic activity with organic dye degradation tests under different light sources. Furthermore, the pharmaceutical drug Carbamazepine, a common water pollutant, is reduced in aqueous solution by up to 48% in 360 minutes. Our approach is scalable and potentially enables future solar-driven wastewater treatment., Comment: 36 pages

Let $n\in\mathbb{N}$ and let $K$ be a field with a henselian discrete valuation of rank $n$ with hereditarily euclidean residue field. Let $F/K$ be an algebraic function field in one variable. We show that the Pythagoras number of $F$ is $2$ or $3$ and we determine the order of the group of nonzero sums of squares modulo sums of two squares in $F$ in terms of the number of equivalence classes of discrete valuations on $F$ of rank at most $n.$ In the case of function fields of hyperelliptic curves of genus $g$, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by $2^{n(g+1)}$. We show in Example 4.6 that this bound is optimal.

We propose a technique to assess the vulnerability of the power system state estimator. We aim at identifying measurements that have a high potential of being the target of false data injection attacks. From an adversary's point of view, such measurements have to show the following characteristics: i) being influential on the variable estimates, and ii) corrupting their measured values is likely to be undetected. Additionally, such characteristics should not change significantly with the system's operating condition. Our technique provides a systematic way of identifying measurements with such characteristics. We illustrate our methodology on a 4-bus system and the New England 39-bus system.

Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider class of functions but still with Lipschitz flavour. As an application we obtain a Banach-Stone-like theorem. On the other hand, we give an extension of a result of Bourgain and Diestel related to limited operators and cosingularity.

It is hereby established that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of $\ell^{\infty}(\mathbb{N})$). This result goes in the line of a previous result of Borwein-Wang. However, while the latter was based on Baire category theorem, our current approach is constructive and is not linked to the uniform convergence. In particular we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions.

In this short note, we develop a method for identifying the spaces $Lip_{0}(U)$ for every nonempty open convex $U$ of $\mathbb{R}^{n}$ and $n\in\mathbb{N}$. Moreover, we show that $\mathcal{F}(U)$ is identified with a quotient of $L^{1}(U;\mathbb{R}^{n})$.