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Some results on the strength of relaxations of multilinear functions

James Luedtke, Mahdi Namazifar, Jeff Linderoth
2012 Mathematical programming  
We then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms.  ...  These results, along with numerical examples we provide, provide insight into how to construct strong relaxations of multilinear functions.  ...  On the computational side, it would be interesting to build on the ideas of [2] and use the insights gained from this paper to devise a relaxation approach for multilinear functions that yields some  ... 
doi:10.1007/s10107-012-0606-z fatcat:ybyi3f2ilnfs3ncrzjlm4c7q2y

Global optimization of nonconvex problems with multilinear intermediates

Xiaowei Bao, Aida Khajavirad, Nikolaos V. Sahinidis, Mohit Tawarmalani
2014 Mathematical Programming Computation  
However, in general, the size of this LP grows exponentially with the number of variables in the multilinear function.  ...  It is well known that the convex hull of a multilinear function over a box is polyhedral, and the facets of this polyhedron can be obtained by solving a linear optimization problem (LP).  ...  The relative strength of the resulting relaxations depends on the variable bounds and, in general, cannot be determined a priori.  ... 
doi:10.1007/s12532-014-0073-z fatcat:xni6phwtgjdszhjdbf72ox7fge

Piecewise Polyhedral Formulations for a Multilinear Term [article]

Kaarthik Sundar, Harsha Nagarajan, Jeff Linderoth, Site Wang, Russell Bent
2020 arXiv   pre-print
We then present computational results showing the effectiveness of proposed formulations on instances of standard benchmarks of nonlinear programs (NLPs) with multilinear terms and compare the proposed  ...  Based on the solution of the PPR, we also present a MILP formulation whose solutions are feasible for nonconvex, multilinear equations.  ...  Acknowledgements The work was funded by the Center for Nonlinear Studies (CNLS) at LANL and LANL's Directed Research and Development projects "POD: A Polyhedral Outer-approximation, Dynamic-discretization  ... 
arXiv:2001.00514v3 fatcat:n2kjrbsaqfb6db5mk2pqj4fk4i


Pierre Bonami, Leo Liberti, Andrew J. Miller, Annick Sartenaer
2012 Mathematical programming  
With the kind permission of Thorsten Gellermann -Some Results on the Strength of Relaxations of Multilinear Functions, by JamesLuedtke, Mahdi Namazifar, and Jeff Linderoth, presents approximation results  ...  for relaxations obtained by approximating the convex hull of general multilinear functions using only the convex hull of the product of two variables.  ... 
doi:10.1007/s10107-012-0607-y fatcat:2bbllxm5rbcjza2mnnrebinkeq

Sequence of Polyhedral Relaxations for Nonlinear Univariate Functions [article]

Kaarthik Sundar and Sujeevraja Sanjeevi and Harsha Nagarajan
2021 arXiv   pre-print
Theoretical convergence of the sequence of relaxations to the graph of the function and its convex hull is established.  ...  Given a nonlinear, univariate, bounded, and differentiable function f(x), this article develops a sequence of Mixed Integer Linear Programming (MILP) and Linear Programming (LP) relaxations that converge  ...  This work was carried out under the U.S. DOE Contract No. DE-AC52-06NA25396.  ... 
arXiv:2005.13445v6 fatcat:f3q2l46dsfef5bismb7qs6nbaq

A constant-factor approximation algorithm for Nash Social Welfare with submodular valuations [article]

Wenzheng Li, Jan Vondrák
2021 arXiv   pre-print
We present a 380-approximation algorithm for the Nash Social Welfare problem with submodular valuations. Our algorithm builds on and extends a recent constant-factor approximation for Rado valuations.  ...  In hindsight, the strength of the approach of [15] is that it is rather modular and isolates the issue of providing at least some nonzero value to each agent as a separate matching problem.  ...  The algorithm of [3] uses a convex relaxation inspired by Gurvits's work on the permanent of doubly stochastic matrices, which relies on properties of real stable polynomials.  ... 
arXiv:2103.10536v2 fatcat:tv5m652i2revti5qt44a5yl2pe

Extended formulations for convex envelopes

Martin Ballerstein, Dennis Michaels
2013 Journal of Global Optimization  
In this work we derive explicit descriptions for the convex envelope of nonlinear functions that are component-wise concave on a subset of the variables and convex on the other variables.  ...  To overcome the combinatorial difficulties in deriving the convex envelope description given by the component-wise concave part of the functions, we consider an extended formulation of the convex envelope  ...  Main parts of this work have been finished while the second author was at the Institute for Operations Research at ETH Zurich and financially supported by DFG through the CRC/Transregio 63.  ... 
doi:10.1007/s10898-013-0104-8 fatcat:y4pqzanrsjhsfbrt6dzj27456q

A multi-term, polyhedral relaxation of a 0–1 multilinear function for Boolean logical pattern generation

Kedong Yan, Hong Seo Ryoo
2018 Journal of Global Optimization  
This paper studies a multi-term relaxation of the objective function of the pattern generation MP for a tight polyhedral relaxation in terms of a small number of stronger 0-1 linear inequalities.  ...  In brief, they yield a set of facet-defining inequalities for the 0-1 multilinear polytope associated with the McCormick inequalities that they replace.  ...  In brief, this table pictorially depicts benefits of the results of this paper in regard to: -algebraic lifting and strengthening of simple, McCormick envelopes for a 0-1 multilinear objective function  ... 
doi:10.1007/s10898-018-0680-8 fatcat:lucuparbwbdsbc566fxkfisvfa

Recursive McCormick Linearization of Multilinear Programs [article]

Arvind U Raghunathan, Carlos Cardonha, David Bergman, Carlos J Nohra
2022 arXiv   pre-print
Our numerical results on a collection of benchmarks indicate that our algorithms outperform the RML strategy implemented in state-of-the-art global optimization solvers.  ...  One example is the family of Recursive McCormick Linearization (RML) strategies, where bilinear products are substituted for artificial variables, which deliver a relaxation of the original problem when  ...  Hence, it is desirable to find a relaxation that combines the strengths of the RML-based LP relaxation and the convex hull-based LP relaxation.  ... 
arXiv:2207.08955v1 fatcat:47elluoeejdppmozajmjinkiza

Compact Relaxations for Polynomial Programming Problems [chapter]

Sonia Cafieri, Pierre Hansen, Lucas Létocart, Leo Liberti, Frédéric Messine
2012 Lecture Notes in Computer Science  
We present some computational results validating our approach.  ...  We present an extension to the general case of polynomial programming problems and discuss the derived convex relaxation.  ...  Sect. 5 discusses some computational experiments on randomly generated instances. rRLT for Polynomial Programming The results presented herein extend [1] to the general polynomial case.  ... 
doi:10.1007/978-3-642-30850-5_8 fatcat:qrw3nwweyfc2rax4ogmfz2lrym

Error bounds for monomial convexification in polynomial optimization

Warren Adams, Akshay Gupte, Yibo Xu
2018 Mathematical programming  
Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints  ...  As part of this analysis, we also derive the convex hull of a multilinear monomial over [-1,1]^n.  ...  Acknowledgements The first author was supported in part by ONR grant N00014-16-1-2168. The second author was supported in part by ONR grant N00014-16-1-2725.  ... 
doi:10.1007/s10107-018-1246-8 fatcat:5fa6mjtaxrelhl57jow673c4ci

Multidimensional NMR inversion without Kronecker products: Multilinear inversion

David Medellín, Vivek R. Ravi, Carlos Torres-Verdín
2016 Journal of magnetic resonance (San Diego, Calif. 1997 : Print)  
The new method is memory efficient, requiring less than 0.1% of the memory required by the LH or BRD methods.  ...  Additionally, it is easy to implement because only a cost function and its first derivative are required to perform the inversion.  ...  We also thank the anonymous reviewers for their suggestions and their meticulous and thorough revision of the manuscript.  ... 
doi:10.1016/j.jmr.2016.05.009 pmid:27209370 fatcat:njdzkd3hrjevnmkgps6ezzwj3a

Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow [article]

Mowen Lu, Harsha Nagarajan, Russell Bent, Sandra D. Eksioglu, Scott J. Mason
2018 arXiv   pre-print
We illustrate the strengths of our algorithm using benchmark ACOPF test cases from the literature.  ...  Computational results show that our novel algorithm reduces the best-known optimality gaps for some hard ACOPF cases.  ...  We note that this formulation generalizes to any multilinear function and is equivalent to the standard McCormick relaxation for bilinear functions.  ... 
arXiv:1803.04633v1 fatcat:2wverrprfnaslb5n2jgftjbe5y

Designing and Implementing Algorithms for Mixed-Integer Nonlinear Optimization (Dagstuhl Seminar 18081)

Pierre Bonami, Ambros M. Gleixner, Jeff Linderoth, Ruth Misener, Michael Wagner
2018 Dagstuhl Reports  
These mixed-integer nonlinear programs (MINLP) may be used to optimize the energy use of large industrial plants, integrate renewable sources into energy networks, design biological and biomedical systems  ...  Current obstacles include characterizing the computability boundary, effectively exploiting known optimization technologies for specialized classes of MINLP, and effectively using logical formulas holistically  ...  We also compare the strength of some binary extended formulations from the literature.  ... 
doi:10.4230/dagrep.8.2.64 dblp:journals/dagstuhl-reports/BonamiGLM18 fatcat:fn6llvricbevzjsm4teuf5xuha

Simple Hard Instances for Low-Depth Algebraic Proofs [article]

Nashlen Govindasamy, Tuomas Hakoniemi, Iddo Tzameret
2022 arXiv   pre-print
Our argument relies on extending the recent breakthrough lower bounds against constant-depth algebraic circuits by Limaye, Srinivasan and Tavenas (FOCS'21) to the functional lower bound framework of Forbes  ...  We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic.  ...  [17] technique to consider set-multilinear polynomials, as well as the use of the functional lower bound approach from [8] which focuses on functions computed on the Boolean cube alone.  ... 
arXiv:2205.07175v1 fatcat:5rpfawkfqrco7bkbbkwcao4mgu
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