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Convex Hull Formulations for Mixed-Integer Multilinear Functions [article]

Harsha Nagarajan, Kaarthik Sundar, Hassan Hijazi, Russell Bent
2018 arXiv   pre-print
In this paper, we present convex hull formulations for a mixed-integer, multilinear term/function (MIMF) that features products of multiple continuous and binary variables.  ...  We numerically observe that the proposed formulations consistently perform better than state-of-the-art relaxation approaches.  ...  Acknowledgements The work was funded by the Center for Nonlinear Studies (CNLS) at LANL and the LANL's directed research and development project "POD: A Polyhedral Outer-approximation, Dynamic-discretization  ... 
arXiv:1807.11007v1 fatcat:rqtxufyexvagfgsg5wo5eegaqu

Piecewise Polyhedral Formulations for a Multilinear Term [article]

Kaarthik Sundar, Harsha Nagarajan, Jeff Linderoth, Site Wang, Russell Bent
2020 arXiv   pre-print
In this paper, we present a mixed-integer linear programming (MILP) formulation of a piecewise, polyhedral relaxation (PPR) of a multilinear term using its convex hull representation.  ...  Based on the solution of the PPR, we also present a MILP formulation whose solutions are feasible for nonconvex, multilinear equations.  ...  optimization solver" and "Discrete Optimization Algorithms for Provably Optimal Quantum Circuit Design".  ... 
arXiv:2001.00514v3 fatcat:n2kjrbsaqfb6db5mk2pqj4fk4i

Preface

Pierre Bonami, Leo Liberti, Andrew J. Miller, Annick Sartenaer
2012 Mathematical programming  
for relaxations obtained by approximating the convex hull of general multilinear functions using only the convex hull of the product of two variables.  ...  Sargent used the phrase "mixed-integer nonlinear programming" verbally; at the time (mid-seventies), there were no practical methods for solving MINLP problems.  ... 
doi:10.1007/s10107-012-0607-y fatcat:2bbllxm5rbcjza2mnnrebinkeq

Relaxations of Multilinear Convex Envelopes: Dual Is Better Than Primal [chapter]

Alberto Costa, Leo Liberti
2012 Lecture Notes in Computer Science  
Convex envelopes are known explicitly for the bilinear case, the trilinear case, and some instances of the quadrilinear case.  ...  This requires a convex relaxation of the original problem, obtained by replacing each multilinear term by appropriately tight convex relaxations.  ...  A sequence of k − 1 Fortet's linearizations will then yield a Mixed-Integer Linear Programming (MILP) formulation of the original multilinear term.  ... 
doi:10.1007/978-3-642-30850-5_9 fatcat:2acrmdbcrjanbjbbhjmmogzsc4

Applications and algorithms for mixed integer nonlinear programming

Sven Leyffer, Jeff Linderoth, James Luedtke, Andrew Miller, Todd Munson
2009 Journal of Physics, Conference Series  
Problems involving both discrete variables and nonlinear constraint functions are known as mixed-integer nonlinear programs (MINLPs) and are among the most challenging computational optimization problems  ...  Nonlinear functions are required to accurately represent physical properties such as pressure, stress, temperature, and equilibrium.  ...  Finding a pattern for how to move the elements to maximize fuel efficiency, subject to power safety constraints, leads to a mixed-integer nonlinear problem.  ... 
doi:10.1088/1742-6596/180/1/012014 fatcat:i4dltuwbczcnllkfapsamigowa

Resource Graph Games: A Compact Representation for Games with Structured Strategy Spaces

Albert Jiang, Hau Chan, Kevin Leyton-Brown
2017 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
We leverage recent results about multilinearity, a key property of games that allows us to represent the mixed strategies compactly, and, as a result, to compute various equilibrium concepts efficiently  ...  While not all RGGs are multilinear, we provide a general method of converting RGGs to those that are multilinear, and identify subclasses of RGGs whose converted version allow efficient computation.  ...  The extreme points (i.e. vertices) of P i are integer vectors, which implies that P i = conv(S i ), the convex hull of S i .  ... 
doi:10.1609/aaai.v31i1.10618 fatcat:ldd5gppbrfg5pmsbroh5vi7mo4

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
Based on these approaches, we develop tight piecewise convex relaxations with convex-hull representations, an adaptive, multivariate partitioning algorithm with bound tightening that progressively improves  ...  Computational results show that our novel algorithm reduces the best-known optimality gaps for some hard ACOPF cases.  ...  In the optimization literature, specifically for a trilinear function, [16] (Meyer-Floudas envelopes) describes the convex hull by deriving all it's facets, and for a generic multilinear function, convex  ... 
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  ...  Mathematical models for optimal decisions often require both nonlinear and discrete components.  ...  Utilizing an equivalent hypergraph representation for the Multilinear set, we derive various types of facet defining inequalities for its polyhedral convex hull and present a number of tightness results  ... 
doi:10.4230/dagrep.8.2.64 dblp:journals/dagstuhl-reports/BonamiGLM18 fatcat:fn6llvricbevzjsm4teuf5xuha

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  ...  LANL's Directed Research and Development (LDRD) projects, "20170201ER: POD: A Polyhedral Outer-approximation, Dynamic-discretization optimization solver" and "20200603ECR: Dis-12 tributed Algorithms for  ... 
arXiv:2005.13445v6 fatcat:f3q2l46dsfef5bismb7qs6nbaq

Multilinear Games [chapter]

Hau Chan, Albert Xin Jiang, Kevin Leyton-Brown, Ruta Mehta
2016 Lecture Notes in Computer Science  
Given a multilinear game, with polynomial time subroutines for these two tasks, we show the following: (a) We can construct a polynomially-computable and polynomiallycontinuous fixed-point formulation,  ...  To address this, we define multilinear games generalizing all. Informally, a game is multilinear if its utility functions are linear in each player's strategy, while fixing other players' strategies.  ...  , and obtaining a convex quadratic formulation for function evaluation, respectively.  ... 
doi:10.1007/978-3-662-54110-4_4 fatcat:hytwfl3p3vf5hhbh5x3lgmm7ri

A hierarchy of relaxations and convex hull characterizations for mixed-integer zero—one programming problems

Hanif D. Sherali, Warren P. Adams
1994 Discrete Applied Mathematics  
The methodology readily extends to multilinear mixed-integer zero-one polynomial programming problems in which the continuous variables appear linearly in the problem.  ...  For the linear case, we propose a technique which first converts the problem into a nonlinear, polynomial mixed-integer zero-one problem by multiplying the constraints with some suitable d-degree polynomial  ...  [4] have proposed another hierarchy leading to the convex hull representation for linear, mixed-integer zero-one problems.  ... 
doi:10.1016/0166-218x(92)00190-w fatcat:uujo7d4r6vak7d23t7v4mf6k6e

A review of recent advances in global optimization

C. A. Floudas, C. E. Gounaris
2008 Journal of Global Optimization  
It covers the areas of twice continuously differentiable nonlinear optimization, mixed-integer nonlinear optimization, optimization with differential-algebraic models, semi-infinite programming, optimization  ...  Ghosh et al. (2005) studied mixed-integer formulations for the identifications of metabolic network fluxes via NMR data.  ...  are mixed integer nonlinear in the outer variables and linear, polynomial, or multilinear in the inner integer variables, and linear in inner continuous variables; and applied it to several challenging  ... 
doi:10.1007/s10898-008-9332-8 fatcat:72fpfq72hrdzhf6mxqyc6ssezm

On the Composition of Convex Envelopes for Quadrilinear Terms [chapter]

Pietro Belotti, Sonia Cafieri, Jon Lee, Leo Liberti, Andrew J. Miller
2012 Optimization, Simulation, and Control  
Within the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity  ...  The two groupings ((x 1 x 2 )x 3 )x 4 and (x 1 x 2 x 3 )x 4 of a quadrilinear term, for example, give rise to two different convex relaxations.  ...  Introduction One of the most crucial steps of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programming (MINLP) problems is the lower bound computation.  ... 
doi:10.1007/978-1-4614-5131-0_1 fatcat:xak4mwjulzaqtc7xnujl73i5ly

Global optimization in the 21st century: Advances and challenges

C.A. Floudas, I.G. Akrotirianakis, S. Caratzoulas, C.A. Meyer, J. Kallrath
2005 Computers and Chemical Engineering  
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.  ...  functions which are mixed integer nonlinear in the outer variables and linear, polynomial, or multilinear in the inner integer variables, and linear in inner continuous variables; and applied it to several  ... 
doi:10.1016/j.compchemeng.2005.02.006 fatcat:egaic2jncfc6zfsbfzcvqnbtk4

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  ...  We are grateful to Stefan Vigerske for his continued support for SCIP.  ... 
doi:10.1007/s10898-013-0104-8 fatcat:y4pqzanrsjhsfbrt6dzj27456q
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