A global optimization method for nonconvex separable programming problems

Li, Hanlin; Yu, Chian Son

doi:10.1016/s0377-2217(98)00243-4

Cited by 35 publications

(10 citation statements)

References 8 publications

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“…Li and Yu [33] proposed another global optimization method for nonlinear programming problems where the objective function and the constraints might be nonconvex. A univariate function is initially expressed by a piecewise…”

Section: Methods 1 Considermentioning

confidence: 99%

“…if − −1 ≥ 0, and otherwise ( ) is a non-convex function which needs to be linearized by adding extra binary variables. By linearizing the absolute terms, Li and Yu [33] converted the nonlinear function into a piecewise linear function as shown below.…”

Section: Methods 1 Considermentioning

confidence: 99%

“…According to the deterministic optimization methods for solving nonconvex nonlinear problems [29,33,38,39,48,49,[53][54][55][56], the inverse or logarithmic transformation is required to be approximated by the piecewise linearization function. For example, the function = ln or = −1 is required to be piecewisely linearized by using an appropriate breakpoint selection strategy, if a new break point is added at the midpoint of each interval of existing break points or at the point with largest approximation error, the number of line segments becomes double in each iteration.…”

Section: Error Evaluationmentioning

confidence: 99%

“…Various methods of piecewisely linearizing a nonlinear function have been proposed in the literature [26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Two well-known mixed-integer formulations for piecewise linear functions are the incremental cost [40] and the convex combination [41] formulations.…”

Section: Introductionmentioning

confidence: 99%

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A Review of Piecewise Linearization Methods

Lin

Carlsson

et al. 2013

Mathematical Problems in Engineering

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Various optimization problems in engineering and management are formulated as nonlinear programming problems. Because of the nonconvexity nature of this kind of problems, no efficient approach is available to derive the global optimum of the problems. How to locate a global optimal solution of a nonlinear programming problem is an important issue in optimization theory. In the last few decades, piecewise linearization methods have been widely applied to convert a nonlinear programming problem into a linear programming problem or a mixed-integer convex programming problem for obtaining an approximated global optimal solution. In the transformation process, extra binary variables, continuous variables, and constraints are introduced to reformulate the original problem. These extra variables and constraints mainly determine the solution efficiency of the converted problem. This study therefore provides a review of piecewise linearization methods and analyzes the computational efficiency of various piecewise linearization methods.

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Section: Methods 1 Considermentioning

confidence: 99%

Section: Methods 1 Considermentioning

confidence: 99%

Section: Error Evaluationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Review of Piecewise Linearization Methods

Lin

Carlsson

et al. 2013

Mathematical Problems in Engineering

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“…Li and Yu [51] proposed a linearization technique to convert the piecewise linear function to a linear form. In this way, firstly the piecewise linear function is converted to an equivalent mathematical programming model.…”

Section: Linearization Of the Non-linear Terms In The Considered Objementioning

confidence: 99%

A sustainable second-generation biodiesel supply chain network design problem under risk

Babazadeh

Razmi

Pishvaee

et al. 2017

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162

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This paper presents a multi-objective possibilistic programming model to design a second-generation biodiesel supply chain network under risk. The proposed model minimizes the total costs of biodiesel supply chain from feedstock supply centers to customer centers besides minimizing the environmental impact (EI) of all involved processes under a well-to-wheel perspective. Non-edible feedstocks are considered for biodiesel production. Variable cultivation cost of non-edible feedstock is assumed to be non-linear and dependent upon the amount of cultivated area. New formulation of possibilistic programming method is developed which is able to minimize the total mean and risk values of problems with possibilistic-based uncertainty. To solve the proposed multi-objective model, a hybrid solution approach based on flexible lexicographic and augmented ɛ-constraint methods is proposed which is capable to find appropriate efficient solutions from the Pareto-optimal set. The performance of the proposed possibilistic programming method as well as the developed solution approach are evaluated and validated through conducting a real case study in Iran. The outcome of this study demonstrates that high investment cost is required for improving the environmental impact and risk of sustainable biodiesel supply chain network design under risk. Decision maker preferences are required for suitable trade-off among total costs, risk values and environmental impact.

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