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A Convex Envelope Formula for Multilinear Functions
Journal of Global Optimization
Analysis of Bounds for Multilinear Functions
Journal of Global Optimization
Convex Envelopes of Monomials of Odd Degree
Journal of Global Optimization
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Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes
Journal of Global Optimization
Journal of Global Optimization
Tight convex underestimators for $${{\mathcal C}^2}$$-continuous problems: I. univariate functions
Journal of Global Optimization
Double variable neighbourhood search with smoothing for the molecular distance geometry problem
Journal of Global Optimization
Reformulation in mathematical programming: An application to quantum chemistry
Discrete Applied Mathematics
Branching and bounds tighteningtechniques for non-convex MINLP
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The Convex Envelope of ($n$-1)-Convex Functions
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The reformulation-optimization software engine
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Feasibility-based bounds tightening via fixed points
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Compact relaxations for polynomial programming problems
SEA'12 Proceedings of the 11th international conference on Experimental Algorithms
Relaxations of multilinear convex envelopes: dual is better than primal
SEA'12 Proceedings of the 11th international conference on Experimental Algorithms
Deterministic global optimization in ab-initio quantum chemistry
Journal of Global Optimization
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The best known method to find exact or at least 驴-approximate solutions to polynomial-programming problems is the spatial Branch-and-Bound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the original program. Although convex envelopes are explicitly known (via linear inequalities) for bilinear and trilinear terms on arbitrary boxes, such a description is unknown, in general, for multilinear terms of higher order. In this paper, we study convex relaxations of quadrilinear terms. We exploit associativity to rewrite such terms as products of bilinear and trilinear terms. Using a general technique, we formally establish the intuitive fact that any relaxation for k-linear terms that employs a successive use of relaxing bilinear terms (via the bilinear convex envelope) can be improved by employing instead a relaxation of a trilinear term (via the trilinear convex envelope). We present a computational analysis which helps establish which relaxations are strictly tighter, and we apply our findings to two well-studied applications: the Molecular Distance Geometry Problem and the Hartree---Fock Problem.