On the existence of polyhedral convex envelopes

Tardella, Fabio

doi:10.1007/978-1-4613-0251-3_30

Cited by 43 publications

(50 citation statements)

References 26 publications

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“…The GloMIQO reformulation uses the observation that disaggregating bilinear terms tightens the relaxation of MIQCQP and actively takes advantage of any redundant linear constraints added to the model. It is standard to use termwise convex/concave envelopes [11,91] to relax MIQCQP, but many tighter relaxations have been developed based on: polyhedral facets of edge-concave multivariable term aggregations [17,26,34,94,95,96,99,111,130,131,132], eigenvector projections [38,106,113,122], piecewise-linear underestimators [29,65,66,73,93,98,99,100,101,107,119,139], outer approximation of convex terms [32,48,47], and semidefinite programming (SDP) relaxations [16,25,35,122,121]. GloMIQO incorporates several of these advanced relaxations.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Because separable functions are sum decomposable, the convex envelope of a sum of separable functions coincides with the sum of the convex envelopes of the separable functions [96,99,130,131,132]. In other words, finding the convex envelope of the equation represented by Figure 3(a) is equivalent to finding the convex envelopes of the four separable multivariable terms in Figure 3(b).…”

Section: Equation/equationmentioning

confidence: 99%

“…where q i, i , q j, j , q k, k are non-positive scalars, we used results of Tardella [130,131,132] and Meyer and Floudas [96] to generate the convex hull of (EC-AGG) and elucidate the facets of (EC-AGG) that strictly dominate the termwise relaxation of MIQCQP.…”

Section: Low-dimensional Edge-concave Relaxationsmentioning

confidence: 99%

“…-We only augment the MILP relaxation of MIQCQP with eigenvector projections of multivariable terms that are not sum decomposable as defined by Tardella [130,131,132] and Meyer and Floudas [96]. Differently stated, we only integrate the eigenvector projection relaxation for multivariable terms where the convex envelope of the aggregate term may be different than the sum of the individual McCormick relaxations.…”

Section: Eigenvector Projectionsmentioning

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%

Section: Equation/equationmentioning

confidence: 99%

Section: Low-dimensional Edge-concave Relaxationsmentioning

confidence: 99%

Section: Eigenvector Projectionsmentioning

confidence: 99%

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GloMIQO: Global mixed-integer quadratic optimizer

2012

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“…Meyer and Floudas (2004) presented explicit expressions for the facets of convex and concave envelopes of trilinear monomials with mixed-sign domains. Tardella (2003) studied the class of functions whose convex envelope on a polyhedron coincides with the convex envelope based on the polyhedron vertices, and proved important conditions for a vertex polyhedral convex envelope. Caratzoulas and Floudas (2005) proposed novel convex underestimators for trigonometric functions which are trigonometric functions themselves.…”

Section: Twice Continuously Differentiable Nlpsmentioning

confidence: 99%

Global optimization in the 21st century: Advances and challenges

Floudas

Akrotirianakis

Caratzoulas

et al. 2005

Computers & Chemical Engineering

199

117

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This paper presents an overview of the research progress in global optimization during the last 5 years (1998)(1999)(2000)(2001)(2002)(2003), and a brief account of our recent research contributions. The review part covers the areas of (a) twice continuously differentiable nonlinear optimization, (b) mixedinteger nonlinear optimization, (c) optimization with differential-algebraic models, (d) optimization with grey-box/black-box/nonfactorable models, and (e) bilevel nonlinear optimization. Our research contributions part focuses on (i) improved convex underestimation approaches that include convex envelope results for multilinear functions, convex relaxation results for trigonometric functions, and a piecewise quadratic convex underestimator for twice continuously differentiable functions, and (ii) the recently proposed novel generalized ␣BB framework. Computational studies will illustrate the potential of these advances.

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Data‐driven approximation of thermodynamic phase equilibria

et al. 2022

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We present a new data‐driven approach for both accurate and computationally efficient approximation of vapor liquid equilibria (VLE) models. Our method is able to provide guaranteed enclosure to limit the approximation errors over the entire domain of interest, all just by sampling only at select points. The approximation relies on a mixed‐integer linear programming (MILP) formulation that exploits vertex polyhedral properties of theoretically guaranteed lower and upper bounds to enclose nonlinear and nonconvex equations of state (EOS) and empirical models. Another advantage is that, unlike traditional full simulation‐based data‐driven approaches, we do not solve nonlinear system of equations (f(x) = 0) for sampling. Instead of looking for only feasible samples, we evaluate f(x) over x‐domain. This functional evaluation eliminates the need for computationally‐demanding full‐scale simulations and the associated convergence issues. We demonstrate excellent performance of the proposed MILP formulation in predicting the solubility of hydrofluorocarbon (HFC) refrigerants in ionic liquids (IL).

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On the existence of polyhedral convex envelopes

Cited by 43 publications

References 26 publications

GloMIQO: Global mixed-integer quadratic optimizer

GloMIQO: Global mixed-integer quadratic optimizer

Global optimization in the 21st century: Advances and challenges

Data‐driven approximation of thermodynamic phase equilibria

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