Solving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programmes

Burlacu, Robert; Geißler, Björn; Schewe, Lars

doi:10.1080/10556788.2018.1556661

Cited by 24 publications

(50 citation statements)

References 25 publications

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“…Generalized Benders Decomposition (GBD) [13,15] solves a convex MINLP by iteratively solving NLP and MIP sub-problems. The adaptive MIP OAmethod is based on the refinement of MIP relaxations by projecting infeasible points onto a feasible set, see [5,8].…”

Section: Convex Minlp-methodsmentioning

confidence: 99%

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The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming

2020

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This paper presents a new two-phase method for solving convex mixed-integer nonlinear programming (MINLP) problems, called Decomposition-based Outer Approximation Algorithm (DECOA). In the first phase, a sequence of linear integer relaxed sub-problems (LP phase) is solved in order to rapidly generate a good linear relaxation of the original MINLP problem. In the second phase, the algorithm solves a sequence of mixed integer linear programming sub-problems (MIP phase). In both phases the outer approximation is improved iteratively by adding new supporting hyperplanes by solving many easier sub-problems in parallel. DECOA is implemented as a part of Decogo (Decomposition-based Global Optimizer), a parallel decomposition-based MINLP solver implemented in Python and Pyomo. Preliminary numerical results based on 70 convex MINLP instances up to 2700 variables show that due to the generated cuts in the LP phase, on average only 2-3 MIP problems have to be solved in the MIP phase.Many optimization problems arising in engineering and science contain both combinatorial and nonlinear relations. Such optimization problems are modeled by mixed-integer nonlinear programming (MINLP), which combines capabilities of mixed-integer linear programming (MILP) and nonlinear programming (NLP). The ability to accurately model real-world problems has made MINLP an active research area with a large number of industrial applications. B Pavlo MutsDecomposition is a general method that can be applied to convex optimization, as well as nonconvex optimization. The idea of decomposition is based on dividing an original problem into smaller and easier sub-problems. Mainly the approach of these methods consists in solving small sub-problems and then feeding the result to a global master problem. In this case, the sub-problems can be solved simultaneously, which makes decomposition methods very attractive in terms of computational demand. Decomposition can be applied along a number of dimensions, like time-windows, resources or system components. Most decomposition methods are based on solving a Lagrangian relaxation of the decomposed problem [11,14,21], e.g. Column Generation (CG) [24]. Rapid Branching is an efficient CG-based heuristic for solving large-scale transport planning problems [3,28]. The new solution approachThis paper describes a decomposition-based successive outer approximation algorithm (DECOA) for convex MINLP problems. Like the OA method, the ESH algorithm, the ECP algorithm, and the adaptive MIP algorithm, DECOA constructs MIP outer approximations

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Section: Convex Minlp-methodsmentioning

confidence: 99%

“…because of data or model errors. Since most MIP solvers, like SCIP, are able to detect the infeasibility of a model, a feasibility flag can be returned after solving (5), which can be used to stop DECOA, if the MINLP model (1) is infeasible.…”

Section: Oa Master Problemmentioning

confidence: 99%

The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming

2020

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“…Martin and Möller [115], Martin et al [116], and Möller [120] investigate this approach and Geißler et al [61][62][63][64] and Morsi [122] show how to maintain the relaxation property. After checking for feasibility again, possible refinement steps are investigated by Burlacu et al [24] and Geißler [60]. The linearization methods itself are presented in Dantzig [41] and Markowitz and Manne [114].…”

Section: Convex Mixed-integer Nonlinear Programming Convexmentioning

confidence: 99%

“…Furthermore, (penalty) alternating direction methods are used to combine mixed-integer and continuous solution strategies in a hybrid approach in Geißler et al [65][66][67]. By far the most common approaches in the literature are (piecewise) linearization techniques for obtaining MIP models that can be solved with state-of-the-art MIP solvers to global optimality; see, e.g., Burlacu et al [24], Geißler [60], Geißler et al [62][63][64], Martin and Möller [115], Martin et al [116], Möller [120], and Morsi [122]. Besides that, we also use these (piecewise) linearization techniques to couple the gas network with an electricity system, where the latter is a unit commitment problem: The content of the article is not part of this thesis.…”

Section: Literature Survey: Solving Gas Transport Optimization Problemsmentioning

confidence: 99%

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Sirvent

2020

Operations Research Proceedings

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List of Figures List of Tables 1.3 Scientific Contribution and Results challenge is to find formulations and corresponding algorithms that are able to tackle transient gas transport optimization problems. Note that the first challenge seeks for general algorithms that are then used for stationary gas transport optimization, whereas the second challenge is tailored to the application of transient gas transport optimization. Our roadmap to cope with the challenges is inspired by two guidelines. We want to decompose our problems and we want to use mixed-integer linear programs. The simple reason for the first point is that decomposing problems into smaller ones has a successful history and is common sense for solving big problems. Second, mixed-integer linear programs underwent an enormous amount of research in the last decades. Consequently, fast, stable, robust, and reliable solvers as Gurobi, Cplex, or Scip are available and we use these solvers, in our case Gurobi, as a workhorse; see Gurobi [80], Cplex [36], and Maher et al. [110], respectively. 1.3 Scientific Contribution and Results For the first challenge, we consider the case where the constraint functions in (1.1c) are not given analytically. We build up a new hierarchy of assumptions that restricts the knowledge about the nonlinear functions. Depending on the assumptions, we develop three new global decomposition algorithms that decompose the difficult constraints (1.1c), i.e., we rely on our first guideline. Specifically, we build upon relaxation strategies that are motivated by developments in the field of (mixed-integer) (non)linear programming and Lipschitz optimization. As a result, mixed-integer linear programs according to our second guideline are obtained. We prove correct and finite terminations and clarify the limitations of our approaches. Note that the presented framework is a general approach for problems in the form of (1.1). In particular, we consider stationary gas transport optimization, i.e., Problem (1.1) with the Euler equations in the form of ODEs in (1.1c). Moreover, we show promising numerical results for the real-world gas network of Greece. Additionally, we consider the second challenge of transient gas transport optimization, i.e., Problem (1.1) with PDEs (1.1c). We use an instantaneous control algorithm and adapt it to the Euler equations for the first time. Specifically, we decompose the problem into single time steps and present new discretization techniques that ensure the mixed-integer linear program property. In other words, we build upon our guidelines again. Finally, we

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“…Geißler, Morsi, Schewe, and Schmidt (2015) use piecewise linear approximation to relax the mixed-integer nonlinear model with a given tolerance error. Burlacu, Geißler, and Schewe (2019) base their work on Geißler et al (2012) and solve an MINLP by solving a sequence of MIP relaxations with gradually increasing accuracy. They use the German network with more than 500 nodes to show the advantages of their work compared with the existing MINLP solvers.…”

Section: Related Literaturementioning

confidence: 99%

Optimizing natural gas pipeline transmission with nonuniform elevation: A new initialization approach

Xue

Deng

Shen

2019

Naval Research Logistics

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In this paper, we develop an iterative piecewise linear approximation approach with a novel initialization method to solve natural gas pipeline transmission problems with the nonuniform network elevation. Previous approaches, such as energy minimization methods, cannot be applied to solve problems with the nonuniform network elevation because they exclude pressure range constraints, and thus provide solutions far from optimum. We propose a new initialization model that considers pressure range constraints and improves the optimality of the solutions and the computational efficiency. Furthermore, we extend the energy minimization methods and provide the necessary conditions under which the extended methods operate in networks with the nonuniform elevation. We test the performances of the methods with previously reported pipeline networks from the literature, with the open data set GasLib, and with our industrial collaborator. The initialization approach is shown to be more efficient than the method with fixed initial breakpoints. The newly proposed initialization approach generates solutions with a higher accuracy than the extended energy minimization methods, especially in large‐size networks. The proposed method has been applied to natural gas transmission planning by the China National Petroleum Corporation and has brought a direct profit increase of 330 million U.S. dollars in 2015‐2017.

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Solving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programmes

Cited by 24 publications

References 25 publications

The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming

The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Optimizing natural gas pipeline transmission with nonuniform elevation: A new initialization approach

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