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Review of mixed-integer nonlinear and Generalized Disjunctive programming Methods

CHEMIE INGENIEUR TECHNIK 2014年第7期86卷 991-1012页

作者： Trespalacios, Francisco Grossmann, Ignacio E. Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

This work presents a review of the main deterministic mixed-integer nonlinear programming (MINLP) solution methods for problems with convex and nonconvex functions. An overview for deriving MINLP formulations through generalized disjunctive programming (GDP), which is an alternative higher-level representation of MINLP problems, is also presented. A review of solution methods for GDP problems is provided. Some relevant applications of MINLP and GDP in process systems engineering are described in this work.

关键词： Generalized disjunctive programming mixed-integer nonlinear programming nonlinear programming Optimization

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Convex mixed-integer nonlinear programs derived from generalized disjunctive programming using cones

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2024年第1期88卷 251-312页

作者： Neira, David E. Bernal Grossmann, Ignacio E. Purdue Univ Davidson Sch Chem Engn 480 Stadium Mall Dr W Lafayette IN 47907 USA Univ Space Res Assoc Res Inst Adv Comp Sci 425 3rd St SWSuite 950 Washington DC 20024 USA NASA Ames Res Ctr Quantum Artificial Intelligence Lab 5000 Forbes Ave Moffett Field CA 94035 USA Carnegie Mellon Univ Chem Engn Dept 5000 Forbes Ave Pittsburgh PA 15213 USA

We propose the formulation of convex Generalized Disjunctive programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into mixed-integer Conic programming (MICP) problems through both the big-M and hull reformulations. These reformulations have the advantage that they are representable using the same cones as the original conic GDP. In the case of the hull reformulation, they require no approximation of the perspective function. Moreover, the MICP problems derived can be solved by specialized conic solvers and offer a natural extended formulation amenable to both conic and gradient-based solvers. We present the closed form of several convex functions and their respective perspectives in conic sets, allowing users to formulate their conic GDP problems easily. We finally implement a large set of conic GDP examples and solve them via the scalar nonlinear and conic mixed-integer reformulations. These examples include applications from Process Systems Engineering, Machine learning, and randomly generated instances. Our results show that the conic structure can be exploited to solve these challenging MICP problems more efficiently. Our main contribution is providing the reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems.

关键词： mixed-integer nonlinear programming Conic programming Generalized disjunctive programming Perspective function

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mixed integer programming with a class of nonlinear convex constraints

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DISCRETE OPTIMIZATION 2017年 24卷 66-86页

作者： Vinel, Alexander Krokhmal, Pavlo A. Auburn Univ Dept Ind & Syst Engn Shelby Ctr 3301 Auburn AL 36849 USA Univ Arizona Dept Syst & Ind Engn 1127 E James E Rogers Way Tucson AZ 85721 USA

We study solution approaches to a class of mixed-integer nonlinear programming problems that arise from recent developments in risk-averse stochastic optimization and contain second- and p-order cone programming as special cases. We explore possible applications of some of the solution techniques that have been successfully used in mixed-integer conic programming and show how they can be generalized to the problems under consideration. Particularly, we consider a branch-and-bound method based on outer polyhedral approximations, lifted nonlinear cuts, and linear disjunctive cuts. Results of numerical experiments with discrete portfolio optimization models are presented. (C) 2016 Elsevier B.V. All rights reserved.

关键词： mixed-integer nonlinear programming Measures of risk Quasi-arithmetic average Valid inequalities Conic programming

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Lift-and-project cuts for convex mixed integer nonlinear programs Linear programming based separation and extended formulations

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MATHEMATICAL programming COMPUTATION 2017年第4期9卷 499-526页

作者： Kilinc, Mustafa R. Linderoth, Jeff Luedtke, James Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA Univ Wisconsin Wisconsin Inst Discovery Dept Ind & Syst Engn Madison WI USA

We describe a computationally effective method for generating lift-andproject cuts for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs and in the limit generates an inequality as strong as the lift-and-project cut that can be obtained from solving a cut-generating nonlinear program. Using this procedure, we are able to approximately optimize over the rank one lift-and-project closure for a variety of convex MINLP instances. The results indicate that lift-and-project cuts have the potential to close a significant portion of the integrality gap for convex MINLPs. In addition, we find that using this procedure within a branch-and-cut solver for convex MINLPs significantly reduces the total solution time for many instances. We also demonstrate that combining lift-and-project cuts with an extended formulation that exploits separability of convex functions yields significant improvements in both relaxation bounds and the time to calculate the relaxation. Overall, these results suggest that with an effective separation routine, like the one proposed here, lift-and-project cuts may be as effective for solving convex MINLPs as they have been for solving mixed-integer linear programs.

关键词： Lift-and-project cuts Extended formulations mixed-integer nonlinear programming

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Design and optimization of steam power systems in industrial parks based on the distributed steam turbine system

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Chinese Journal of Chemical Engineering 2025年第1期77卷 259-272页

作者： Lingwei Zhang Ziyuan Cui Yufei Wang School of Chemical Engineering and Environment China University of Petroleum(Beijing)Beijing 102249China

Steam power systems(SPSs)in industrial parks are the typical utility systems for heat and electricity *** SPSs,electricity is generated by steam turbines,and steam is generally produced and supplied at multiple levels to serve the heat demands of consumers with different temperature grades,so that energy is utilized in *** a large number of steam levels enhances energy utilization efficiency,it also tends to cause a complex steam pipeline network in the industrial *** practice,a moderate number of steam levels is always adopted in SPSs,leading to temperature mismatches between heat supply and demand for some *** study proposes a distributed steam turbine system(DSTS)consisting of main steam turbines on the energy supply side and auxiliary steam turbines on the energy consumption side,aiming to balance the heat production costs,the distance-related costs,and the electricity generation of SPSs in industrial parks.A mixed-integer nonlinear programming model is established for the optimization of SPSs,with the objective of minimizing the total annual cost(TAC).The optimal number of steam levels and the optimal configuration of DSTS for an industrial park can be determined by solving the model.A case study demonstrates that the TAC of the SPS is reduced by 220.6×10^(3)USD(2.21%)through the arrangement of auxiliary steam *** sub-optimal number of steam levels and a non-optimal operating condition slightly increase the TAC by 0.46%and 0.28%,*** sensitivity analysis indicates that the optimal number of steam levels tends to decrease from 3 to 2 as electricity price declines.

关键词： Industrial parks Steam power systems Distributed steam turbine system mixed-integer nonlinear programming Optimization Enthalpy

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Granularity for mixed-integer Polynomial Optimization Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2025年第2期205卷 1-24页

作者： Eggen, Carl Stein, Oliver Volkwein, Stefan Univ Konstanz Dept Math & Stat Universitatsstr 10 D-78464 Constance Germany Karlsruhe Inst Technol KIT Inst Operat Res Blucherstr 17 D-76185 Karlsruhe Germany

Finding good feasible points is crucial in mixed-integer programming. For this purpose we combine a sufficient condition for consistency, called granularity, with the moment-/sum-of-squares-hierarchy from polynomial optimization. If the mixed-integer problem is granular, we obtain feasible points by solving continuous polynomial problems and rounding their optimal points. The moment-/sum-of-squares-hierarchy is hereby used to solve those continuous polynomial problems, which generalizes known methods from the literature. Numerical examples from the MINLPLib illustrate our approach.

关键词： mixed-integer nonlinear programming Granularity Rounding Polynomial optimization Semidefinite programming

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Drone-Delivery Network for Opioid Overdose: nonlinear integer Queueing-Optimization Models and Methods

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OPERATIONS RESEARCH 2025年第1期73卷 iii-vii, 1-582, C2-C3页

作者： Lejeune, Miguel A. Ma, Wenbo George Washington Univ Dept Decis Sci Washington DC 20052 USA

We propose a new stochastic emergency network design model that uses a fleet of drones to quickly deliver naloxone in response to opioid overdoses. The network is represented as a collection of M/G/K / G / K queueing systems in which the capacity K of each system is a decision variable, and the service time is modeled as a decision -dependent random variable. The model is a queuing -based optimization problem which locates fixed (drone bases) and mobile (drones) servers and determines the drone dispatching decisions and takes the form of a nonlinear integer problem intractable in its original form. We develop an efficient reformulation and algorithmic framework. Our approach reformulates the multiple nonlinearities (fractional, polynomial, exponential, factorial terms) to give a mixed -integer linear programming (MILP) formulation. We demonstrate its generalizability and show that the problem of minimizing the average response time of a collection of M/G/K / G / K queueing systems with unknown capacity K is always MILP-representable. We design an outer approximation branch -and -cut algorithmic framework that is computationally efficient and scales well. The analysis based on real -life data reveals that drones can in Virginia Beach: (1) decrease the response time by 82%, (2) increase the survival chance by more than 273%, (3) save up to 33 additional lives per year, and (4) provide annually up to 279 additional quality -adjusted life years.

关键词： opioid overdose drone delivery optimization-based queueing model survival chance mixed-integer nonlinear programming stochastic network design

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Outer Approximation Method for Discrete AC Optimal Power Flow

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IEEE TRANSACTIONS ON POWER SYSTEMS 2025年第2期40卷 1943-1954页

作者： Jabr, Rabih A. Amer Univ Beirut Dept Elect & Comp Engn Beirut 11072020 Lebanon

This paper introduces an Outer Approximation (OA) method for solving discrete AC Optimal Power Flow (OPF) problems that account for switching decisions. The OPF problem is formulated via the extended conic quadratic format. It includes selecting generators for dispatch and controlling switched-type capacitor/inductor banks. The OA method is based on relaxing nonlinear equality constraints into conic and linear inequalities with penalty slack variables. The resulting mixed-integer Second-Order Cone programming (MISOCP) solution is iterated with a continuous variable nonlinear programming (NLP) solver that holds the discrete decisions constant at their most recent MISOCP values. The OA method can utilize polyhedral approximations and is scalable to test instances with more than 3000 nodes. Comparisons are reported with globally optimal discrete AC OPF results from the recent literature and a commercial solver for nonconvex mixed-integer nonlinear programming (MINLP) utilizing dynamic outer approximation and piecewise linear approximation with spatial branching. Comparative analysis shows that the proposed OA method provides global solutions for previously studied test instances while offering significant speed-up. It also applies to larger test networks where the other methods fail to calculate a feasible solution.

关键词： Switches Reactive power programming Load flow Renewable energy sources Mathematical models Generators Convex optimization iterative algorithms mixed-integer nonlinear programming optimization methods power system analysis computing power system management

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A mixed-integer nonlinear Model to Support the Operation of Distribution Systems With Hidden DERs

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IEEE TRANSACTIONS ON POWER DELIVERY 2025年第1期40卷 484-496页

作者： Vasconcelos, Pedro N. Trindade, Fernanda C. L. Venkatesh, Bala Freitas, Walmir de Souza, Antonio C. Zambroni Taranto, Glauco N. Univ Fed Itajuba BR-37500903 Itajuba Brazil Univ Estadual Campinas BR-13083970 Campinas Brazil Toronto Metropolitan Univ Toronto ON M5B 2K3 Canada Univ Fed Rio de Janeiro BR-21945970 Rio De Janeiro RJ Brazil

With impending deep electrification powered by innumerable Distributed Energy Resources (DERs), modeling each DER individually is becoming a critical challenge to the Distribution System Operators (DSOs). Even though the DSOs know the total DER installed capacity at the feeder level, the exact individual type, location, and size of such generators may remain unknown. This paper proposes a mixed-integer nonlinear programming formulation to support the operation of distribution systems under massive DER integration. The proposal accurately estimates distribution system power flows, relying on a limited set of measurements. It aims to establish equivalent DER models representing hidden resources and improve the representation of limited-visibility networks. The only DER information required is the total solar and wind installed capacity at the feeder level. The performance is assessed by comparing estimated and measured values of bus voltage magnitudes and branch power flows. Results demonstrate the efficacy of the proposed formulation in accurately replicating measurements, achieving an accuracy of over 90% when estimating active power flows in unmetered branches.

关键词： Voltage measurement Load flow Distributed power generation Power measurement Accuracy Reactive power Wind speed Aggregates State estimation Sparse matrices Deep electrification distributed generation hidden DERs mixed-integer nonlinear programming radial distribution systems

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Transitioning to electrification: Optimal strategies for ship fleet replacement

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TRANSPORTATION RESEARCH PART C-EMERGING TECHNOLOGIES 2025年 174卷

作者： Cui, Ziyu Fu, Xiaowen Jin, Huan Chen, Anthony Wang, Kun Hong Kong Polytech Univ Dept Ind & Syst Engn Kowloon Hong Kong Peoples R China Univ Nottingham Dept Comp Sci Ningbo Peoples R China Hong Kong Polytech Univ Dept Civil & Environm Engn Kowloon Hong Kong Peoples R China

To reduce carbon emissions, some shipping companies are strategically transitioning their fleets from diesel-powered ships to electric ones. However, the question of how to cost-effectively design this fleet conversion and the construction of charging stations for electric ships remains a critical issue. In this paper, we present a mixed-integer nonlinear programming (MINLP) model to minimize the total cost of designing the ship fleet replacement plan within a given planning horizon. At the planning level, the yearly strategic transition schedules for purchasing and salvaging ships are obtained, while the optimal strategy for charging station construction is also determined to satisfy annual cargo demand and charging demand. Then, our model integrates the operational level to determine optimal fleet deployment strategies and operational decisions. Meanwhile, the impact of real-world factors such as waterway depth and emission cost are discussed, respectively. Moreover, we introduce two subsidy schemes and compare their effectiveness on the fleet electrification process. Finally, the proposed model is applied to the Yangtze River shipping network in China using real data. The results show that the shipping company will initially invest in lower-cost electric ships with a limited budget, followed by the acquisition of larger ships. The charging stations are strategically established in the downstream areas, coinciding with the initial deployment of electric ships. Moreover, the impacts of fuel prices and emission cost of diesel-powered ships on the electrification process are compared, with the results indicating that the increase in emission cost has a more pronounced effect on the total cost compared to the rise in fuel prices. Additionally, the subsidy schemes' effect is explored, revealing that financial incentives markedly accelerate the fleet electrification process.

关键词： Electrification Electric ships Fleet replacement Charging stations mixed-integer nonlinear programming

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