Solving global optimization problems using reformulations and signomial transformations
•A framework for convexifying MINLP problems containing signomial functions and twice-differentiable functions is presented.•Lifting transformation schemes based on power, exponential and αBB-type reformulations is used.•The transformations used in the framework are obtained by solving an optimizati...
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| Vydáno v: | Computers & chemical engineering Ročník 116; s. 122 - 134 |
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| Jazyk: | angličtina |
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Elsevier Ltd
04.08.2018
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| ISSN: | 0098-1354, 1873-4375 |
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| Abstract | •A framework for convexifying MINLP problems containing signomial functions and twice-differentiable functions is presented.•Lifting transformation schemes based on power, exponential and αBB-type reformulations is used.•The transformations used in the framework are obtained by solving an optimization problem in a preprocessing step.•The reformulations technique is included in a global algorithm that can solve any twice-differentiable MINLP problem.
In this paper, a framework for reformulating nonconvex mixed-integer nonlinear programming (MINLP) problems containing twice-differentiable (C2) functions to convex relaxed form is discussed. To provide flexibility and for utilizing more effective transformation strategies, the twice-differentiable functions can be partitioned into convex, signomial and general nonconvex functions. The latter two can then be convexified using lifting transformations in combination with approximations using piecewise linear functions (PLFs). However, since there are many degrees of freedom in how to select the set of transformations, an optimization-based method is proposed for finding an optimal set. The lifting transformations are based on single-variable power and exponential transformations for signomials. For nonconvex C2-functions the α reformulation (αR) technique as well as more generally the method of difference of convex functions can be applied. In the αR, the αBB convex underestimator can be used. The framework is utilized in the α signomial global optimization (αSGO) algorithm to find the ϵ-global solution to a nonconvex problem by iteratively updating the approximations provided by the PLFs. The framework can also be used to directly obtain a convex relaxation of any nonconvex MINLP problem of the specified type to a determined accuracy. |
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| AbstractList | •A framework for convexifying MINLP problems containing signomial functions and twice-differentiable functions is presented.•Lifting transformation schemes based on power, exponential and αBB-type reformulations is used.•The transformations used in the framework are obtained by solving an optimization problem in a preprocessing step.•The reformulations technique is included in a global algorithm that can solve any twice-differentiable MINLP problem.
In this paper, a framework for reformulating nonconvex mixed-integer nonlinear programming (MINLP) problems containing twice-differentiable (C2) functions to convex relaxed form is discussed. To provide flexibility and for utilizing more effective transformation strategies, the twice-differentiable functions can be partitioned into convex, signomial and general nonconvex functions. The latter two can then be convexified using lifting transformations in combination with approximations using piecewise linear functions (PLFs). However, since there are many degrees of freedom in how to select the set of transformations, an optimization-based method is proposed for finding an optimal set. The lifting transformations are based on single-variable power and exponential transformations for signomials. For nonconvex C2-functions the α reformulation (αR) technique as well as more generally the method of difference of convex functions can be applied. In the αR, the αBB convex underestimator can be used. The framework is utilized in the α signomial global optimization (αSGO) algorithm to find the ϵ-global solution to a nonconvex problem by iteratively updating the approximations provided by the PLFs. The framework can also be used to directly obtain a convex relaxation of any nonconvex MINLP problem of the specified type to a determined accuracy. |
| Author | Westerlund, T. Lundell, A. |
| Author_xml | – sequence: 1 givenname: A. orcidid: 0000-0002-1548-1893 surname: Lundell fullname: Lundell, A. email: andreas.lundell@abo.fi organization: Mathematics and Statistics, Åbo Akademi University, Tuomiokirkontori 1, FIN-20500 Turku, Finland – sequence: 2 givenname: T. surname: Westerlund fullname: Westerlund, T. email: tapio.westerlund@abo.fi organization: Process Design and Systems Engineering, Åbo Akademi University, Piispankatu 8, FIN-20500 Turku, Finland |
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| Keywords | Global optimization Twice-differentiable nonconvex functions Reformulation techniques Signomial functions α reformulation Power transformations Difference of convex functions Exponential transformations αBB convex underestimator Nonconvex MINLP |
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| SubjectTerms | Difference of convex functions Exponential transformations Global optimization Nonconvex MINLP Power transformations Reformulation techniques Signomial functions Twice-differentiable nonconvex functions α reformulation αBB convex underestimator |
| Title | Solving global optimization problems using reformulations and signomial transformations |
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