A review of recent advances in global optimization

Floudas, Christodoulos A.; Gounaris, Chrysanthos E.

doi:10.1007/s10898-008-9332-8

Cited by 402 publications

(217 citation statements)

References 330 publications

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“…We refer the reader to the reviews of Floudas and co-workers [55,60] for background on the state-of-the-art in global optimization, to the recent review of Bussieck and Vigerske [36] for detailed descriptions of generic MINLP solver software, and to an array of excellent texts [51,52,68,105,125,133]. Hereafter, we narrow our focus to the contributions most relevant to this paper.…”

Section: Literature Reviewmentioning

confidence: 99%

“…GloMIQO falls broadly into the category of branch-and-bound global optimization because it [6,7,27,51,52,55,57,60,125,133]: generates and solves convex relaxations of the nonconvex MIQCQP that rigorously guarantee lower bounds on the global solution, finds feasible solutions via local optimization to bound the global solution from above, and divides and conquers the feasible set to generate a sequence of convex relaxations converging to the global optimum.…”

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confidence: 99%

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GloMIQO: Global mixed-integer quadratic optimizer

2012

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This paper introduces the Global Mixed-Integer Quadratic Optimizer, GloMIQO, a numerical solver addressing mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-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,

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Section: Literature Reviewmentioning

confidence: 99%

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confidence: 99%

GloMIQO: Global mixed-integer quadratic optimizer

2012

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“…For a broader perspective on the range of global optimization methods, we refer the reader to the reviews of Floudas and co-workers [23][24][25] and to a variety of texts [26][27][28][29][30][31]. The software implementation of the MISO global optimization framework addresses a class of mixed-integer nonlinear programs (MINLP).…”

Section: Problem Definition and Literature Reviewmentioning

confidence: 99%

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Misener

Floudas

2013

J Optim Theory Appl

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Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO[1], this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulationlinearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers.

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“…Similarly, if the function is convex nowhere on the box and the box lies inside the feasible set, then the box can be removed as it contains no local, and hence also no global, minimum. Convexity also plays an important role in the global optimization αBB method [3,4,5,12,39], which is based in constructing a convex underestimator of the objective function by appending an additional convex quadratic term.…”

Section: Introductionmentioning

confidence: 99%

Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

Hladík

2017

Studies in Systems, Decision and Control

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We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient / necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques.

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A review of recent advances in global optimization

Cited by 402 publications

References 330 publications

GloMIQO: Global mixed-integer quadratic optimizer

GloMIQO: Global mixed-integer quadratic optimizer

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

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