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global optimization of general non-convex problems with intermediate bilinear substructures

optimization METHODS & SOFTWARE 2014年第3期29卷 442-462页

作者： Zorn, Keith Sahinidis, Nikolaos V. Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

This work considers the global optimization of general non-convex nonlinear and mixed-integer nonlinear programming (MINLP) problems with underlying bilinear substructures. We combine reformulation-linearization techniques and advanced convex envelope construction techniques to produce tight subproblem formulations for these underlying structures. When incorporated as linear cutting planes, these relaxation strengthening strategies are highly effective at tightening standard linear programming relaxations generated by factorable programming techniques. Because the size of these augmented linear relaxations increases exponentially with the number of variables, we employ cut filtering and selection strategies to ensure that the tightened subproblems are solved efficiently. We introduce algorithms for bilinear substructure detection, cutting plane identification, cut filtering, and cut selection and embed the proposed implementation in branch-and-Reduce optimization Navigator at every node in the branch-and-bound tree. A computational study including problem instances from standard literature test libraries is included to assess the performance of the proposed implementation. Results show that underlying bilinear substructures are identified in 30% of the problems in globalLib and MINLPLib and that the exploitation of these structures significantly reduces computational time, branch-and-bound tree size, and required memory.

关键词： branch-and-bound global optimization factorable polyhedral relaxation reformulation-linearization techniques mixed-integer nonlinear programming

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global optimization of general nonconvex problems with intermediate polynomial substructures

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JOURNAL OF global optimization 2014年第2-3期59卷 673-693页

作者： Zorn, Keith Sahinidis, Nikolaos V. Carnegie Mellon Univ Dept Chem Engn Pittsburgh PA 15213 USA

This work considers the global optimization of general nonconvex nonlinear and mixed-integer nonlinear programming problems with underlying polynomial substructures. We incorporate linear cutting planes inspired by reformulation-linearization techniques to produce tight subproblem formulations that exploit these underlying structures. These cutting plane strategies simultaneously convexify linear and nonlinear terms from multiple constraints and are highly effective at tightening standard linear programming relaxations generated by sequential factorable programming techniques. Because the number of available cutting planes increases exponentially with the number of variables, we implement cut filtering and selection strategies to prevent an exponential increase in relaxation size. We introduce algorithms for polynomial substructure detection, cutting plane identification, cut filtering, and cut selection and embed the proposed implementation in BARON at every node in the branch-and-bound tree. A computational study including randomly generated problems of varying size and complexity demonstrates that the exploitation of underlying polynomial substructures significantly reduces computational time, branch-and-bound tree size, and required memory.

关键词： branch-and-bound global optimization Factorable polyhedral relaxation Reformulation-linearization techniques Polynomial programming

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GloMIQO: global mixed-integer quadratic optimizer

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JOURNAL OF global optimization 2013年第1期57卷 3-50页

作者： Misener, Ruth Floudas, Christodoulos A. Princeton Univ Dept Chem & Biol Engn Princeton NJ 08544 USA

This paper introduces the global mixed-integer quadratic optimizer, GloMIQO, a numerical solver addressing mixed-integer quadratically-constrained quadratic programs to -global optimality. The algorithmic components are presented for: reformulating user input, detecting special structure including convexity and edge-concavity, generating tight convex relaxations, partitioning the search space, bounding the variables, and finding good feasible solutions. To demonstrate the capacity of GloMIQO, we extensively tested its performance on a test suite of 399 problems of diverse size and structure. The test cases are taken from process networks applications, computational geometry problems, globalLib, MINLPLib, and the Bonmin test set. We compare the performance of GloMIQO with respect to four state-of-the-art global optimization solvers: BARON 10.1.2, Couenne 0.4, Lindoglobal 6.1.1.588, and SCIP 2.1.0.

关键词： Mixed-integer quadratically-constrained quadratic programs Numerical optimization software Mathematical programming reformulations branch-and-bound global optimization

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