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Polyhedral Approaches to Mixed Integer Linear Programming [chapter]

Michele Conforti, Gérard Cornuéjols, Giacomo Zambelli
2009 50 Years of Integer Programming 1958-2008  
It applies them to the study of valid inequalities for mixed integer linear sets, such as Gomory's mixed integer cuts. .  ...  This survey presents tools from polyhedral theory that are used in integer programming.  ...  The polyhedral approach is a powerful tool for solving mixed integer linear programs (1) . This is the topic of this tutorial.  ... 
doi:10.1007/978-3-540-68279-0_11 fatcat:zfpkhg5oyveajpzk746wnfzh5u

Conic mixed-integer rounding cuts

Alper Atamtürk, Vishnu Narayanan
2008 Mathematical programming  
Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space.  ...  This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming.  ...  Our approach extends the successful polyhedral method for linear integer programming, in which one studies the facial structure of simpler substructures to secondorder conic integer programming.  ... 
doi:10.1007/s10107-008-0239-4 fatcat:cgfxok6ydfga5mll2amqb5vgoa

Page 1769 of Mathematical Reviews Vol. , Issue 86d [page]

1986 Mathematical Reviews  
Their idea was first to solve the corresponding linear program and from the resulting solution to obtain an optimal solution to the integer program by making a simple transformation.  ...  A sufficient condition is derived for the existence of a feasible solu- tion to pure integer linear programming problems.  ... 

Page 5548 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
F. (2-KIEV) A successive approach to the solution of problems of linear partially integer programming of special structure. (Russian. English summary) Dokl. Akad. Nauk Ukrain. SSR Ser.  ...  Summary: “We discuss methods for the solution of a nonlin- ear mixed integer programming problem.  ... 

Combinatorial Optimization: Theory and Computation The Aussois Workshop 2004

Thomas M. Liebling, Denis Naddef, Laurence A. Wolsey
2005 Mathematical programming  
In particular several papers describe advances in mixed integer programming, others include novel applications in biology and finance as well as the more traditional but in no-way easier ones stemming  ...  The first includes contributions to the advancement of the theory of combinatorial optimization and related fields, in particular polyhedral combinatorics, optimization on graphs and other finite structures  ...  Upon formulating the relevant optimization problem as a nonlinear mixed integer program, the authors use piecewise linear approximations to derive a linear mixed integer program whose formulation is then  ... 
doi:10.1007/s10107-005-0646-8 fatcat:4bz5pe2l6befdcciyr4szet6oe

A unified approach for price directive decomposition procedures in integer programming

Søren Holm, Jørgen Tind
1988 Discrete Applied Mathematics  
This paper shows how to generalize the Dantzig-Wolfe decomposition principle to integer programming.  ...  In both instances the authority at the central level issues price dirkctives in the form of a polyhedral, concave price function, where the purpose is to charge the sublevels for the use of central resources  ...  Additionally, it should be noted that the above framework can be extended to include a mixed integer programming version of (2.1) as well.  ... 
doi:10.1016/0166-218x(88)90077-7 fatcat:kqlue6tsgzh7jfx7fjr4rwguja

Page 2243 of Mathematical Reviews Vol. , Issue 82e [page]

1982 Mathematical Reviews  
the binary mixed integer linear program: minc{x,+0c5x, subject to D,x,+D,x,<d, x,20, x;;=0 or | for each i=1,---,n where D, is of dimension m Xn; the extreme point optimization problem: minc'x subject  ...  To obtain a stronger bound, many authors have resorted to various approaches, using the Lagrange multipliers theory for integer problems.  ... 

How Could Polyhedral Theory Harness Deep Learning? [article]

Thiago Serra and Christian Tjandraatmadja and Srikumar Ramalingam
2018 arXiv   pre-print
We outline promising research directions based on polyhedral theory and mixed-integer representability that may offer an analytical approach to this question, in contrast to the empirical techniques often  ...  The holy grail of deep learning is to come up with an automatic method to design optimal architectures for different applications.  ...  This led to a novel stream of work fueled by integer programming and its related polyhedral studies, a contrasting area where some theory waited decades for computers that could exploit it [4] .  ... 
arXiv:1806.06365v1 fatcat:oj4v3ypbenfo3ljn2ft75ajiu4

Page 7480 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
“We apply these results to provide tight formulations for cer- tain piecewise-linear convex objective integer programs, and in a companion paper we exploit them to provide polyhedral descrip- tions and  ...  After this, the paper under review is dedicated to the question: How can this mixed integer Farkas-like lemma be uti- lized to derive cutting planes for algorithmic solving of mixed integer programs?  ... 

Polyhedral approximation in mixed-integer convex optimization

Miles Lubin, Emre Yamangil, Russell Bent, Juan Pablo Vielma
2017 Mathematical programming  
Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex  ...  Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers  ...  The aim of this paper is to develop methodologies for solving the more general class of mixed-integer convex optimization, or mixed-integer convex programming (MICP), problems by reducing them to a sequence  ... 
doi:10.1007/s10107-017-1191-y fatcat:zpzgtx2xfjgpjfg2qyoitquiwy

Page 435 of The Journal of the Operational Research Society Vol. 43, Issue 5 [page]

1992 The Journal of the Operational Research Society  
Solving directly as an integer program leads to prohibitively large branch and bound trees.  ...  We choose to study this model for several reasons. (1) Start-up cost and times are an important element in many problems that are not adequately treated by standard mixed integer programming formulations  ... 

Valid Inequalities for the Lasdon-Terjung Production Model

Francois Vanderbeck, Laurence A. Wolsey
1992 Journal of the Operational Research Society  
Solving directly as an integer program leads to prohibitively large branch and bound trees.  ...  We choose to study this model for several reasons. (1) Start-up cost and times are an important element in many problems that are not adequately treated by standard mixed integer programming formulations  ... 
doi:10.2307/2583563 fatcat:wqdxmfrjavdibj7ccqavuzfbja

Page 1189 of Mathematical Reviews Vol. , Issue 81C [page]

1981 Mathematical Reviews  
objective functions in the form of mixed-integer program- ming problems.  ...  P. 81c:90061 Rational mixed-integer and polyhedral union minimization models. Math, Oper. Res. 5 (1980), no. 1, 135-146.  ... 

A cell decomposition approach to cooperative path planning and collision avoidance via disjunctive programming

Ashleigh Swingler, Silvia Ferrari
2010 49th IEEE Conference on Decision and Control (CDC)  
The approach utilizes approximate cell decomposition to obtain a disjunctive program representation of C-obstacles for obstacles that are not necessarily convex polyhedrons, and robot geometries that are  ...  In order to produce programs that are computationally tractable, this approach derives a subset of all possible inequality constraints by pruning the connectivity graph based on adjacency relationships  ...  mixed-integer program [14] , [15] .  ... 
doi:10.1109/cdc.2010.5717137 dblp:conf/cdc/SwinglerF10 fatcat:3cg5v3lzine7dp222kjmhz6rii

On Valid Inequalities for Mixed Integer p-Order Cone Programming

Alexander Vinel, Pavlo Krokhmal
2013 Journal of Optimization Theory and Applications  
We discuss two families of valid inequalities for linear mixed integer programming problems with p-order cone constraints that arise in the context of stochastic optimization with downside risk measures  ...  In particular, we extend the results of Atamtürk and Narayanan (2010, 2011) who developed mixed integer rounding cuts and lifted cuts for for mixed integer programming problems with second-order cone constraints  ...  In many cases, the proposed approaches attempt to extend some of the techniques developed for mixed integer linear programming (MILP).  ... 
doi:10.1007/s10957-013-0315-7 fatcat:slreqcx77zcevoodcu24a4dxjq
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