Explicit convex and concave envelopes through polyhedral subdivisions

Tawarmalani, Mohit; Richard, Jean‐Philippe P.; Xiong, Chuanhui

doi:10.1007/s10107-012-0581-4

Cited by 74 publications

(66 citation statements)

References 29 publications

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“…For large classes of univariate and bivariate functions, tightest convex extensions have been characterized by Tawarmalani et al (2013) and Locatelli (2014). Convex envelopes of multivariate functions that are convex in all but one variable have been characterized by Jach et al (2008).…”

Section: Convex Extensionsmentioning

confidence: 99%

Convexification of Learning from Constraints

Shcherbatyi

Andres

2016

Lecture Notes in Computer Science

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Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. In this form, the problem is resistant to standard optimization techniques. We construct MIPs with the same solutions whose objective functions are convex. Specifically, we characterize the tightest convex extension of the objective function, given by the Legendre-Fenchel biconjugate. Computing values of this tightest convex extension is NP-hard. However, by applying our characterization to every function in an additive decomposition of the objective function, we obtain a class of looser convex extensions that can be computed efficiently. For some decompositions, common loss and regularization functions, we derive a closed form.

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Section: Convex Extensionsmentioning

confidence: 99%

Convexification of Learning from Constraints

Shcherbatyi

Andres

2016

Lecture Notes in Computer Science

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“…Corollary 3.13 is related to Proposition 2 from Meyer and Floudas [73]. Submodular functions are useful in the context of relaxing general nonlinear functions [107], but Corollary 3.15 shows that submodularity-based cuts cannot tighten the termwise relaxation of MIQCQP; the advantage for using submodularity in MIQCQP would have to stem from using a reduced number of constraints to represent the relaxation. Example 3.16 is equivalent to the test of Conforti et al [33] for an unbalanced hole of length 6.…”

Section: Illustration 32mentioning

confidence: 99%

“…Every cycle in bipartite Q m has an even number or elements, so finding a cycle with an odd number of positively-weighted edges requires that some nonzeros in the cycle be negative. 2 Definition 3.14 [34,107]: A bilinear function f (x 1 , . .…”

Section: Corollary 311mentioning

confidence: 99%

Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

Misener

Smadbeck

Floudas

2014

Optimization Methods and Software

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The Global Mixed-Integer Quadratic Optimizer, GloMIQO, addresses mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. This paper documents the branch-and-cut framework integrated into GloMIQO 2. Cutting planes are derived from reformulation-linearisation technique equations, convex multivariable terms, αBB convexifications, and low-and high-dimensional edge-concave aggregations. Cuts are based on both individual equations and collections of nonlinear terms in MIQCQP. Novel contributions of this paper include: development of a corollary to Crama's [35] necessary and sufficient condition for the existence of a cut dominating the termwise relaxation of a bilinear expression; algorithmic descriptions for deriving each class of cut; presentation of a branch-and-cut framework integrating the cuts. Computational results are presented along with comparison of the GloMIQO 2 performance to several state-of-the-art solvers.

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“…The complementary horizontal, term-based data structures easily admit multivariable relaxations that are specifically designed for particular functional forms. For example, beyond the convex, bilinear, trilinear, fractional, fractional trilinear, univariate concave, and general nonconvex terms as introduced by Adjiman et al [33,34], underestimators have been introduced or improved for: fractional terms [31,36,54]; trilinear terms [55,56]; quadrilinear terms [57]; odd degree monomials [58]; signomial terms [8,48,50]; low-dimensional edge-concave terms [59][60][61][62]; submodular functions [63]; and interesting products [36,[64][65][66][67].…”

Section: Problem Definition and Literature Reviewmentioning

confidence: 99%

“…Specialized underestimators have been designed for a variety of functional forms (e.g., [8,31,36,48,50,[54][55][56][57][58][59][60][61][62][63][64][65][66][67]). One advantage of the flattening transformations is that it exposes a variety of multivariable terms; flexible software design allows easy integration of new relaxations corresponding to further research advances.…”

Section: Establishing Underestimatorsmentioning

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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Explicit convex and concave envelopes through polyhedral subdivisions

Cited by 74 publications

References 29 publications

Convexification of Learning from Constraints

Convexification of Learning from Constraints

Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

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