Russell Bent

Harsha Nagarajan, Mowen Lu, Site Wang et al.

In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of piecewise polyhedral relaxation approaches via disjunctive formulations to solve MINLPs to global optimality in contrast to the conventional spatial branch-and-bound approaches. In order to maintain relatively small-scale mixed-integer linear programs at every iteration of the algorithm, we adaptively partition the variable domains appearing in the multi-linear terms. We also provide proofs on convergence guarantees of the proposed algorithm to a global solution. Further, we discuss a few algorithmic enhancements based on the sequential bound-tightening procedure as a presolve step, where we observe the importance of solving piecewise relaxations compared to basic convex relaxations to speed-up the convergence of the algorithm to global optimality. We demonstrate the effectiveness of our disjunctive formulations and the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare with state-of-the-art global optimization solvers. With this novel approach, we solve several large-scale instances which are, in some cases, intractable by the global optimization solver. We also shrink the best known optimality gap for one of the hard, generalized pooling problem instance.

Mowen Lu, Harsha Nagarajan, Emre Yamangil et al.

In recent years, there have been increasing concerns about how geomagnetic disturbances (GMDs) impact electrical power systems. Geomagnetically-induced currents (GICs) can saturate transformers, induce hot spot heating and increase reactive power losses. These effects can potentially cause catastrophic damage to transformers and severely impact the ability of a power system to deliver power. To address this problem, we develop a model of GIC impacts to power systems that includes 1) GIC thermal capacity of transformers as a function of normal Alternating Current (AC) and 2) reactive power losses as a function of GIC. We use this model to derive an optimization problem that protects power systems from GIC impacts through line switching, generator redispatch, and load shedding. We employ state-of-the-art convex relaxations of AC power flow equations to lower bound the objective. We demonstrate the approach on a modified RTS96 system and the UIUC 150-bus system and show that line switching is an effective means to mitigate GIC impacts. We also provide a sensitivity analysis of optimal switching decisions with respect to GMD direction.

Joan Vendrell Gallart, Russell Bent, Michael Grosskopf

Large Language Models are increasingly deployed inside agentic systems, where they must follow structured protocols, adapt to evolving states, and operate under memory, latency, and cost constraints. In such regimes, prompt extension is unreliable: growing contexts can push compact models outside their effective prompt domain, while deployment-time fine-tuning remains limited by scarce data and compute. We propose a hierarchical control-and-learning framework in which a compact model is first distilled to learn the required output schema, then supervised online by an oracle-controller loop. The controller monitors protocol validity and semantic performance, projects accumulated histories into a feasible prompt domain, and triggers lightweight oracle-supervised fine-tuning under drift. This separates schema learning for communication compatibility from semantic adaptation for task-level correction. We formalize prompt-domain feasibility and attention-induced saturation, motivating control of the effective prompt state rather than reliance on nominal context length. Using Multi-Fidelity Bayesian Optimization as a controlled sequential testbed, we characterize a core deployment failure mode and show improved reliability and cost-efficiency over non-hierarchical, distillation-only, and non-distilled baselines.

Joan Vendrell Gallart, Solmaz Kia, Russell Bent et al.

We introduce CAROL (Chain-based Adaptive Reconfiguration Over Lattices), a probabilistic framework for test-time hallucination reduction in large language models. Rather than relying on token-level uncertainty, CAROL defines a semantic uncertainty measure based on the consistency between generated responses and a trusted context, inducing a string-submodular objective over a lattice of textual sequences. This formulation enables hallucination mitigation to be cast as a Markov chain accept-reject process with provable convergence and near-optimality guarantees, allowing the model to iteratively refine outputs toward semantic consistency. By operating at the level of meaning, CAROL unifies hallucination detection and mitigation within a single framework. Empirical results on question answering and multi-agent reasoning benchmarks show that CAROL significantly reduces hallucinations and improves reliability and interpretability compared to likelihood-based and retrieval-augmented baselines, while maintaining competitive computational efficiency.

Mowen Lu, Harsha Nagarajan, Russell Bent et al.

Since the alternating current optimal power flow (ACOPF) problem was introduced in 1962, developing efficient solution algorithms for the problem has been an active field of research. In recent years, there has been increasing interest in convex relaxations-based solution approaches that are often tight in practice. Based on these approaches, we develop tight piecewise convex relaxations with convex-hull representations, an adaptive, multivariate partitioning algorithm with bound tightening that progressively improves these relaxations and, given sufficient time, converges to the globally optimal solution. We illustrate the strengths of our algorithm using benchmark ACOPF test cases from the literature. Computational results show that our novel algorithm reduces the best-known optimality gaps for some hard ACOPF cases.

Harsha Nagarajan, Russell Bent, Pascal Van Hentenryck et al.

As illustrated in recent years (Superstorm Sandy, the Northeast Ice Storm of 1998, etc.), extreme weather events pose an enormous threat to the electric power transmission systems and the associated socio-economic systems that depend on reliable delivery of electric power. Besides inevitable malfunction of power grid components, deliberate malicious attacks can cause high risks to the service. These threats motivate the need for approaches and methods that improve the resilience of power systems. In this paper, we develop a model and tractable methods for optimizing the upgrade of transmission systems through a combination of hardening existing components, adding redundant lines, switches, generators, and FACTS and phase-shifting devices. While many of these controllable components are included in traditional design (expansion planning) problems, we uniquely assess their benefits from a resiliency point of view. More importantly, perhaps, we evaluate the suitability of using state-of-the-art AC power flow relaxations versus the common DC approximation in resilience improvement studies. The resiliency model and algorithms are tested on a modified version of the RTS-96 (single area) system.

Kaarthik Sundar, Harsha Nagarajan, Miles Lubin et al.

As renewable wind energy penetration rates continue to increase, one of the major challenges facing grid operators is the question of how to control transmission grids in a reliable and a cost-efficient manner. The stochastic nature of wind forces an alteration of traditional methods for solving day-ahead and look-ahead unit commitment and dispatch. In particular, uncontrollable wind generation increases the risk of random component failures. To address these questions, we present an N-1 Security and Chance-Constrained Unit Commitment (SCCUC) that includes the modeling of generation reserves that respond to wind fluctuations and tertiary reserves to account for single component outages. The basic formulation is reformulated as a mixed-integer second-order cone problem to limit the probability of failure. We develop three different algorithms to solve the problem to optimality and present a detailed case study on the IEEE RTS-96 single area system. The case study assesses the economic impacts due to contingencies and various degrees of wind power penetration into the system and also corroborates the effectiveness of the algorithms.

Kaarthik Sundar, Carleton Coffrin, Harsha Nagarajan et al.

This paper considers a probabilistic generalization of the $N$-$k$ failure-identification problem in power transmission networks, where the probability of failure of each component in the network is known a priori and the goal of the problem is to find a set of $k$ components that maximizes disruption to the system loads weighted by the probability of simultaneous failure of the $k$ components. The resulting problem is formulated as a bilevel mixed-integer nonlinear program. Convex relaxations, linear approximations, and heuristics are developed to obtain feasible solutions that are close to the optimum. A general cutting-plane algorithm is proposed to solve the convex relaxation and linear approximations of the $N$-$k$ problem. Extensive numerical results corroborate the effectiveness of the proposed algorithms on small-, medium-, and large-scale test instances, the test instances include the IEEE 14-bus system, the IEEE single-area and three-area RTS96 systems, the IEEE 118-bus system, the WECC 240-bus test system, the 1354-bus PEGASE system, and the 2383-bus Polish winter-peak test system.

Kaarthik Sundar, Harsha Nagarajan, Sidhant Misra et al.

This article develops a strengthened convex quadratic convex (QC) relaxation of the AC Optimal Power Flow (AC-OPF) problem and presents an optimization-based bound-tightening (OBBT) algorithm to compute tight, feasible bounds on the voltage magnitude variables for each bus and the phase angle difference variables for each branch in the network. Theoretical properties of the strengthened QC relaxation that show its dominance over the other variants of the QC relaxation studied in the literature are also derived. The effectiveness of the strengthened QC relaxation is corroborated via extensive numerical results on benchmark AC-OPF test networks. In particular, the results demonstrate that the proposed relaxation consistently provides the tightest variable bounds and optimality gaps with negligible impacts on runtime performance.

Sai Krishna Kanth Hari, Kaarthik Sundar, Harsha Nagarajan et al.

In recent years, microgrids, i.e., disconnected distribution systems, have received increasing interest from power system utilities to support the economic and resiliency posture of their systems. The economics of long distance transmission lines prevent many remote communities from connecting to bulk transmission systems and these communities rely on off-grid microgrid technology. Furthermore, communities that are connected to the bulk transmission system are investigating microgrid technologies that will support their ability to disconnect and operate independently during extreme events. In each of these cases, it is important to develop methodologies that support the capability to design and operate microgrids in the absence of transmission over long periods of time. Unfortunately, such planning problems tend to be computationally difficult to solve and those that are straightforward to solve often lack the modeling fidelity that inspires confidence in the results. To address these issues, we first develop a high fidelity model for design and operations of a microgrid that include component efficiencies, component operating limits, battery modeling, unit commitment, capacity expansion, and power flow physics; the resulting model is a mixed-integer quadratically-constrained quadratic program (MIQCQP). We then develop an iterative algorithm, referred to as the Model Predictive Control (MPC) algorithm, that allows us to solve the resulting MIQCQP. We show, through extensive computational experiments, that the MPC-based method can scale to problems that have a very long planning horizon and provide high quality solutions that lie within 5\% of optimal.

Harsha Nagarajan, Mowen Lu, Emre Yamangil et al.

In this work, we propose a two-stage approach to strengthen piecewise McCormick relaxations for mixed-integer nonlinear programs (MINLP) with multi-linear terms. In the first stage, we exploit Constraint Programing (CP) techniques to contract the variable bounds. In the second stage we partition the variables domains using a dynamic multivariate partitioning scheme. Instead of equally partitioning the domains of variables appearing in multi-linear terms, we construct sparser partitions yet tighter relax- ations by iteratively partitioning the variable domains in regions of interest. This approach decouples the number of partitions from the size of the variable domains, leads to a significant reduction in computation time, and limits the number of binary variables that are introduced by the partitioning. We demonstrate the performance of our algorithm on well-known benchmark problems from MINLPLIB and discuss the computational benefits of CP-based bound tightening procedures.

David M. Fobes, Harsha Nagarajan, Manuel Garcia et al.

The distribution system restoration (DSR) problem has received considerable attention over the last decade or more. Solutions to the DSR problem identify the best set or sequence of actions to perform on a distribution circuit to restore service after a disruption. The problem is challenging from a computational perspective, with engineering constraints specific to distribution systems, such as radial operations, that are difficult to effectively model. In this paper, we revisit the model for how specific loads are shed, energized and restored--and develop a formulation that more accurately models the requirements of load shedding, load energizing and restoration in distribution systems.

Andrew W. Rosemberg, Joaquim Dias Garcia, François Pacaud et al.

Differentiating through constrained optimization problems is increasingly central to learning, control, and large-scale decision-making systems, yet practical integration remains challenging due to solver specialization and interface mismatches. This paper presents a general and streamlined framework-an updated DiffOpt.jl-that unifies modeling and differentiation within the Julia optimization stack. The framework computes forward - and reverse-mode solution and objective sensitivities for smooth, potentially nonconvex programs by differentiating the KKT system under standard regularity assumptions. A first-class, JuMP-native parameter-centric API allows users to declare named parameters and obtain derivatives directly with respect to them - even when a parameter appears in multiple constraints and objectives - eliminating brittle bookkeeping from coefficient-level interfaces. We illustrate these capabilities on convex and nonconvex models, including economic dispatch, mean-variance portfolio selection with conic risk constraints, and nonlinear robot inverse kinematics. Two companion studies further demonstrate impact at scale: gradient-based iterative methods for strategic bidding in energy markets and Sobolev-style training of end-to-end optimization proxies using solver-accurate sensitivities. Together, these results demonstrate that differentiable optimization can be deployed as a routine tool for experimentation, learning, calibration, and design-without deviating from standard JuMP modeling practices and while retaining access to a broad ecosystem of solvers.

Miao Li, Michael Klamkin, Pascal Van Hentenryck et al.

This paper studies optimization proxies, machine learning (ML) models trained to efficiently predict optimal solutions for AC Optimal Power Flow (ACOPF) problems. While promising, optimization proxy performance heavily depends on training data quality. To address this limitation, this paper introduces a novel active sampling framework for ACOPF optimization proxies designed to generate realistic and diverse training data. The framework actively explores varied, flexible problem specifications reflecting plausible operational realities. More importantly, the approach uses optimization-specific quantities (active constraint sets) that better capture the salient features of an ACOPF that lead to the optimal solution. Numerical results show superior generalization over existing sampling methods with an equivalent training budget, significantly advancing the state-of-practice for trustworthy ACOPF optimization proxies.

Smitha Gopinath, Hassan L. Hijazi, Adam Collins et al.

In airborne light detection and ranging (LiDAR) systems, misalignments between the LiDAR-scanner and the inertial navigation system (INS) mounted on an unmanned aerial vehicle (UAV)'s frame can lead to inaccurate 3D point clouds. Determining the orientation offset, or boresight error is key to many LiDAR-based applications. In this work, we introduce a mixed-integer quadratically constrained quadratic program (MIQCQP) that can globally solve this misalignment problem. We also propose a nested spatial branch and bound (nsBB) algorithm that improves computational performance. The nsBB relies on novel preprocessing steps that progressively reduce the problem size. In addition, an adaptive grid search (aGS) allowing us to obtain quick heuristic solutions is presented. Our algorithms are open-source, multi-threaded and multi-machine compatible.

Duong Thuy Anh Nguyen, Harsha Nagarajan, Robert Ferrando et al.

This paper presents an optimal network topology control framework using cutting-plane methods for efficient network partitioning with controllable edges. The objective is to enable real-time reconfiguration of interconnected sub-networks while ensuring radial connectivity, resource feasibility, and structured leader allocation, which are essential for distributed control, stability, and coordination. The problem is formulated as a mixed-integer program that integrates graph-theoretic constraints, resource flow, and network structural properties to enforce an operational hierarchy. To address the combinatorial complexity of cycle elimination and leader assignment, we propose an iterative cutting-plane framework that ensures convergence to an optimal and feasible network topology. Theoretical guarantees on optimality preservation, feasibility, and convergence are established, ensuring systematic elimination of infeasible configurations while maintaining distributed controllability. Simulations on a modified Iowa 240-bus power distribution grid demonstrate the framework's effectiveness in network reconfiguration under resource constraints. The approach achieves median and best-case speedups of 57.5x and over 64x in a 46-switch configuration, highlighting its applicability to other networked control systems.

Mohamad Fares El Hajj Chehade, Wenting Li, Brian W. Bell et al.

The robustness of neural networks is crucial in safety-critical applications, where identifying a reliable input space is essential for effective model selection, robustness evaluation, and the development of reliable control strategies. Most existing robustness verification methods assess the worst-case output under the assumption that the input space is known. However, precisely identifying a verifiable input space \(\mathcal{C}\), where no adversarial examples exist, is challenging due to the possible high dimensionality, discontinuity, and non-convex nature of the input space. To address this challenge, we propose a novel framework, **LEVIS**, consisting of **LEVIS-α** and **LEVIS-\b{eta}**. **LEVIS-α** identifies a single, large verifiable ball that intersects at least two boundaries of a bounded region \(\mathcal{C}\), while **LEVIS-\b{eta}** systematically captures the entirety of the verifiable space by integrating multiple verifiable balls. Our contributions include: (1) introducing a verification framework that uses mixed-integer programming (MIP) to compute nearest and directional adversarial points, (2) integrating complementarity-constrained (CC) optimization with a reduced MIP formulation for scalability, achieving up to a 6 times runtime reduction, (3) theoretically characterizing the properties of the verifiable balls obtained by **LEVIS-α**, and (4) validating the approach across applications including electrical power flow regression and image classification, with demonstrated performance gains and geometric insights into the verifiable region.

Hongwei Jin, Prasanna Balaprakash, Allen Zou et al.

The threat of geomagnetic disturbances (GMDs) to the reliable operation of the bulk energy system has spurred the development of effective strategies for mitigating their impacts. One such approach involves placing transformer neutral blocking devices, which interrupt the path of geomagnetically induced currents (GICs) to limit their impact. The high cost of these devices and the sparsity of transformers that experience high GICs during GMD events, however, calls for a sparse placement strategy that involves high computational cost. To address this challenge, we developed a physics-informed heterogeneous graph neural network (PIHGNN) for solving the graph-based dc-blocker placement problem. Our approach combines a heterogeneous graph neural network (HGNN) with a physics-informed neural network (PINN) to capture the diverse types of nodes and edges in ac/dc networks and incorporates the physical laws of the power grid. We train the PIHGNN model using a surrogate power flow model and validate it using case studies. Results demonstrate that PIHGNN can effectively and efficiently support the deployment of GIC dc-current blockers, ensuring the continued supply of electricity to meet societal demands. Our approach has the potential to contribute to the development of more reliable and resilient power grids capable of withstanding the growing threat that GMDs pose.

Robert Parker, Oscar Dowson, Nicole LoGiudice et al.

We propose a reduced-space formulation for optimizing over trained neural networks where the network's outputs and derivatives are evaluated on a GPU. To do this, we treat the neural network as a "gray box" where intermediate variables and constraints are not exposed to the optimization solver. Compared to the full-space formulation, in which intermediate variables and constraints are exposed to the optimization solver, the reduced-space formulation leads to faster solves and fewer iterations in an interior point method. We demonstrate the benefits of this method on two optimization problems: Adversarial generation for a classifier trained on MNIST images and security-constrained optimal power flow with transient feasibility enforced using a neural network surrogate.

Wenting Li, Saif R. Kazi, Russell Bent et al.

Neural networks achieve strong empirical performance, but robustness concerns still hinder deployment in safety-critical applications. Formal verification provides robustness guarantees, but current methods face a scalability-completeness trade-off. We propose a hybrid verifier in a branch-and-bound (BaB) framework that efficiently tightens both upper and lower bounds until an $ε-$global optimum is reached or early stop is triggered. The key is an exact nonlinear program with complementarity constraints (NLP-CC) for upper bounding that preserves the ReLU input-output graph, so any feasible solution yields a valid counterexample and enables rapid pruning of unsafe subproblems. We further accelerate verification with (i) warm-started NLP solves requiring minimal constraint-matrix updates and (ii) pattern-aligned strong branching that prioritizes splits most effective at tightening relaxations. We also provide conditions under which NLP-CC upper bounds are tight. Experiments on MNIST and CIFAR-10 show markedly tighter upper bounds than PGD across perturbation radii spanning up to three orders of magnitude, fast per-node solves in practice, and substantial end-to-end speedups over MIP-based verification, amplified by warm-starting, GPU batching, and pattern-aligned branching.

Michael Grosskopf, Russell Bent, Rahul Somasundaram et al.

Large language models (LLMs) have moved far beyond their initial form as simple chatbots, now carrying out complex reasoning, planning, writing, coding, and research tasks. These skills overlap significantly with those that human scientists use day-to-day to solve complex problems that drive the cutting edge of research. Using LLMs in "agentic" AI has the potential to revolutionize modern science and remove bottlenecks to progress. In this work, we present URSA, a scientific agent ecosystem for accelerating research tasks. URSA consists of a set of modular agents and tools, including coupling to advanced physics simulation codes, that can be combined to address scientific problems of varied complexity and impact. This work highlights the architecture of URSA, as well as examples that highlight the potential of the system.

Robert B. Parker, Oscar Dowson, Nicole LoGiudice et al.

We compare full-space, reduced-space, and gray-box formulations for representing trained neural networks in nonlinear constrained optimization problems. We test these formulations on a transient stability-constrained, security-constrained alternating current optimal power flow (SCOPF) problem where the transient stability criteria are represented by a trained neural network surrogate. Optimization problems are implemented in JuMP and trained neural networks are embedded using a new Julia package: MathOptAI.jl. To study the bottlenecks of the three formulations, we use neural networks with up to 590 million trained parameters. The full-space formulation is bottlenecked by the linear solver used by the optimization algorithm, while the reduced-space formulation is bottlenecked by the algebraic modeling environment and derivative computations. The gray-box formulation is the most scalable and is capable of solving with the largest neural networks tested. It is bottlenecked by evaluation of the neural network's outputs and their derivatives, which may be accelerated with a graphics processing unit (GPU). Leveraging the gray-box formulation and GPU acceleration, we solve our test problem with our largest neural network surrogate in 2.5$\times$ the time required for a simpler SCOPF problem without the stability constraint.

Kaarthik Sundar, Harsha Nagarajan, Line Roald et al.

As renewable wind energy penetration rates continue to increase, one of the major challenges facing grid operators is the question of how to control transmission grids in a reliable and a cost-efficient manner. The stochastic nature of wind forces an alteration of traditional methods for solving day-ahead and look-ahead unit commitment and dispatch. In particular, the variability of wind generation increases the risk of unexpected overloads and cascading events. To address these questions, we present an N-1 Security and Chance-Constrained Unit Commitment (SCCUC) that includes models of generation reserves that respond to wind fluctuations and component outages. We formulate the SCCUC as a mixed-integer, second-order cone problem that limits the probability of failure. We develop a modified Benders decomposition algorithm to solve the problem to optimality and present detailed case studies on the IEEE RTS-96 three-area and the IEEE 300 NESTA test systems. The case studies assess the economic impacts of contingencies and various degrees of wind power penetration and demonstrate the effectiveness and scalability of the algorithm.

Carleton Coffrin, Harsha Nagarajan, Russell Bent

The recent emergence of novel computational devices, such as adiabatic quantum computers, CMOS annealers, and optical parametric oscillators, present new opportunities for hybrid-optimization algorithms that are hardware accelerated by these devices. In this work, we propose the idea of an Ising processing unit as a computational abstraction for reasoning about these emerging devices. The challenges involved in using and benchmarking these devices are presented and commercial mixed integer programming solvers are proposed as a valuable tool for the validation of these disparate hardware platforms. The proposed validation methodology is demonstrated on a D-Wave 2X adiabatic quantum computer, one example of an Ising processing unit. The computational results demonstrate that the D-Wave hardware consistently produces high-quality solutions and suggests that as IPU technology matures it could become a valuable co-processor in hybrid-optimization algorithms.

Arthur Barnes, Harsha Nagarajan, Emre Yamangil et al.

In the electrical grid, the distribution system is themost vulnerable to severe weather events. Well-placed and coordinatedupgrades, such as the combination of microgrids, systemhardening and additional line redundancy, can greatly reduce thenumber of electrical outages during extreme events. Indeed, ithas been suggested that resilience is one of the primary benefitsof networked microgrids. We formulate a resilient distributiongrid design problem as a two-stage stochastic program andmake use of decomposition-based heuristic algorithms to scaleto problems of practical size. We demonstrate the feasibilityof a resilient distribution design tool on a model of an actualdistribution network. We vary the study parameters, i.e., thecapital cost of microgrid generation relative to system hardeningand target system resilience metrics, and find regions in thisparametric space corresponding to different distribution systemarchitectures, such as individual microgrids, hardened networks,and a transition region that suggests the benefits of microgridsnetworked via hardened circuit segments.

Scott Backhaus, Russell Bent, James Bono et al.

Recent years have seen increased interest in the design and deployment of smart grid devices and control algorithms. Each of these smart communicating devices represents a potential access point for an intruder spurring research into intruder prevention and detection. However, no security measures are complete, and intruding attackers will compromise smart grid devices leading to the attacker and the system operator interacting via the grid and its control systems. The outcome of these machine-mediated human-human interactions will depend on the design of the physical and control systems mediating the interactions. If these outcomes can be predicted via simulation, they can be used as a tool for designing attack-resilient grids and control systems. However, accurate predictions require good models of not just the physical and control systems, but also of the human decision making. In this manuscript, we present an approach to develop such tools, i.e. models of the decisions of the cyber-physical intruder who is attacking the systems and the system operator who is defending it, and demonstrate its usefulness for design.