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From combinatorial optimization to real algebraic geometry and back

Janez Povh
2014 Croatian Operational Research Review  
The latter formulation enables a hierarchy of approximations which rely on results from polynomial optimization, a sub-field of real algebraic geometry.  ...  We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices.  ...  bound for the optimal value of (QAP).  ... 
doi:10.17535/crorr.2014.0001 fatcat:fngmhkpzyza5jjhe4ssc7k5st4

Dagstuhl Report 13082: Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices [article]

LeRoy Beasely and Troy Lee and Hartmut Klauck and Dirk Oliver Theis
2013 arXiv   pre-print
This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices", held in February 2013  ...  For certain applications, in particular for the extended formulation size lower bounds for approximation problems, nonnegative rank lower bounds need to be shown for matrices that are strictly positive  ...  Some lower bounds can be proved for matrices which arise from vertex/facet slack matrices of polytopes.  ... 
arXiv:1305.4147v1 fatcat:guody7xpqbdohbet6kef72u5qi

Probabilistic Combinatorial Optimization: Moments, Semidefinite Programming, and Asymptotic Bounds

Dimitris Bertsimas, Karthik Natarajan, Chung-Piaw Teo
2004 SIAM Journal on Optimization  
We show that for a fairly general class of marginal information, a tight upper (lower) bound on the expected optimal objective value of a 0-1 maximization (minimization) problem can be computed in polynomial  ...  We provide an efficiently solvable semidefinite programming formulation to compute this tight bound.  ...  As the size of the problem N approaches infinity, the tight lower bound clearly converges to C * 1 = 1: For the spanning tree problem, lim N →∞ Z * min = 1.  ... 
doi:10.1137/s1052623403430610 fatcat:y3fewqmfhjazzluzulzmp3nv6y

On the bridge between combinatorial optimization and nonlinear optimization: a family of semidefinite bounds for 0–1 quadratic problems leading to quasi-Newton methods

Jérôme Malick, Frédéric Roupin
2012 Mathematical programming  
On the bridge between combinatorial optimization and nonlinear optimization: a family of semidefinite bounds for 0-1 quadratic problems leading to quasi-Newton methods Jérôme Malick · Frédéric Roupin the  ...  Abstract This article presents a family of semidefinite programming bounds, obtained by Lagrangian duality, for 0-1 quadratic optimization problems with linear or quadratic constraints.  ...  Acknowledgement We are grateful to Sourour Elloumi who provided us a lot of material for the numerical tests, and to Sofia Zaourar who helped us to develop parts of the solver for bisection problems. hal  ... 
doi:10.1007/s10107-012-0628-6 fatcat:rspngkezl5gupilwejxvb6xwza

A Convex Relaxation Bound for Subgraph Isomorphism

Christian Schellewald
2012 International Journal of Combinatorics  
The bound is based on a semidefinite programming relaxation of a combinatorial optimisation formulation for subgraph isomorphism and is explained in detail.  ...  The bound is based on the fact that a subgraph isomorphism always leads to zero as lowest possible optimal objective value in the combinatorial problem formulation.  ...  research was partly supported by Marie Curie Intra-European Fellowships within the 6th European Community Framework Programme and an Alain Bensoussan Fellowship from the European Research Consortium for  ... 
doi:10.1155/2012/908356 fatcat:sf77jmkh4ffcxglc3f2audkmzq

Semidefinite approximations for quadratic programs over orthogonal matrices

Janez Povh
2009 Journal of Global Optimization  
This opens new possibilities to get good lower bounds for several problems from combinatorial optimization, like the Quadratic Assignment Problem (QAP) and the Graph Partitioning Problem (GPP).  ...  In the last part of the paper we show that the copositive strengthening of the semidefinite lower bounds for QAP and GPP yields the exact values.  ...  Anstreicher and Wolkowicz [1] formulated this lower bound as the optimal value of a semidefinite program.  ... 
doi:10.1007/s10898-009-9499-7 fatcat:nsjxutr56bdylcgs63wl6e755y

Tight lower bounds by semidefinite relaxations for the discrete lot-sizing and scheduling problem with sequence-dependent changeover costs [chapter]

Celine Gicquel, Abdel Lisser
2012 Operations Research Proceedings  
In the present paper, we propose to compute a tight lower bound of the optimal solution value by using a semidefinite relaxation of the problem rather than a standard linear relaxation.  ...  Tight lower bounds by semidefinite relaxation for the discrete lot-sizing and scheduling problem with sequence-dependent changeover costs. 9th International ABSTRACT: We study a production planning problem  ...  In particular, semidefinite relaxations were proved to provide tight bounds for some well-known quadratic combinatorial optimization problems such as the max-cut problem, the quadratic assignment problem  ... 
doi:10.1007/978-3-642-29210-1_67 dblp:conf/or/GicquelL11 fatcat:l4u3qofn3jea5h6xduuebswlvq

Practical comparison of approximation algorithms for scheduling problems

Eduardo Candido Xavier, Flávio K. Miyazawa
2004 Pesquisa Operacional  
The first one is a semidefinite formulation for the problem R||sigmaw jCj and the other one is a linear formulation for the problem R|r j|sigmaw jCj.  ...  We also made an experimental comparison on two lower bounds based on the formulations used by the algorithms.  ...  It is interesting to note that this lower bound may be far away from the integer optimal, since the value of an optimal integer solution for the program is already a lower bound for the original problem  ... 
doi:10.1590/s0101-74382004000200002 fatcat:ee5ffgsz4ng4jivimvpwq7ouda

Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix [article]

Troy Lee, Dirk Oliver Theis
2013 arXiv   pre-print
The most important, lower bound technique for nonnegative rank is solely based on the support of the matrix S, i.e., its zero/non-zero pattern.  ...  In this paper, we characterize the power of lower bounds on positive semidefinite rank based on solely on the support.  ...  Letchford for discussions about specific positive semidefinite formulations of various optimization problems.  ... 
arXiv:1203.3961v4 fatcat:xmzv2c6gafbazc2anuk63xm2wm

Lifts of convex sets in optimization

Volker Kaibel, Rekha Thomas
2015 Mathematical programming  
Yannakakis also showed that there is a precise connection between the nonnegative rank of the slack matrix of a polytope and the size of the smallest polyhedral lift that is possible for this polytope.  ...  In particular, positive semidefinite lifts of polytopes have received a fair bit of attention lately, while the older notion of B Volker Kaibel  ...  Known exponential lower bounds on the extension complexity of the cut polytope of the complete graph are used to establish corresponding results for other polytopes associated with hard optimization problems  ... 
doi:10.1007/s10107-015-0940-z fatcat:dojppko74nhpzjvuzbvp5gerlu

BiqCrunch

Nathan Krislock, Jérôme Malick, Frédéric Roupin
2017 ACM Transactions on Mathematical Software  
BiqCrunch: a semidefinite branch-and-bound method for solving binary quadratic problems.  ...  It has been successfully tested on a variety of well-known combinatorial optimization problems, such as Max-Cut, Max-k-Cluster, and Max-Independent-Set.  ...  These computational results provide strong evidence that BiqCrunch is among the best solvers for solving to optimality combinatorial optimization problems that can be formulated using quadratic terms.  ... 
doi:10.1145/3005345 fatcat:pbgqg24elnculiecrr2zfxzl4i

Beating the SDP bound for the floor layout problem: a simple combinatorial idea

Joey Huchette, Santanu S. Dey, Juan Pablo Vielma
2017 INFOR. Information systems and operational research  
For many Mixed-Integer Programming (MIP) problems, high-quality dual bounds can obtained either through advanced formulation techniques coupled with a state-of-the-art MIP solver, or through Semidefinite  ...  In this paper, we introduce an alternative bounding approach that exploits the "combinatorial implosion" effect by solving portions of the original problem and aggregating this information to obtain a  ...  Matching dual bounds (for minimization, a lower bound on the optimal cost) and primal bounds (e.g. the objective cost of a feasible solution) immediately provide a certificate of optimality, and for difficult  ... 
doi:10.1080/03155986.2017.1363592 fatcat:sczowvs4ubfkjapig2zafmjnzu

A Semidefinite Relaxation for Air Traffic Flow Scheduling [article]

Alexandre d'Aspremont, Laurent El Ghaoui
2006 arXiv   pre-print
We first formulate the problem of optimally scheduling air traffic low with sector capacity constraints as a mixed integer linear program.  ...  Because of the specific structure of the air traffic flow problem, the relaxation has a single semidefinite constraint of size dn where d is the maximum delay and n the number of flights.  ...  It can be solved efficiently and gives a lower bound on the optimal value of the nonconvex QCQP.  ... 
arXiv:cs/0609145v1 fatcat:t662szgk55hytgtfsiv4lxmdge

Beating the SDP bound for the floor layout problem: A simple combinatorial idea [article]

Joey Huchette, Santanu S. Dey, Juan Pablo Vielma
2017 arXiv   pre-print
For many mixed-integer programming (MIP) problems, high-quality dual bounds can be obtained either through advanced formulation techniques coupled with a state-of-the-art MIP solver, or through semidefinite  ...  In this paper, we introduce an alternative bounding approach that exploits the "combinatorial implosion" effect by solving portions of the original problem and aggregating this information to obtain a  ...  Matching dual bounds (for minimization, a lower bound on the optimal cost) and primal bounds (e.g. the objective cost of a feasible solution) immediately provide a certificate of optimality, and for difficult  ... 
arXiv:1602.07802v2 fatcat:cgre74lfmvb4fiwtgajhmfd4ri

The RPR 2 Rounding Technique for Semidefinite Programs [chapter]

Uriel Feige, Michael Langberg
2001 Lecture Notes in Computer Science  
Several combinatorial optimization problems can be approximated using algorithms based on semidefinite programming.  ...  In many of these algorithms a semidefinite relaxation of the underlying problem is solved yielding an optimal vector configuration v 1 . . . vn.  ...  A common method for obtaining an approximation algorithm for a combinatorial optimization problem is based on linear programming: 1. Formulate the problem as an integer linear program. 2.  ... 
doi:10.1007/3-540-48224-5_18 fatcat:xmmnchn2kjcm7ltgzanblosvou
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