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Variable aggregation for nonlinear optimization problems

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arXiv 2025年

作者： Naik, Sakshi Biegler, Lorenz Bent, Russell Parker, Robert Department of Chemical Engineering Carnegie Mellon University PittsburghPA United States Applied Math and Plasma Physics Group Los Alamos National Laboratory Los AlamosNM United States

Variable aggregation has been largely studied as an important presolve algorithm for optimization of linear and mixed-integer programs. Although some nonlinear solvers and algebraic modeling languages implement variable aggregation as a pre-solve reduction, the impact it can have on constrained nonlinear programs is unexplored. In this work, we formalize variable aggregation as a pre-solve algorithm to develop reduced-space formulations of nonlinear programs. A novel approximate maximum variable aggregation strategy is developed to aggregate as many variables as possible. Furthermore, aggregation strategies that preserve the problem structure are compared against approximate maximum aggregation. Our results show that variable aggregation can generally help to improve the convergence reliability of nonlinear programs. It can also help in reducing total solve time. However, Hessian evaluation can become a bottleneck if aggregation significantly increases the number of variables appearing nonlinearly in many constraints. © 2025, CC BY.

关键词： mixed-integer linear programming

Integrated Optimization and Game Theory Framework for Fair Cost Allocation in Community Microgrids

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arXiv 2025年

作者： Babu, K. Victor Sam Moses Chakraborty, Pratyush Pal, Mayukha ABB Ability Innovation Center Hyderabad500084 India Department of Electrical and Electronics Engineering BITS Pilani Hyderabad Campus Hyderabad500078 India

Fair cost allocation in community microgrids remains a significant challenge due to the complex interactions between multiple participants with varying load profiles, distributed energy resources, and storage systems. Traditional cost allocation methods often fail to adequately address the dynamic nature of participant contributions and benefits, leading to inequitable distribution of costs and reduced participant satisfaction. This paper presents a novel framework integrating multi-objective optimization with cooperative game theory for fair and efficient microgrid operation and cost allocation. The proposed approach combines mixed-integer linear programming for optimal resource dispatch with Shapley value analysis for equitable benefit distribution, ensuring both system efficiency and participant satisfaction. The framework was validated using real-world data across six distinct operational scenarios, demonstrating significant improvements in both technical and economic performance. Results show peak demand reductions ranging from 7.8% to 62.6%, solar utilization rates reaching 114.8% through effective storage integration, and cooperative gains of up to $1,801.01 per day. The Shapley value-based allocation achieved balanced benefit-cost distributions, with net positions ranging from -16.0% to +14.2% across different load categories, ensuring sustainable participant cooperation. © 2025, CC BY-NC-ND.

关键词： mixed-integer linear programming

Sample-Based Piecewise linear Power Flow Approximations Using Second-Order Sensitivities

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arXiv 2025年

作者： Buason, Paprapee Misra, Sidhant Molzahn, Daniel K. Los Alamos National Laboratory Los AlamosNM United States School of Electrical and Computer Engineering Georgia Institute of Technology AtlantaGA United States

The inherent nonlinearity of the power flow equations poses significant challenges in accurately modeling power systems, particularly when employing linearized approximations. Although power flow linearizations provide computational efficiency, they can fail to fully capture nonlinear behavior across diverse operating conditions. To improve approximation accuracy, we propose conservative piecewise linear approximations (CPLA) of the power flow equations, which are designed to consistently over- or under-estimate the quantity of interest, ensuring conservative behavior in optimization. The flexibility provided by piecewise linear functions can yield improved accuracy relative to standard linear approximations. However, applying CPLA across all dimensions of the power flow equations could introduce significant computational complexity, especially for large-scale optimization problems. In this paper, we propose a strategy that selectively targets dimensions exhibiting significant nonlinearities. Using a second-order sensitivity analysis, we identify the directions where the power flow equations exhibit the most significant curvature and tailor the CPLAs to improve accuracy in these specific directions. This approach reduces the computational burden while maintaining high accuracy, making it particularly well-suited for mixed-integer programming problems involving the power flow equations. © 2025, CC BY-NC-ND.

关键词： mixed-integer linear programming

Optimal User and Target Scheduling, User-Target Pairing, and Low-Resolution Phase-Only Beamforming for ISAC Systems

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arXiv 2025年

作者： Abanto-Leon, Luis F. Maghsudi, Setareh Ruhr University Bochum Germany

We investigate the joint user and target scheduling, user-target pairing, and low-resolution phase-only beamforming design for integrated sensing and commmunications (ISAC). Scheduling determines which users and targets are served, while pairing specifies which users and targets are grouped into pairs. Additionally, the beamformers are designed using few-bit constant-modulus phase shifts. This resource allocation problem is a nonconvex mixed-integer nonlinear program (MINLP) and challenging to solve. To address it, we propose an exact mixed-integer linear program (MILP) reformulation, which leads to a globally optimal solution. Our results demonstrate the superiority of an optimal joint design compared to heuristic stage-wise approaches, which are highly sensitive to scenario characteristics. © 2025, CC BY.

关键词： mixed-integer linear programming

Deterministic Global Optimization over trained Kolmogorov Arnold Networks

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arXiv 2025年

作者： Karia, Tanuj Lastrucci, Giacomo Schweidtmann, Artur M. Department of Chemical Engineering Delft University of Tecnology Van der Maasweg 9 Delft2629 HZ Netherlands

To address the challenge of tractability for optimizing mathematical models in science and engineering, surrogate models are often employed. Recently, a new class of machine learning models named Kolmogorov Arnold Networks (KANs) have been proposed. It was reported that KANs can approximate a given input/output relationship with a high level of accuracy, requiring significantly fewer parameters than multilayer perceptrons. Hence, we aim to assess the suitability of deterministic global optimization of trained KANs by proposing their mixed-integer Nonlinear programming (MINLP) formulation. We conduct extensive computational experiments for different KAN architectures. Additionally, we propose alternative convex hull reformulation, local support and redundant constraints for the formulation aimed at improving the effectiveness of the MINLP formulation of the KAN. KANs demonstrate high accuracy while requiring relatively modest computational effort to optimize them, particularly for cases with less than five inputs or outputs. For cases with higher inputs or outputs, carefully considering the KAN architecture during training may improve its effectiveness while optimizing over a trained KAN. Overall, we observe that KANs offer a promising alternative as surrogate models for deterministic global optimization. Copyright © 2025, The Authors. All rights reserved.

关键词： mixed-integer linear programming

Sharp Hybrid Zonotopes: Set Operations and the Reformulation-linearization Technique

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arXiv 2025年

作者： Glunt, Jonah J. Robbins, Joshua A. Silvestre, Daniel Pangborn, Herschel C. Department of Mechanical Engineering The Pennsylvania State University University Park PA16802 United States School of Science and Technology NOVA University of Lisbon Lisbon1049-001 Portugal COPELABS Lusófona University Lisbon1749-024 Portugal Institute for Systems and Robotics Instituto Superior Técnico University of Lisbon Lisbon1649-004 Portugal

mixed integer set representations, and specifically hybrid zonotopes, have enabled new techniques for reachability and verification of nonlinear and hybrid systems. mixed-integer sets which have the property that their convex relaxation is equal to their convex hull are said to be sharp. This property allows the convex hull to be computed with minimal overhead, and is known to be important for improving the convergence rates of mixed-integer optimization algorithms that rely on convex relaxations. This paper examines methods for formulating sharp hybrid zonotopes and provides sharpness-preserving methods for performing several key set operations. The paper then shows how the reformulation-linearization technique can be applied to create a sharp realization of a hybrid zonotope that is initially not sharp. A numerical example applies this technique to find the convex hull of a level set of a feedforward ReLU neural network. Copyright © 2025, The Authors. All rights reserved.

关键词： mixed-integer linear programming

Scenario Reduction for Distributionally Robust Optimization

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arXiv 2025年

作者： Aigner, Kevin-Martin Denzler, Sebastian Liers, Frauke Pokutta, Sebastian Sharma, Kartikey Friedrich-Alexander-Universität Erlangen-Nürnberg Germany Zuse Institute Berlin Germany

Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the proposed approximation significantly reduces solution time while maintaining high solution quality with only minor *** Codes 49 © 2025, CC BY.

关键词： mixed-integer linear programming

Distribution Network Reconfiguration under Modeling of AC Optimal Power Flow Equations: A mixed-integer programming Approach

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Distribution Network Reconfiguration under Modeling of AC Op...

IEEE PES 2nd Latin American Conference on Innovative Smart Grid Technologies (ISGT Latin America)

作者： Ferreira, R. S. Borges, C. L. T. Pereira, M. V. F. PSR Rio De Janeiro RJ Brazil

ISBN: (纸本)9781467352741;9781467352727

One of the most common applications of the AC optimal power flow (ACOPF) in distribution systems is the network reconfiguration problem, which consists of altering the topology of the network in order to optimize a given objective function - usually, minimizing ohmic losses. We propose a mixed-integer linear programming reformulation of the network reconfiguration problem for distribution systems, under full modeling of the ACOPF equations. The proposed formulation captures the non-linear behavior of the electrical network via approximations of arbitrary accuracy, allows the representation of discrete decisions via integer decision variables, and can be solved to global optimality with commercially available optimization solvers. The applicability of the proposed formulation is indicated with help of case studies.

关键词： Distribution systems network reconfiguration AC optimal power flow mixed-integer linear programming

A Mathematical Model for a Realistic Course Scheduling at Universities 4th

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A Mathematical Model for a Realistic Course Scheduling at...

4th International Conference on Advanced Research in Technologies, Information, Innovation and Sustainability 2024, ARTIIS 2024

作者： Vidal, Nicolas Hernández, Felipe Morillo-Torres, Daniel Gatica, Gustavo Department of Civil and Industrial Engineering Faculty of Engineering Pontificia Universidad Javeriana Cali Cali Colombia Faculty of Engineering Universidad Andres Bello Santiago Chile

ISBN: (纸本)9783031832062

Timetabling is a combinatorial and challenging problem in different fields where it is required to allocate scarce resources. One of the main variants is the University Course Timetabling Problem (UCTP), where courses must take a location and the academic sessions must be distributed avoiding overlapping schedules or preference conflicts. Despite the high complexity of its solution, it is increasingly necessary to include more realistic characteristics and formulate efficient models. In this paper, new realistic features focused on the development of student’s curriculum are included and a mixed-integer linear programming model is proposed to address this problem. The model’s objective is to maximize all teacher’s preferences and reduce the overlaps between courses that can be viewed either in advance or delayed. The proposed model was validated in the timetabling in the Industrial Engineering Program at the Pontificia Universidad Javeriana Cali, Colombia. The results show an increase of 47.7% regarding the preferences of the professors, as well as a reduction of 99.37% of the time associated with the semester scheduling process and currently, the perception of students is being assessed as they progress through the semesters, with preliminary results indicating an increase in satisfaction. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

关键词： mixed-integer linear programming