arXiv subjects

Kaarthik Sundar

Publications and source records attributed to Kaarthik Sundar.

At least 19 recordsLinked to original sources

arXiv2026

Conformal Margins for Electrical Transmission Capacity Under Fixed Balancing Policies

This article presents a data-driven approach to minimally adjust forecast-based steady-state generation dispatch in electrical transmission networks. The goal is to manage congestion caused by uncertainty in demand and renewable generation under a fixed automatic generation control (AGC) policy operating without saturation. Under this policy, we first show that the post-AGC residual injection distribution is invariant to the scheduled generation dispatch. This result separates line loading into a dispatch-dependent baseline and an uncertainty-induced residual component. Using historical forecast errors, we then apply split conformal prediction to construct transmission-line margins with distribution-free, finite-sample linewise or joint coverage guarantees. We also calibrate the aggregate pre-AGC mismatch to determine the generation headroom required for the prescribed AGC response. We incorporate these calibrated quantities into a convex secure re-dispatch problem that minimally adjusts the nominal schedule while reserving sufficient transmission capacity and AGC headroom. The resulting corrected dispatch provides finite-sample probabilistic guarantees on per-line security and unsaturated AGC operation. Case studies on the RTS-GMLC system demonstrate substantial reductions in out-of-sample line-limit violations with modest operating-cost increases.

math.OC↗

arXiv2026

Learning Discrete Decisions for MIPs with Constraint-Aware Diffusion

This paper proposes a novel learning-based approach to approximately solve instances of mixed-integer optimization problems. These problems are computationally challenging, as they require jointly determining discrete and continuous decisions while satisfying complex combinatorial constraints. The proposed method relies on a graph-based generative diffusion model that learns the discrete component of mixed-integer optimization problems while integrating a training-free feasibility projection operator directly into the reverse diffusion process to steer intermediate samples toward the feasible set throughout generation. Once the discrete decisions are generated, the remaining optimization reduces to a continuous problem that can be solved efficiently (relative to the original problem) using existing numerical methods. The resulting framework named Constrained Graph Diffusion (CGD), is problem-agnostic and can accommodate a broad class of mixed-integer optimization problems through suitable projection operators. We evaluate CGD on optimal transmission switching for ACOPF and discrete portfolio optimization, demonstrating substantial improvements in feasibility and solution quality over learning-based baselines while achieving speedups of up to $425\times$ over state-of-the-art numerical solvers for MINLPs.

cs.LG↗

arXiv2026

Stability-Constrained AC Optimal Power Flow--A Gaussian Process-Based Approach

The Alternating Current Optimal Power Flow (ACOPF) problem is a core task in power system operations, aimed at determining cost-effective generation dispatch while satisfying physical and operational constraints. However, conventional ACOPF formulations rely on steady-state models and neglect generator dynamics, which can result in operating points that are economically optimal but dynamically unstable. This paper proposes a novel, data-driven approach to incorporate generator dynamics into the ACOPF using Gaussian Process (GP) models. Specifically, it introduces an exponential surrogate function to characterize the stability of solutions to the differential equations governing synchronous generator dynamics. The exponent, which indicates whether system trajectories decay (stable) or grow (unstable), is learned as a function of the bus voltage using GP regression. Crucially, the framework enables probabilistic stability assessment to be integrated directly into the optimization process. The resulting dynamics-aware ACOPF formulation identifies operating points that satisfy both operational safety and dynamic stability criteria. Numerical experiments on the IEEE 39-bus, 57-bus, and 118-bus systems demonstrate that, compared with existing data-driven approaches, the proposed method efficiently captures generator dynamics with limited training data, yielding more reliable and robust decisions across a wide range of operating conditions.

math.OC↗

arXiv2026

A Family of Convex Models to Achieve Fairness through Dispersion Control

Controlling the dispersion of a subset of decision variables in an optimization problem is crucial for enforcing fairness or load-balancing across a wide range of applications. Building on the well-known equivalence of finite-dimensional norms, the article develops a family of parameterized convex models that regulate the dispersion of a vector of decision-variable values through its coefficient of variation. Each model has a single parameter taking values in the interval $[0,1]$. When the parameter is set to zero, the model imposes only a trivial constraint on the optimization problem; when set to one, it enforces equality of all the decision variables. As the parameter varies, the coefficient of variation is provably bounded above by a monotonic function of that parameter. The article also presents theoretical results relating the space of feasible solutions across all models. Finally, it compares the models' solution quality on a variant of the assignment problem that regulates the dispersion in the assignment costs.

math.OC↗

arXiv2026

Equitable Routing--Rethinking the Multiple Traveling Salesman Problem

The Multiple Traveling Salesman Problem (MTSP) extends the traveling salesman problem by assigning multiple salesmen to visit a set of targets from a common depot, with each target visited exactly once while minimizing total tour length. A common variant, the min-max MTSP, focuses on workload balance by minimizing the longest tour, but it is difficult to solve optimally due to weak linear relaxation bounds. This paper introduces two new parametric fairness-driven variants of the MTSP: the $\varepsilon$-Fair-MTSP and the $Δ$-Fair-MTSP, which promote equitable distribution of tour lengths while controlling overall cost. The $\varepsilon$-Fair-MTSP is formulated as a mixed-integer second-order cone program, while the $Δ$-Fair-MTSP is modeled as a mixed-integer linear program. We develop algorithms that guarantee global optimality for both formulations. Computational experiments on benchmark instances and real-world applications, including electric vehicle fleet routing, demonstrate their effectiveness. Furthermore, we show that the algorithms presented for the fairness-constrained MTSP variants can be used to obtain the Pareto front of a bi-objective optimization problem in which one objective minimizes the total tour length and the other balances the lengths of the individual tours. Overall, these fairness-constrained MTSP variants provide a practical and flexible alternative to the min-max MTSP.

math.OC↗

arXiv2026

Simulating Arbitrage Optimization for Market Monitoring in Gas and Electricity Transmission Networks

We examine market outcomes in energy transport networks with a focus on gas-fired generators, which are producers in a wholesale electricity market and consumers in the natural gas market. Market administrators monitor bids to determine whether a participant wields market power to manipulate the price of energy, reserves, or financial transmission rights. If economic or physical withholding of generation from the market is detected, mitigation is imposed by replacing excessive bids with reference level bids to prevent artificial supply shortages. We review market monitoring processes in the power grid, and present scenarios in small interpretable test networks to show how gas-fired generators can bid in the gas market to alter outcomes in a power market. We develop a framework based on DC optimal power flow (OPF) and steady-state optimal gas flow (OGF) formulations to represent two interacting markets with structured exchange of price and quantity bids. We formulate optimization-based methods to identify market power in a power grid, as well as to identify market conditions that indicate market power being exerted by a generator using gas market bids.

eess.SY↗

arXiv2026

The Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations via Graph Partitioning

The solution of potential-driven steady-state flow in large networks is required in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and its solution grows more challenging as the network size increases. We present an algorithm that utilizes a given partition of a network into tractable sizes to compute a global solution for the full nonlinear system through local solution of smaller subsystems induced by the partitions. When the partitions are induced by interconnects or transfer points corresponding to networks owned by different operators, the method ensures data is shared solely at the interconnects, leaving network operators free to solve the network flow system corresponding to their own domain in any manner of their choosing. The proposed method is shown to be connected to the Schur complement and the method's viability demonstrated on some challenging test cases.

physics.comp-ph↗

arXiv2026

Steady Flow of Natural Gas in Pipeline Networks via Solution of a Nonlinear Differential-Algebraic System of Equations

In the consideration of steady-state flow of gas in pipeline networks, the exclusion of gravity and nonlinear inertial effects (convective acceleration) leads to a fortuitous simplification in the governing equations to yield a system of nonlinear algebraic equations. Consequently, there are no studies that quantify the effect of gravity and inertial effects on the flow of gas in pipeline networks or delineate regimes of flow conditions wherein the effects are significant or negligible. In addressing this need, we consider the steady-state flow equations in pipeline networks without neglecting the gravitational and inertial terms and in place of a system of algebraic equations (one for each pipe), this approach results in a nonlinear system of first-order ordinary differential equations (ODEs) which are coupled through algebraic equations that appear in the form of boundary conditions on the pressure and balance of mass flows at either end. One of our main contributions in this article is to demonstrate how the Newton-Raphson algorithm can still be used to solve the coupled nonlinear differential-algebraic system by utilizing the appropriate forward sensitivity ODEs to evaluate the Jacobian terms arising in the iterative scheme. We also propose a variable transformation that alleviates the poor scaling of the ODE, and we introduce a two-point collocation scheme as a coarse approximation of the system from which to find initial guesses for the Newton iterations. Simulation studies were conducted for a single pipe as well as a large-scale pipeline network with real data. From these studies, we concluded that while the effect of gravity is important, the inertial effect was negligible in all cases. The proposed methodology is applicable to a wide class of pipeline and thermal-fluid networks beyond natural gas, including liquid pipelines and hydrogen transport.

math.GM↗

arXiv2025

A General and Streamlined Differentiable Optimization Framework

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.

cs.LG↗

arXiv2025

Global Optimization Algorithm for Mixed-Integer Nonlinear Programs with Trigonometric Functions

This article presents the first mixed-integer linear programming (MILP)-based iterative algorithm to solve factorable mixed-integer nonlinear programs (MINLPs) with bounded, differentiable periodic functions to global optimality with an emphasis on trigonometric functions. At each iteration, the algorithm solves a MILP relaxation of the original MINLP to obtain a bound on the optimal objective value. The relaxations are constructed using partitions of variables involved in each nonlinear term and across successive iterations, the solution of the relaxations is used to refine these partitions further leading to tighter relaxations. Also, at each iteration, a heuristic/local solve on the MINLP is used to obtain a feasible solution to the MINLP. The iterative algorithm terminates till the optimality gap is sufficiently small. This article proposes novel refinement strategies that first choose a subset of variables whose domain is refined, refinement schemes that specify the manner in which the variable domains are refined, and MILP relaxations that exploit the principal domain of the periodic functions. We also show how solving the resulting MILP relaxation may be accelerated when two or more periodic functions are related by a linking constraint. This is especially useful as any periodic function may be approximated to arbitrary precision by a Fourier series. Finally, we examine the effectiveness of the proposed approach by solving a path planning problem for a single fixed-wing aerial vehicle and present extensive numerical results comparing the various refinement schemes and techniques.

math.OC↗

arXiv2025

Optimization Proxies using Limited Labeled Data and Training Time -- A Semi-Supervised Bayesian Neural Network Approach

Constrained optimization problems arise in various engineering systems such as inventory management and power grids. Standard deep neural network (DNN) based machine learning proxies are ineffective in practical settings where labeled data is scarce and training times are limited. We propose a semi-supervised Bayesian Neural Networks (BNNs) based optimization proxy for this complex regime, wherein training commences in a sandwiched fashion, alternating between a supervised learning step for minimizing cost, and an unsupervised learning step for enforcing constraint feasibility. We show that the proposed semi-supervised BNN outperforms DNN architectures on important non-convex constrained optimization problems from energy network operations, achieving up to a tenfold reduction in expected maximum equality gap and halving the inequality gaps. Further, the BNN's ability to provide posterior samples is leveraged to construct practically meaningful probabilistic confidence bounds on performance using a limited validation data, unlike prior methods. The implementation code for this study is available at: https://github.com/kaarthiksundar/BNN-OPF/.

cs.LG↗

arXiv2025

$A^*$ for Graphs of Convex Sets

We present a novel algorithm that fuses the existing convex-programming based approach with heuristic information to find optimality guarantees and near-optimal paths for the Shortest Path Problem in the Graph of Convex Sets (SPP-GCS). Our method, inspired by $A^*$, initiates a best-first-like procedure from a designated subset of vertices and iteratively expands it until further growth is neither possible nor beneficial. Traditionally, obtaining solutions with bounds for an optimization problem involves solving a relaxation, modifying the relaxed solution to a feasible one, and then comparing the two solutions to establish bounds. However, for SPP-GCS, we demonstrate that reversing this process can be more advantageous, especially with Euclidean travel costs. In other words, we initially employ $A^*$ to find a feasible solution for SPP-GCS, then solve a convex relaxation restricted to the vertices explored by $A^*$ to obtain a relaxed solution, and finally, compare the solutions to derive bounds. We present numerical results to highlight the advantages of our algorithm over the existing approach in terms of the sizes of the convex programs solved and computation time.

math.OC↗

arXiv2025

Hierarchical Network Partitioning for Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations

The solution of potential-driven steady-state flow in large networks is a task which manifests in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology and in general there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions, but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. This article proposes and describes such a procedure, an hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

math.NA↗

arXiv2024

A Parametric, Second-Order Cone Representable Model of Fairness for Decision-Making Problems

The article develops a parametric model of fairness called "$\varepsilon$-fairness" that can be represented using a single second-order cone constraint and incorporated into existing decision-making problem formulations without impacting the complexity of solution techniques. We develop the model from the fundamental result of finite-dimensional norm equivalence in linear algebra and show that this model has a closed-form relationship to an existing metric for measuring fairness widely used in the literature. Finally, a simple case study on the optimal operation of a damaged power transmission network illustrates its effectiveness.

math.OC↗

arXiv2024

Mitigating the Impact of Uncertain Wildfire Risk on Power Grids through Topology Control

Wildfires pose a significant threat to the safe and reliable operations of the electric grid. To mitigate wildfire risk, system operators resort to public safety power shutoffs, or PSPS, that shed load for a subset of customers. As wildfire risk forecasts are stochastic, such decision-making may often be sub-optimal. This paper proposes a two-stage topology control problem that jointly minimizes generation and load-shedding costs in the face of uncertain fire risk. Compared to existing work, we include preand post-event topology control actions and consider scenarios where the wildfire risk is known with low and high confidence. The effectiveness of the proposed approach is demonstrated using a benchmark test system, artificially geo-located in Southern California, and using stochastic wildfire risk data that exists in the literature. Our work provides a crucial study of the comparative benefits of pre-event versus post-event control and the effects of wildfire risk accuracy on each control strategy.

eess.SY↗

arXiv2024

Locational Marginal Pricing of Energy in Pipeline Transport of Natural Gas and Hydrogen with Carbon Offset Incentives

We propose an optimization formulation for locational pricing of energy transported through a pipeline network that carries mixtures of natural gas and hydrogen from distributed sources to consumers. The objective includes the economic value provided by the pipeline to consumers of energy and suppliers of natural gas and green hydrogen, as well as incentives to lower carbon emissions by consuming the latter instead of the former. The optimization is subject to the physics of gas flow and mixing in the pipeline network as well as engineering limits. In addition to formulating this mathematical program, we synthesize the Lagrangian and derive analytical expressions for the dual variables. We propose that the dual solution can be used to derive locational marginal prices of natural gas, hydrogen, and energy, as well as the decarbonization premium paid by consumers that receive hydrogen. We derive several properties of solutions obtained using the proposed market mechanism, and demonstrate them using case studies for standard 8-node and 40-node pipeline test networks. Finally, we show that optimization-based analysis of the type proposed here is critical for making sound decisions about economic policy and infrastructure expansion for blending green hydrogen into existing natural gas pipelines.

math.OC↗

arXiv2024

Deep Reinforcement Learning-Based Approach for a Single Vehicle Persistent Surveillance Problem with Fuel Constraints

This article presents a deep reinforcement learning-based approach to tackle a persistent surveillance mission requiring a single unmanned aerial vehicle initially stationed at a depot with fuel or time-of-flight constraints to repeatedly visit a set of targets with equal priority. Owing to the vehicle's fuel or time-of-flight constraints, the vehicle must be regularly refueled, or its battery must be recharged at the depot. The objective of the problem is to determine an optimal sequence of visits to the targets that minimizes the maximum time elapsed between successive visits to any target while ensuring that the vehicle never runs out of fuel or charge. We present a deep reinforcement learning algorithm to solve this problem and present the results of numerical experiments that corroborate the effectiveness of this approach in comparison with common-sense greedy heuristics.

cs.RO↗

arXiv2024

Optimization of District Heating Network Parameters in Steady-State Operation

We examine the modeling, simulation, and optimization of district heating systems, which are widely used for thermal transport using steam or hot water as a carrier. We propose a generalizable framework to specify network models and scenario parameters, and develop an optimization method for evaluating system states including pressures, fluid flow rates, and temperatures throughout the network. The network modeling includes pipes, thermal plants, pumps, and passive or controllable loads as system components. We propose basic models for thermodynamic fluid transport and enforce the balance of physical quantities in steady-state flow over co-located outgoing and return networks. We formulate an optimization problem with steam and hot water as the outgoing and return carriers, as in legacy 20th century systems. The physical laws and engineering limitations are specified for each component type, and the thermal network flow optimization (TNFO) problem is formulated and solved for a realistic test network under several scenarios.

math.OC↗