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Transformation and Linearization Techniques in Optimization: A State-of-the-Art Survey

Author

Listed:
  • Mohammad Asghari

    (Department of Industrial Engineering, Dalhousie University, 5269 Morris Street, Halifax, NS B3H 4R2, Canada)

  • Amir M. Fathollahi-Fard

    (Department of Electrical Engineering, École de Technologie Supérieure, University of Québec, Montréal, QC H3C 1K3, Canada)

  • S. M. J. Mirzapour Al-e-hashem

    (Department of Industrial Engineering and Management Systems, Amirkabir University of Technology (Tehran Polytechnic), Tehran 15875-4413, Iran)

  • Maxim A. Dulebenets

    (Department of Civil & Environmental Engineering, College of Engineering, Florida A&M University-Florida State University (FAMU-FSU), 2525 Pottsdamer Street, Building A, Suite A124, Tallahassee, FL 32310-6046, USA)

Abstract

To formulate a real-world optimization problem, it is sometimes necessary to adopt a set of non-linear terms in the mathematical formulation to capture specific operational characteristics of that decision problem. However, the use of non-linear terms generally increases computational complexity of the optimization model and the computational time required to solve it. This motivates the scientific community to develop efficient transformation and linearization approaches for the optimization models that have non-linear terms. Such transformations and linearizations are expected to decrease the computational complexity of the original non-linear optimization models and, ultimately, facilitate decision making. This study provides a detailed state-of-the-art review focusing on the existing transformation and linearization techniques that have been used for solving optimization models with non-linear terms within the objective functions and/or constraint sets. The existing transformation approaches are analyzed for a wide range of scenarios (multiplication of binary variables, multiplication of binary and continuous variables, multiplication of continuous variables, maximum/minimum operators, absolute value function, floor and ceiling functions, square root function, and multiple breakpoint function). Furthermore, a detailed review of piecewise approximating functions and log-linearization via Taylor series approximation is presented. Along with a review of the existing methods, this study proposes a new technique for linearizing the square root terms by means of transformation. The outcomes of this research are anticipated to reveal some important insights to researchers and practitioners, who are closely working with non-linear optimization models, and assist with effective decision making.

Suggested Citation

  • Mohammad Asghari & Amir M. Fathollahi-Fard & S. M. J. Mirzapour Al-e-hashem & Maxim A. Dulebenets, 2022. "Transformation and Linearization Techniques in Optimization: A State-of-the-Art Survey," Mathematics, MDPI, vol. 10(2), pages 1-26, January.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:2:p:283-:d:726907
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    References listed on IDEAS

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    1. Camille Negrello & Pierre Gosselet & Christian Rey, 2021. "Nonlinearly Preconditioned FETI Solver for Substructured Formulations of Nonlinear Problems," Mathematics, MDPI, vol. 9(24), pages 1-25, December.
    2. Konstantinos Petridis & Georgios Drogalas & Eleni Zografidou, 2021. "Internal auditor selection using a TOPSIS/non-linear programming model," Annals of Operations Research, Springer, vol. 296(1), pages 513-539, January.
    3. Monique Guignard, 2020. "Strong RLT1 bounds from decomposable Lagrangean relaxation for some quadratic 0–1 optimization problems with linear constraints," Annals of Operations Research, Springer, vol. 286(1), pages 173-200, March.
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