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How to convexify the intersection of a second order cone and a nonconvex quadratic

MATHEMATICAL programming 2017年第1-2期162卷 393-429页

作者： Burer, Samuel Kilinc-Karzan, Fatma Univ Iowa Dept Management Sci Iowa City IA 52242 USA Carnegie Mellon Univ Tepper Sch Business Pittsburgh PA 15213 USA

A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown-by several authors using different techniques-that the convex hull of the intersection of an ellipsoid, , and a split disjunction, with , equals the intersection of with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form and , where is a SOCr cone, is a nonconvex cone defined by a single homogeneous quadratic, and H is an affine hyperplane. Under several easy-to-verify conditions, we derive simple, computable convex relaxations and , where is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.

关键词： Convex hull disjunctive programming Mixed-integer linear programming Mixed-integer nonlinear programming Mixed-integer quadratic programming Nonconvex quadratic programming Second-order-cone programming Trust-region subproblem

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Model reformulations for Work and Heat Exchange Network (WHEN) synthesis problems

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COMPUTERS & CHEMICAL ENGINEERING 2019年 125卷 89-97页

作者： Yu, Haoshui Vikse, Matias Anantharaman, Rahul Gundersen, Truls Norwegian Univ Sci & Technol Dept Energy & Proc Engn Kolbjoern Hejes V 1A NO-7491 Trondheim Norway SINTEF Energy Res Kolbjoern Hejes V 1A NO-7491 Trondheim Norway

The Duran-Grossmann model can deal with heat integration problems with variable process streams. Work and Heat Exchange Networks (WHENs) represent an extension of Heat Exchange Networks. In WHEN problems, the identities of streams (hot/cold) are regarded as variables. The original Duran-Grossmann model has been extended and applied to WHENs without knowing the identity of streams a priori. In the original Duran-Grossmann model, the max operator is a challenge for solving the model. This paper analyzes four ways to reformulate the Duran-Grossmann model. Smooth Approximation, Explicit Disjunctions, Direct Disjunctions and Intermediate Temperature strategy are reviewed and compared. The Extended Duran-Grossmann model for WHEN problems consists of both binary variables and non-smooth functions. The Extended Duran-Grossmann model can be reformulated in similar ways. In this study, the performance of different reformulations of the Extended Duran-Grossmann model for WHEN problems are compared based on a small case study in this paper. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Work and heat exchange networks Duran-Grossmann model Reformulations disjunctive programming MINLP

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Design of Ionic Liquid as Carbon Capture Solvent for a Bioenergy System: Integration of Bioenergy and Carbon Capture Systems

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ACS SUSTAINABLE CHEMISTRY & ENGINEERING 2017年第6期5卷 5241-5252页

作者： Chong, Fah Keen Andiappan, Viknesh Ng, Denny K. S. Foo, Dominic C. Y. Eljack, Fadwa T. Atilhan, Mert Chemmangattuvalappil, Nishanth G. Univ Nottingham Ctr Excellence Green Technol Dept Chem & Environm Engn Malaysia CampusBroga Rd Semenyih 43500 Selangor Malaysia Taylors Univ Sch Engn Energy & Environm Res Grp Lakeside Campus1 Jalan Taylors Subang Jaya 47500 Selangor Malaysia Qatar Univ Coll Engn Dept Chem Engn POB 2713 Doha Qatar Texas A&M Univ Dept Chem Engn Doha Qatar

Current atmospheric carbon dioxide (CO2) concentration has exceeded the safe limit of 350 ppm. One potential technology to remove CO2 from the atmosphere is the integrated bioenergy production and carbon capture system. A bioenergy production system produces multiple energy products from biomass, resulting in zero net increment of CO2 amount in the atmosphere. Meanwhile, CO2 produced from bioenergy production is separated for other purposes through carbon capture. To ensure the entire system is environmental friendly, an efficient and green carbon capture solvent should be utilized. Ionic liquids (ILs) are the potential solvents for this purpose, as they have negligible vapor pressure and high thermal stability. However, there are up to a million possible combinations of cations and anions that may make up ILs. This work presents a systematic approach to identify an optimal IL solvent to separate CO2 produced from a bioenergy system at the optimal conditions of carbon capture process. Following that, the bioenergy system is retrofitted to provide sufficient utilities to a carbon capture system to make sure that the entire process is self-sustainable. A case study involving an existing palm-based bioenergy system, integrated with carbon capture to produce CO2, is used to demonstrate the presented approach.

关键词： Bioenergy production system Computer-aided molecular design Decision-making tool disjunctive programming Input-output modeling

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Relaxations and cutting planes for linear programs with complementarity constraints

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JOURNAL OF GLOBAL OPTIMIZATION 2024年第1期90卷 27-51页

作者： Del Pia, Alberto Linderoth, Jeff Zhu, Haoran Univ Wisconsin Madison Dept Ind & Syst Engn Madison WI 53706 USA Univ Wisconsin Madison Wisconsin Inst Discovery Madison WI 53715 USA

We study relaxations for linear programs with complementarity constraints, especially instances whose complementary pairs of variables are not independent. Our formulation is based on identifying vertex covers of the conflict graph of the instance and contains the extended formulation obtained from the ERLT introduced by Nguyen, Richard, and Tawarmalani as a special case. We demonstrate how to obtain strong cutting planes for our formulation from both the stable set polytope and the boolean quadric polytope associated with a complete bipartite graph. Through an extensive computational study for three types of practical problems, we assess the performance of our proposed linear relaxation and new cutting-planes in terms of the optimality gap closed.

关键词： Complementarity constraints Cutting-planes Convex hull disjunctive programming Boolean quadric polytope

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Active and Reactive Power Coordinated Two-Stage MG Scheduling for Resilient Distribution Systems Under Uncertainties

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IEEE TRANSACTIONS ON SMART GRID 2022年第4期13卷 2986-2998页

作者： Cai, Sheng Xie, Yunyun Wu, Qiuwei Zhang, Menglin Jin, Xiaolong Xiang, Zhengrong Nanjing Univ Sci & Technol Sch Automat Nanjing 210094 Peoples R China Tsinghua Univ Tsinghua Shenzhen Int Grad Sch Tsinghua Berkeley Shenzhen Inst Shenzhen 518055 Peoples R China China Univ Geosci Sch Automat Wuhan 430074 Peoples R China Tianjin Univ Key Lab Smart Grid Minist Educ Tianjin 30072 Peoples R China

This paper proposes a two-stage microgrid (MG) scheduling approach that considers the dynamic MG formation and the coordinated optimization of active and reactive powers to enhance the distribution system resilience after blackouts. It aims to dispatch multiple devices operating in different timescales to sectionalize the distribution system into self-supplied MGs and supply the critical loads continuously with reliable power. In the hour-ahead operation stage, switches, capacitor banks and energy storage charging/discharging decisions are optimized based on hour-ahead interval predictions of renewable energy generations and load demands. In the intra-hour operation stage, the output of distributed energy resources and static Var compensators are redispatched to compensate the hour-ahead operation decisions after the uncertainty realization. Compared with the existing methods, the proposed MG scheduling method can restore more critical loads and sustain secure operation under random renewable energy generation and load demand. In addition, to address uncertainties, the MG scheduling problem is modeled as a robust optimization model, which is decomposed into a master problem and a subproblem by adopting the column-and-constraint generation algorithm. Furthermore, the subproblem is reformulated via the disjunctive programming method to reduce the computational complexity. The effectiveness of the proposed method is verified on the modified IEEE test systems.

关键词： Uncertainty Reactive power Renewable energy sources Load modeling Resilience Computational modeling Static VAr compensators Microgrid scheduling resilience enhancement two-stage optimization uncertainties disjunctive programming

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ON THE CONVEX-HULL OF THE UNION OF CERTAIN POLYHEDRA

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OPERATIONS RESEARCH LETTERS 1988年第6期7卷 279-283页

作者： BALAS, E Management Science Research Group Graduate School of Industrial Administration Carnegie Mellon University Pittsburgh PA 15213 USA

We consider a finite collection of polyhedra whose defining linear systems differ only in their right hand sides. Jeroslow [5] and Blair [4] specified conditions under which the convex hull of the union of these polyhedra is defined by a system whose left hand side is the common lefthand side of the individual systems, and whose right hand side is a convex combination of the individual right hand sides. We give a new sufficient condition for this property to hold, which is often easier to recognize. In particular, we show that the condition is satisfied for polyhedra whose defining systems involve the node-arc incidence matrices of directed graphs, with certain righ hand sides. We also derive as a special case the compact linear characterization of the two terminal Steiner tree polytope given in Ball, Liu and Pulleyblank [3].

关键词： convex hull union of polyhedra disjunctive programming mixed integer representation

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A complete characterization of disjunctive conic cuts for mixed integer second order cone optimization

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DISCRETE OPTIMIZATION 2017年 24卷 3-31页

作者： Belotti, Pietro Goez, Julio C. Polik, Imre Ralphs, Ted K. Terlaky, Minas FICO Xpress Optimizer Team Birmingham W Midlands England NHH Norwegian Sch Econ Dept Business & Management Sci N-5045 Bergen Norway SAS Inst Cary NC 27513 USA Lehigh Univ Dept Ind & Syst Engn Bethlehem PA 18015 USA

We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization (MISOCO) problem. We extend our prior work on disjunctive conic cuts (DCCs), which has thus far been restricted to the case in which the intersection of the hyperplanes and the feasible set is bounded. Using a similar technique, we show that one can extend our previous results to the case in which that intersection is unbounded. We provide a complete characterization in closed form of the conic inequalities required to describe the convex hull when the hyperplanes defining the disjunction are parallel. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Second order cone optimization Mixed integer optimization disjunctive programming disjunctive conic cuts

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Logic-based outer approximation for globally optimal synthesis of process networks

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COMPUTERS & CHEMICAL ENGINEERING 2005年第9期29卷 1914-1933页

作者： Bergamini, ML Aguirre, P Grossmann, I Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA Inst Desarrollo & Diseno INGAR Santa Fe Argentina

Process network problems can be formulated as generalized disjunctive programs where a logic-based representation is used to deal with the discrete and continuous decisions. A new deterministic algorithm for the global optimization of process networks is presented in this work. The proposed algorithm, which does not rely on spatial branch-and-bound, is based on the logic-based outer approximation that exploits the special structure of flowsheet synthesis models. The method is capable of considering non-convexities, while guaranteeing globality in the solution of an optimal synthesis of process network problem. This is accomplished by solving iteratively reduced NLP subproblems to global optimality and MILP master problems, which are valid outer approximations of the original problem. Piecewise linear under and overestimators for bilinear and concave terms have been constructed with the property of having zero gap in a finite set of points. The global optimization of the reduced NLP may be performed either with a suitable global solver or using the inner optimization strategy that is proposed in this work. Theoretical properties are discussed as well as several alternatives for implementing the proposed algorithm. Several examples were successfully solved with this algorithm. Results show that only few iterations are required to solve them to global optimality. (c) 2005 Elsevier Ltd. All rights reserved.

关键词： global optimization disjunctive programming process synthesis

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A comparison of solution approaches for the numerical treatment of or-constrained optimization problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2020年第1期76卷 233-275页

作者： Mehlitz, Patrick Brandenburg Tech Univ Cottbus Senftenberg Inst Math D-03046 Cottbus Germany

Mathematical programs with or-constraints form a new class of disjunctive optimization problems with inherent practical relevance. In this paper, we provide a comparison of three different solution methods for the numerical treatment of this problem class which are inspired by classical approaches from disjunctive programming. First, we study the replacement of the or-constraints as nonlinear inequality constraints using suitable NCP-functions. Second, we transfer the or-constrained program into a mathematical program with switching or complementarity constraints which can be treated with the aid of well-known relaxation methods. Third, a direct Scholtes-type relaxation of the or-constraints is investigated. A numerical comparison of all these approaches which is based on three essentially different model programs from or-constrained optimization closes the paper.

关键词： disjunctive programming Global convergence NCP-functions Or-constrained programming Relaxation methods

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The C³ theorem and a D² algorithm for large scale stochastic mixed-integer programming:: Set convexification

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MATHEMATICAL programming 2005年第1期104卷 1-20页

作者： Sen, S Higle, JL Univ Arizona Dept Syst & Ind Engn Tucson AZ 85721 USA

This paper considers the two-stage stochastic integer programming problem, with an emphasis on instances in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C-3 (Common Cut Coefficients) Theorem. Based on the C-3 Theorem, we develop a decomposition algorithm which we refer to as disjunctive Decomposition (D-2). In this new class of algorithms, we work with master and subproblems that result from convexifications of two coupled disjunctive programs. We show that when the second stage consists of 0-1 MILP problems, we can obtain accurate second stage objective function estimates after finitely many steps. This result implies the convergence of the D2 algorithm.

关键词： stochastic mixed-integer programming disjunctive programming cutting plane algorithms

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