Convex mixed-integer nonlinear programs derived from generalized disjunctive programming using cones

We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into Mixed-Integer Conic Programming (MICP) problems through both the big-M and hull reformulations. These refo...

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Veröffentlicht in:	Computational optimization and applications Jg. 88; H. 1; S. 251 - 312
Hauptverfasser:	Bernal Neira, David E., Grossmann, Ignacio E.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.05.2024 Springer Nature B.V
Schlagworte:	Algebra Approximation Artificial intelligence Chemical engineering Cones Convex analysis Convex and Discrete Geometry Integer programming Linear programming Machine learning Management Science Mathematical analysis Mathematics Mathematics and Statistics Mixed integer Nonlinear programming Operations Research Operations Research/Decision Theory Optimization Solvers Statistics Systems engineering Perspective function Generalized disjunctive programming Mixed-integer nonlinear programming Conic programming
ISSN:	0926-6003, 1573-2894
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Abstract	We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into Mixed-Integer Conic Programming (MICP) problems through both the big-M and hull reformulations. These reformulations have the advantage that they are representable using the same cones as the original conic GDP. In the case of the hull reformulation, they require no approximation of the perspective function. Moreover, the MICP problems derived can be solved by specialized conic solvers and offer a natural extended formulation amenable to both conic and gradient-based solvers. We present the closed form of several convex functions and their respective perspectives in conic sets, allowing users to formulate their conic GDP problems easily. We finally implement a large set of conic GDP examples and solve them via the scalar nonlinear and conic mixed-integer reformulations. These examples include applications from Process Systems Engineering, Machine learning, and randomly generated instances. Our results show that the conic structure can be exploited to solve these challenging MICP problems more efficiently. Our main contribution is providing the reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems.
AbstractList	We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into Mixed-Integer Conic Programming (MICP) problems through both the big-M and hull reformulations. These reformulations have the advantage that they are representable using the same cones as the original conic GDP. In the case of the hull reformulation, they require no approximation of the perspective function. Moreover, the MICP problems derived can be solved by specialized conic solvers and offer a natural extended formulation amenable to both conic and gradient-based solvers. We present the closed form of several convex functions and their respective perspectives in conic sets, allowing users to formulate their conic GDP problems easily. We finally implement a large set of conic GDP examples and solve them via the scalar nonlinear and conic mixed-integer reformulations. These examples include applications from Process Systems Engineering, Machine learning, and randomly generated instances. Our results show that the conic structure can be exploited to solve these challenging MICP problems more efficiently. Our main contribution is providing the reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems.
Author	Bernal Neira, David E. Grossmann, Ignacio E.
Author_xml	– sequence: 1 givenname: David E. surname: Bernal Neira fullname: Bernal Neira, David E. organization: Davidson School of Chemical Engineering, Purdue University, Research Institute of Advanced Computer Science, Universities Space Research Association, Quantum Artificial Intelligence Laboratory, NASA Ames Research Center – sequence: 2 givenname: Ignacio E. orcidid: 0000-0002-7210-084X surname: Grossmann fullname: Grossmann, Ignacio E. email: grossmann@cmu.edu organization: Chemical Engineering Department, Carnegie Mellon University
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Keywords	Perspective function Generalized disjunctive programming Mixed-integer nonlinear programming Conic programming
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Snippet	We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show...
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SubjectTerms	Algebra Approximation Artificial intelligence Chemical engineering Cones Convex analysis Convex and Discrete Geometry Integer programming Linear programming Machine learning Management Science Mathematical analysis Mathematics Mathematics and Statistics Mixed integer Nonlinear programming Operations Research Operations Research/Decision Theory Optimization Solvers Statistics Systems engineering
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Title	Convex mixed-integer nonlinear programs derived from generalized disjunctive programming using cones
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