S-free Sets for Polynomial Optimization
|
|
- Kerry Franklin
- 5 years ago
- Views:
Transcription
1 S-free Sets for and Daniel Bienstock, Chen Chen, Gonzalo Muñoz, IEOR, Columbia University May, 2017 SIAM OPT 2017 S-free sets for PolyOpt 1
2 The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions SIAM OPT 2017 S-free sets for PolyOpt 2
3 The Polyhedral Approach The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions Consider a mathematical program of the following form minc T x subject tox S P. P := {x R n Ax b} is a polyhedral set, ands R n is a closed set. Can we strengthenp with cuts? SIAM OPT 2017 S-free sets for PolyOpt 3
4 The Polyhedral Approach The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions Consider a mathematical program of the following form minc T x subject tox S P. P := {x R n Ax b} is a polyhedral set, ands R n is a closed set. Can we strengthenp with cuts? We shall focus on the geometric approach: cuts vias-free sets. (Many other ways to generate cuts, e.g. disjunctions, algebraic arguments, combinatorics, convex cuts, etc.) SIAM OPT 2017 S-free sets for PolyOpt 3
5 TighteningP with ans-free setc The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions P S LetS be some closed set. Example: a rectanglep. Want to separate the extreme point marked with a red circle froms. SIAM OPT 2017 S-free sets for PolyOpt 4
6 TighteningP with ans-free setc The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions C P S C is ans-free set [Dey and Wolsey 2010]: a closed convex set with an interior that does not intersect withs. SIAM OPT 2017 S-free sets for PolyOpt 4
7 TighteningP with ans-free setc The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions C P S We obtain a cut by subtractingc fromp. SIAM OPT 2017 S-free sets for PolyOpt 4
8 TighteningP with ans-free setc The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions conv(pnint(c)) S Applying the single cut gives us conv(p \C). SIAM OPT 2017 S-free sets for PolyOpt 4
9 conv(p \ int(c)) is tricky The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions More than one cut, possibly an infinite number are needed. Separation is generally NP-Hard, e.g. P a polytope,c a ball models an NP-complete set containment problem. SIAM OPT 2017 S-free sets for PolyOpt 5
10 conv(p \ int(c)) is tricky The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions Balas, 1971 (see also Tuy, 1964): IfP is a simplicial cone then the intersection cut guarantees separation over conv(p \ int(c)) Simplicial cone: n linearly independent linear inequalities Simplicial conic relaxationp P is easily obtained from a basic solution ofp With less ambition we go for conv(p \ int(c)) Intersection cut is described in closed form fast separation of extreme points ofp usingp SIAM OPT 2017 S-free sets for PolyOpt 5
11 Intersection cuts and maximals-free sets The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions BiggerC, deeper cuts. C P C 0 S AnS-free setc is maximal S-free if it is not contained in another S-free set. SIAM OPT 2017 S-free sets for PolyOpt 6
12 (Non-Exhaustive) Additional Literature The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature Our Contributions MaximalS-free sets and minimal valid inequalities: [Basu et al. 2010], [Conforti et al. 2014], [Cornuejols, Wolsey, Yildiz, 2015], [Kilinc-Karzan 2015], etc. Intersection cuts and for mixed-integer conic programs programming: [Atamturk and Narayanan 2010], [Belotti et al., 2013], [Andersen and Jensen, 2013], [Dadush, Dey, Vielma 2011], [Modaresi, Kilinc, Vielma 2015/2016], etc. Intersection cuts for bilevel optimization: [Fischetti, Monaci, Sinni, 2016]. Generalized intersection cut procedures: [Balas and Margot, 2013], [Balas, Kazachkov, Margot 2016] SIAM OPT 2017 S-free sets for PolyOpt 7
13 Our Contributions The Polyhedral Approach TighteningP with an S-free setc conv(p \ int(c)) is tricky Intersection cuts and maximals-free sets (Non-Exhaustive) Additional Literature 1) A simple, generic way to generates-free sets that ensures separation. Also, a corresponding cutting plane method for arbitrary closed sets, guaranteed to converge on bounded problems. 2) A study of maximals-free sets for polynomial optimization Our Contributions SIAM OPT 2017 S-free sets for PolyOpt 8
14 Distance Oracle Cut Closure Convergence SIAM OPT 2017 S-free sets for PolyOpt 9
15 Distance Oracle Distance Oracle Cut Closure Convergence Suppose we have an oracle for a closed sets that gives us the distanced(x,s) from any pointx R n to the nearest point ins. SIAM OPT 2017 S-free sets for PolyOpt 10
16 Distance Oracle Distance Oracle Cut Closure Convergence Suppose we have an oracle for a closed sets that gives us the distanced(x,s) from any pointx R n to the nearest point ins. Examples: Integer programming: ifs is the lattice, then one can round. optimization: distance can be calculated to arbitrary accuracy in polynomial time Cardinality constraints: nearest vector of card(k) can be obtained by rounding. SIAM OPT 2017 S-free sets for PolyOpt 10
17 Distance Oracle Distance Oracle Cut Closure Convergence Suppose we have an oracle for a closed sets that gives us the distanced(x,s) from any pointx R n to the nearest point ins. Examples: Integer programming: ifs is the lattice, then one can round. optimization: distance can be calculated to arbitrary accuracy in polynomial time Cardinality constraints: nearest vector of card(k) can be obtained by rounding. Observation. The ball centered aroundxwith radiusd(x,s) is S-free. Call it B(x, d(x, S)). We shall call the corresponding intersection cut an oracle ball cut. SIAM OPT 2017 S-free sets for PolyOpt 10
18 Cut Closure Convergence Distance Oracle Cut Closure Convergence Start with a polytopep 0. DefineP k+1 asp k intersected with conv(p k \ int(b(x,d(x,s)))) for every extreme pointx. This is the rankk oracle cut closure. SIAM OPT 2017 S-free sets for PolyOpt 11
19 Cut Closure Convergence Distance Oracle Cut Closure Convergence Start with a polytopep 0. DefineP k+1 asp k intersected with conv(p k \ int(b(x,d(x,s)))) for every extreme pointx. This is the rankk oracle cut closure. Theorem: lim k P k = conv(s P). SIAM OPT 2017 S-free sets for PolyOpt 11
20 Cut Closure Convergence Distance Oracle Cut Closure Convergence Start with a polytopep 0. DefineP k+1 asp k intersected with conv(p k \ int(b(x,d(x,s)))) for every extreme pointx. This is the rankk oracle cut closure. Theorem: lim k P k = conv(s P). Corollary: given an inexact but arbitrarily accurate distance oracle, we can obtain arbitrarily close (in terms of Hausdorff distance) polyhedral approximation to conv(s P) in finite time. Borrows from proof technique used in [Averkov 2011]. SIAM OPT 2017 S-free sets for PolyOpt 11
21 Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 12
22 Cuts for Lifted Representation S-free sets for z := inf x S p 0(x) S := {x R n p 1 (x) 0,...,p m (x) 0} Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 13
23 Cuts for Cuts for Lifted Representation S-free sets for [Saxena, Bonami, Lee 2010/2011] Disjunctive cuts from MILP inner-approximation + convex cuts Applies to bounded polynomial optimization [Ghaddar, Vera, Anjos 2011] Projections of moment relaxations. Generalizes Balas, Ceria, Cornuejols lifting. Separation not guaranteed in general. Older literature on convex envelopes of functions, e.g. multilinear. Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 14
24 Cuts for Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i [Saxena, Bonami, Lee 2010/2011] Disjunctive cuts from MILP inner-approximation + convex cuts Applies to bounded polynomial optimization [Ghaddar, Vera, Anjos 2011] Projections of moment relaxations. Generalizes Balas, Ceria, Cornuejols lifting. Separation not guaranteed in general. Older literature on convex envelopes of functions, e.g. multilinear. Our intersection cuts (using e.g. the ball) guarantee polynomial-time separation without boundedness assumptions. The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 14
25 Lifted Representation Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 [Shor 1987], [Lovasz and Schrijver 1991] Define a vector of monomials,m = [1,x 1,...,x n,x 1 x 2,x 1 x 3,...,x k n]. Let M = mm T. optimization can be formulated as min A 0,M s.t. A i,m b i,i = 1,...,m. This is a linear programming relaxation with respect tom. A i,m := a ij m ij is the inner product. Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 15
26 Lifted Representation Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 15 [Shor 1987], [Lovasz and Schrijver 1991] Define a vector of monomials,m = [1,x 1,...,x n,x 1 x 2,x 1 x 3,...,x k n]. Let M = mm T. optimization can be formulated as min A 0,M s.t. A i,m b i,i = 1,...,m. This is a linear programming relaxation with respect tom. A i,m := a ij m ij is the inner product. Equivalency when M 0, rank(m) = 1 and consistency constraints. Dropping the rank constraint gives the moment relaxation [Lasserre, SIAMOPT 2001].
27 S-free sets for Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i Geometric notions are with respect to a vectorized space, e.g. M S 2 2 {M 11,M 12,M 22 } R 3 A convex set (in the appropriate vectorized space) is outer-product-free (OPF) if no point in the interior corresponds to a matrix that can be represented as an outer-product. A set is maximal OPF if no OPF set strictly contains it. The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 16
28 S-free sets for Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i Geometric notions are with respect to a vectorized space, e.g. M S 2 2 {M 11,M 12,M 22 } R 3 A convex set (in the appropriate vectorized space) is outer-product-free (OPF) if no point in the interior corresponds to a matrix that can be represented as an outer-product. A set is maximal OPF if no OPF set strictly contains it. It turns out OPF sets can have nice structure and lead to easily generated intersection cuts. The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 16
29 Violation Ball Cuts for Lifted Representation S-free sets for Using a modification by Dax (2016) of the Eckart-Young-Mirksy theorem, we can find the nearest (by Frobenius norm) outer-product to a given matrix. This uses eigenvalues, and hence can be generated (to specified precision) in polynomial time. Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 17
30 Max Full-Dim OPF Sets are Cones Cuts for Lifted Representation S-free sets for Theorem: LetC S n n be an outer-product-free set with full dimension. Then clcone(c) is outer-product-free. Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 18
31 Negative SemidefiniteP i Cuts for Lifted Representation S-free sets for P i,m 0 Theorem: Every such halfspace withp i NSD is maximal outer-product-free. These halfspaces yield the standard outer approximation cut for SDP:M 0 c T Mc 0 c R n. Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 19
32 The2 2cone Cuts for Lifted Representation S-free sets for DefineC i,j := {M S n n M [i,j] 0} Theorem: C i,j is max OPF for1 i j n. Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 20
33 Characterization for2 2 Cuts for Lifted Representation S-free sets for The 3-dimensional case. Theorem: Forn = 2, the maximal OPF set arec 1,2 and halfspaces of the form P,M 0, wherep is NSD. i.e. the PSD cone and halfspaces with boundaries that support the cone Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 21
34 Cuts Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i Recall: we want every extreme point of our relaxation to be an outer-product, i.e. PSD and rank one. Any non-psd extreme point can be separated by the outer-approximation cutsc T Mc 0. Any PSD extreme point with rank greater than 1 can be separated by the intersection cut given byc i,j for somei,j [Chen, Oren, Atamturk 2016]. The oracle cut can be strengthened by recentering the ball and taking the conic hull (complicated expression but computationally fast). The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 22
35 Experiments: Setup Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i Pure cutting plane algorithm implemented in Python using combinations of the following cuts: OB (Oracle Ball Cuts), SO (Strengthened Oracle Cuts) OA (Outer Approximation Cuts), 2x2 (2x2 Principal Minor Cuts) LP solver: Gurobi Hardware: 20-core server, Intel Xeon 3.10GHz CPU, 264 GB RAM 26 QCQP problems from GLOBALLib (6-63 variables) 99 BoxQP instances ( variables) The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 23
36 Experiments: Results Cuts for Lifted Representation S-free sets for Cut Family Initial Gap End Gap Closed Gap # Cuts Iters Time (s) LPTime (%) OB % % 1.00% % SO % 8.77% % OA % 8.61% % 2x2 + OA % 32.61% % SO+2x2+OA % 31.91% % Table 1: Averages for GLOBALLib instances Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 24 Cut Family Initial Gap End Gap Closed Gap # Cuts Iters Time (s) LPTime (%) OB % % 0.04% % SO % 0.34% % OA 30.88% 75.55% % 2x2 + OA 32.84% 74.52% % SO+2x2+OA 33.43% 74.03% % Table 2: Averages for BoxQP instances
37 BoxQP Cuts for Lifted Representation S-free sets for V2: second-order conic outer-approximation of PSD constraint MIP to derive disjunctive cuts Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 25
38 GLOBALLib Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 26 Instance V2 Gap V2 Time Gap Closed Time Ex % % 0.41 Ex % % 0.13 Ex % % 0.95 Ex % % Ex % % 36.9 Ex % % 0.55 Ex % % 0.04 Ex % % 0.26 Ex5 2 2 case1 0.00% % 0.47 Ex5 2 2 case2 0.00% % 0.26 Ex5 2 2 case3 0.36% % 0.16 Ex % % 5.69 Table 3: Comparison with V2 on GLOBALLib instances
39 GLOBALLib Cuts for Lifted Representation S-free sets for Violation Ball Max Full-Dim OPF Sets are Cones Negative Semidefinite P i Instance V2 Gap V2 Time Gap Closed Time Ex % % 2.33 Ex % % 0.59 Ex % % 0.34 Ex % % Ex % % Ex % % 0.12 Ex % 0.12 The2 2 cone Characterization for 2 2 Cuts Experiments: Setup Experiments: Results BoxQP SIAM OPT 2017 S-free sets for PolyOpt 27
40 Take-homes Thanks! SIAM OPT 2017 S-free sets for PolyOpt 28
41 Take-homes Take-homes Thanks! We introduced an oracle-based intersection cut for closed sets. Furthermore, we constructed a convergent cutting plane algorithm that uses this oracle to ping the sets. All of this is done without using any explicit structure abouts. Outer-product-free sets provide a new way to generate cuts for polynomial optimization. SIAM OPT 2017 S-free sets for PolyOpt 29
42 Thanks! Preprint available: Take-homes Thanks! SIAM OPT 2017 S-free sets for PolyOpt 30
From final point cuts to!-polyhedral cuts
AUSSOIS 2017 From final point cuts to!-polyhedral cuts Egon Balas, Aleksandr M. Kazachkov, François Margot Tepper School of Business, Carnegie Mellon University Overview Background Generalized intersection
More informationOptimality certificates for convex minimization and Helly numbers
Optimality certificates for convex minimization and Helly numbers Amitabh Basu Michele Conforti Gérard Cornuéjols Robert Weismantel Stefan Weltge May 10, 2017 Abstract We consider the problem of minimizing
More informationOptimality certificates for convex minimization and Helly numbers
Optimality certificates for convex minimization and Helly numbers Amitabh Basu Michele Conforti Gérard Cornuéjols Robert Weismantel Stefan Weltge October 20, 2016 Abstract We consider the problem of minimizing
More informationLecture 3. Corner Polyhedron, Intersection Cuts, Maximal Lattice-Free Convex Sets. Tepper School of Business Carnegie Mellon University, Pittsburgh
Lecture 3 Corner Polyhedron, Intersection Cuts, Maximal Lattice-Free Convex Sets Gérard Cornuéjols Tepper School of Business Carnegie Mellon University, Pittsburgh January 2016 Mixed Integer Linear Programming
More informationConvex Optimization - Chapter 1-2. Xiangru Lian August 28, 2015
Convex Optimization - Chapter 1-2 Xiangru Lian August 28, 2015 1 Mathematical optimization minimize f 0 (x) s.t. f j (x) 0, j=1,,m, (1) x S x. (x 1,,x n ). optimization variable. f 0. R n R. objective
More informationConvexity: an introduction
Convexity: an introduction Geir Dahl CMA, Dept. of Mathematics and Dept. of Informatics University of Oslo 1 / 74 1. Introduction 1. Introduction what is convexity where does it arise main concepts and
More informationLecture 2 - Introduction to Polytopes
Lecture 2 - Introduction to Polytopes Optimization and Approximation - ENS M1 Nicolas Bousquet 1 Reminder of Linear Algebra definitions Let x 1,..., x m be points in R n and λ 1,..., λ m be real numbers.
More informationApplied Integer Programming
Applied Integer Programming D.S. Chen; R.G. Batson; Y. Dang Fahimeh 8.2 8.7 April 21, 2015 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces,
More informationThe Chvátal-Gomory Closure of a Strictly Convex Body is a Rational Polyhedron
The Chvátal-Gomory Closure of a Strictly Convex Body is a Rational Polyhedron Juan Pablo Vielma Joint work with Daniel Dadush and Santanu S. Dey July, Atlanta, GA Outline Introduction Proof: Step Step
More informationIntersection Cuts with Infinite Split Rank
Intersection Cuts with Infinite Split Rank Amitabh Basu 1,2, Gérard Cornuéjols 1,3,4 François Margot 1,5 April 2010; revised April 2011; revised October 2011 Abstract We consider mixed integer linear programs
More informationExperiments On General Disjunctions
Experiments On General Disjunctions Some Dumb Ideas We Tried That Didn t Work* and Others We Haven t Tried Yet *But that may provide some insight Ted Ralphs, Serdar Yildiz COR@L Lab, Department of Industrial
More informationTowards Efficient Higher-Order Semidefinite Relaxations for Max-Cut
Towards Efficient Higher-Order Semidefinite Relaxations for Max-Cut Miguel F. Anjos Professor and Canada Research Chair Director, Trottier Energy Institute Joint work with E. Adams (Poly Mtl), F. Rendl,
More informationSplit-Cuts and the Stable Set Polytope of Quasi-Line Graphs
Split-Cuts and the Stable Set Polytope of Quasi-Line Graphs Friedrich Eisenbrand Joint work with G. Oriolo, P. Ventura and G. Stauffer Gomory cutting planes P = {x n Ax b} polyhedron, c T x δ, c n valid
More information60 2 Convex sets. {x a T x b} {x ã T x b}
60 2 Convex sets Exercises Definition of convexity 21 Let C R n be a convex set, with x 1,, x k C, and let θ 1,, θ k R satisfy θ i 0, θ 1 + + θ k = 1 Show that θ 1x 1 + + θ k x k C (The definition of convexity
More information6 Randomized rounding of semidefinite programs
6 Randomized rounding of semidefinite programs We now turn to a new tool which gives substantially improved performance guarantees for some problems We now show how nonlinear programming relaxations can
More informationLecture 5: Properties of convex sets
Lecture 5: Properties of convex sets Rajat Mittal IIT Kanpur This week we will see properties of convex sets. These properties make convex sets special and are the reason why convex optimization problems
More informationLecture 2 September 3
EE 381V: Large Scale Optimization Fall 2012 Lecture 2 September 3 Lecturer: Caramanis & Sanghavi Scribe: Hongbo Si, Qiaoyang Ye 2.1 Overview of the last Lecture The focus of the last lecture was to give
More informationDisjunctive Programming
Chapter 10 Disjunctive Programming Egon Balas Introduction by Egon Balas In April 1967 I and my family arrived into the US as fresh immigrants from behind the Iron Curtain. After a fruitful semester spent
More information1. Lecture notes on bipartite matching February 4th,
1. Lecture notes on bipartite matching February 4th, 2015 6 1.1.1 Hall s Theorem Hall s theorem gives a necessary and sufficient condition for a bipartite graph to have a matching which saturates (or matches)
More informationConic Duality. yyye
Conic Linear Optimization and Appl. MS&E314 Lecture Note #02 1 Conic Duality Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/
More informationInteger Programming ISE 418. Lecture 1. Dr. Ted Ralphs
Integer Programming ISE 418 Lecture 1 Dr. Ted Ralphs ISE 418 Lecture 1 1 Reading for This Lecture N&W Sections I.1.1-I.1.4 Wolsey Chapter 1 CCZ Chapter 2 ISE 418 Lecture 1 2 Mathematical Optimization Problems
More informationMathematical Programming and Research Methods (Part II)
Mathematical Programming and Research Methods (Part II) 4. Convexity and Optimization Massimiliano Pontil (based on previous lecture by Andreas Argyriou) 1 Today s Plan Convex sets and functions Types
More informationPOLYHEDRAL GEOMETRY. Convex functions and sets. Mathematical Programming Niels Lauritzen Recall that a subset C R n is convex if
POLYHEDRAL GEOMETRY Mathematical Programming Niels Lauritzen 7.9.2007 Convex functions and sets Recall that a subset C R n is convex if {λx + (1 λ)y 0 λ 1} C for every x, y C and 0 λ 1. A function f :
More informationMVE165/MMG630, Applied Optimization Lecture 8 Integer linear programming algorithms. Ann-Brith Strömberg
MVE165/MMG630, Integer linear programming algorithms Ann-Brith Strömberg 2009 04 15 Methods for ILP: Overview (Ch. 14.1) Enumeration Implicit enumeration: Branch and bound Relaxations Decomposition methods:
More informationStable sets, corner polyhedra and the Chvátal closure
Stable sets, corner polyhedra and the Chvátal closure Manoel Campêlo Departamento de Estatística e Matemática Aplicada, Universidade Federal do Ceará, Brazil, mcampelo@lia.ufc.br. Gérard Cornuéjols Tepper
More informationFINITE DISJUNCTIVE PROGRAMMING CHARACTERIZATIONS FOR GENERAL MIXED-INTEGER LINEAR PROGRAMS
FINITE DISJUNCTIVE PROGRAMMING CHARACTERIZATIONS FOR GENERAL MIXED-INTEGER LINEAR PROGRAMS BINYUAN CHEN, SİMGE KÜÇÜKYAVUZ, SUVRAJEET SEN Abstract. In this paper, we give a finite disjunctive programming
More informationMath 5593 Linear Programming Lecture Notes
Math 5593 Linear Programming Lecture Notes Unit II: Theory & Foundations (Convex Analysis) University of Colorado Denver, Fall 2013 Topics 1 Convex Sets 1 1.1 Basic Properties (Luenberger-Ye Appendix B.1).........................
More information4 Integer Linear Programming (ILP)
TDA6/DIT37 DISCRETE OPTIMIZATION 17 PERIOD 3 WEEK III 4 Integer Linear Programg (ILP) 14 An integer linear program, ILP for short, has the same form as a linear program (LP). The only difference is that
More informationInvestigating Mixed-Integer Hulls using a MIP-Solver
Investigating Mixed-Integer Hulls using a MIP-Solver Matthias Walter Otto-von-Guericke Universität Magdeburg Joint work with Volker Kaibel (OvGU) Aussois Combinatorial Optimization Workshop 2015 Outline
More informationLP-Modelling. dr.ir. C.A.J. Hurkens Technische Universiteit Eindhoven. January 30, 2008
LP-Modelling dr.ir. C.A.J. Hurkens Technische Universiteit Eindhoven January 30, 2008 1 Linear and Integer Programming After a brief check with the backgrounds of the participants it seems that the following
More informationLecture 2 Convex Sets
Optimization Theory and Applications Lecture 2 Convex Sets Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Fall 2016 2016/9/29 Lecture 2: Convex Sets 1 Outline
More informationConvex Sets (cont.) Convex Functions
Convex Sets (cont.) Convex Functions Optimization - 10725 Carlos Guestrin Carnegie Mellon University February 27 th, 2008 1 Definitions of convex sets Convex v. Non-convex sets Line segment definition:
More informationLecture 2. Topology of Sets in R n. August 27, 2008
Lecture 2 Topology of Sets in R n August 27, 2008 Outline Vectors, Matrices, Norms, Convergence Open and Closed Sets Special Sets: Subspace, Affine Set, Cone, Convex Set Special Convex Sets: Hyperplane,
More informationMathematical and Algorithmic Foundations Linear Programming and Matchings
Adavnced Algorithms Lectures Mathematical and Algorithmic Foundations Linear Programming and Matchings Paul G. Spirakis Department of Computer Science University of Patras and Liverpool Paul G. Spirakis
More informationHowever, this is not always true! For example, this fails if both A and B are closed and unbounded (find an example).
98 CHAPTER 3. PROPERTIES OF CONVEX SETS: A GLIMPSE 3.2 Separation Theorems It seems intuitively rather obvious that if A and B are two nonempty disjoint convex sets in A 2, then there is a line, H, separating
More informationThe Simplex Algorithm for LP, and an Open Problem
The Simplex Algorithm for LP, and an Open Problem Linear Programming: General Formulation Inputs: real-valued m x n matrix A, and vectors c in R n and b in R m Output: n-dimensional vector x There is one
More informationConvex Geometry arising in Optimization
Convex Geometry arising in Optimization Jesús A. De Loera University of California, Davis Berlin Mathematical School Summer 2015 WHAT IS THIS COURSE ABOUT? Combinatorial Convexity and Optimization PLAN
More informationOn the polyhedrality of cross and quadrilateral closures
On the polyhedrality of cross and quadrilateral closures Sanjeeb Dash IBM Research sanjeebd@us.ibm.com Oktay Günlük IBM Research gunluk@us.ibm.com December 9, 2014 Diego A. Morán R. Virginia Tech dmoran@gatech.edu
More information2. Convex sets. x 1. x 2. affine set: contains the line through any two distinct points in the set
2. Convex sets Convex Optimization Boyd & Vandenberghe affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual
More informationA Verification Based Method to Generate Cutting Planes for IPs
A Verification Based Method to Generate Cutting Planes for IPs Santanu S. Dey Sebastian Pokutta Georgia Institute of Technology, USA. Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany. SIAM Conference
More informationCS675: Convex and Combinatorial Optimization Spring 2018 Consequences of the Ellipsoid Algorithm. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Spring 2018 Consequences of the Ellipsoid Algorithm Instructor: Shaddin Dughmi Outline 1 Recapping the Ellipsoid Method 2 Complexity of Convex Optimization
More informationNumerical Optimization
Convex Sets Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Let x 1, x 2 R n, x 1 x 2. Line and line segment Line passing through x 1 and x 2 : {y
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 9 Discrete optimization: theory and algorithms
MVE165/MMG631 Linear and integer optimization with applications Lecture 9 Discrete optimization: theory and algorithms Ann-Brith Strömberg 2018 04 24 Lecture 9 Linear and integer optimization with applications
More informationExploiting Degeneracy in MIP
Exploiting Degeneracy in MIP Tobias Achterberg 9 January 2018 Aussois Performance Impact in Gurobi 7.5+ 35% 32.0% 30% 25% 20% 15% 14.6% 10% 5.7% 7.9% 6.6% 5% 0% 2.9% 1.2% 0.1% 2.6% 2.6% Time limit: 10000
More informationDM545 Linear and Integer Programming. Lecture 2. The Simplex Method. Marco Chiarandini
DM545 Linear and Integer Programming Lecture 2 The Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. 2. 3. 4. Standard Form Basic Feasible Solutions
More informationLecture 4: Convexity
10-725: Convex Optimization Fall 2013 Lecture 4: Convexity Lecturer: Barnabás Póczos Scribes: Jessica Chemali, David Fouhey, Yuxiong Wang Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:
More informationIntroduction to Mathematical Programming IE406. Lecture 20. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 20 Dr. Ted Ralphs IE406 Lecture 20 1 Reading for This Lecture Bertsimas Sections 10.1, 11.4 IE406 Lecture 20 2 Integer Linear Programming An integer
More informationcuts François Margot 1 Chicago, IL Abstract The generalized intersection cut (GIC) paradigm is a recent framework for generating
Partial hyperplane activation for generalized intersection cuts Aleksandr M. Kazachkov 1 Selvaprabu Nadarajah 2 Egon Balas 1 François Margot 1 1 Tepper School of Business, Carnegie Mellon University, Pittsburgh,
More informationEmbedding Formulations, Complexity and Representability for Unions of Convex Sets
, Complexity and Representability for Unions of Convex Sets Juan Pablo Vielma Massachusetts Institute of Technology CMO-BIRS Workshop: Modern Techniques in Discrete Optimization: Mathematics, Algorithms
More informationCS 435, 2018 Lecture 2, Date: 1 March 2018 Instructor: Nisheeth Vishnoi. Convex Programming and Efficiency
CS 435, 2018 Lecture 2, Date: 1 March 2018 Instructor: Nisheeth Vishnoi Convex Programming and Efficiency In this lecture, we formalize convex programming problem, discuss what it means to solve it efficiently
More informationAdvanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
More informationChapter 8. Voronoi Diagrams. 8.1 Post Oce Problem
Chapter 8 Voronoi Diagrams 8.1 Post Oce Problem Suppose there are n post oces p 1,... p n in a city. Someone who is located at a position q within the city would like to know which post oce is closest
More informationIntroduction to Mathematical Programming IE496. Final Review. Dr. Ted Ralphs
Introduction to Mathematical Programming IE496 Final Review Dr. Ted Ralphs IE496 Final Review 1 Course Wrap-up: Chapter 2 In the introduction, we discussed the general framework of mathematical modeling
More information2. Convex sets. affine and convex sets. some important examples. operations that preserve convexity. generalized inequalities
2. Convex sets Convex Optimization Boyd & Vandenberghe affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual
More informationLinear Programming Duality and Algorithms
COMPSCI 330: Design and Analysis of Algorithms 4/5/2016 and 4/7/2016 Linear Programming Duality and Algorithms Lecturer: Debmalya Panigrahi Scribe: Tianqi Song 1 Overview In this lecture, we will cover
More informationResearch Interests Optimization:
Mitchell: Research interests 1 Research Interests Optimization: looking for the best solution from among a number of candidates. Prototypical optimization problem: min f(x) subject to g(x) 0 x X IR n Here,
More informationExact Algorithms for Mixed-Integer Bilevel Linear Programming
Exact Algorithms for Mixed-Integer Bilevel Linear Programming Matteo Fischetti, University of Padova (based on joint work with I. Ljubic, M. Monaci, and M. Sinnl) Lunteren Conference on the Mathematics
More informationConvex sets and convex functions
Convex sets and convex functions Convex optimization problems Convex sets and their examples Separating and supporting hyperplanes Projections on convex sets Convex functions, conjugate functions ECE 602,
More informationCompact Sets. James K. Peterson. September 15, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Compact Sets James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 15, 2017 Outline 1 Closed Sets 2 Compactness 3 Homework Closed Sets
More informationCMU-Q Lecture 9: Optimization II: Constrained,Unconstrained Optimization Convex optimization. Teacher: Gianni A. Di Caro
CMU-Q 15-381 Lecture 9: Optimization II: Constrained,Unconstrained Optimization Convex optimization Teacher: Gianni A. Di Caro GLOBAL FUNCTION OPTIMIZATION Find the global maximum of the function f x (and
More informationConvex Sets. CSCI5254: Convex Optimization & Its Applications. subspaces, affine sets, and convex sets. operations that preserve convexity
CSCI5254: Convex Optimization & Its Applications Convex Sets subspaces, affine sets, and convex sets operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual
More informationConvex Optimization. Chapter 1 - chapter 2.2
Convex Optimization Chapter 1 - chapter 2.2 Introduction In optimization literatures, one will frequently encounter terms like linear programming, convex set convex cone, convex hull, semidefinite cone
More informationImplementing a B&C algorithm for Mixed-Integer Bilevel Linear Programming
Implementing a B&C algorithm for Mixed-Integer Bilevel Linear Programming Matteo Fischetti, University of Padova 8th Cargese-Porquerolles Workshop on Combinatorial Optimization, August 2017 1 Bilevel Optimization
More informationWeek 5. Convex Optimization
Week 5. Convex Optimization Lecturer: Prof. Santosh Vempala Scribe: Xin Wang, Zihao Li Feb. 9 and, 206 Week 5. Convex Optimization. The convex optimization formulation A general optimization problem is
More information3 No-Wait Job Shops with Variable Processing Times
3 No-Wait Job Shops with Variable Processing Times In this chapter we assume that, on top of the classical no-wait job shop setting, we are given a set of processing times for each operation. We may select
More information/ Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang
600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang 9.1 Linear Programming Suppose we are trying to approximate a minimization
More informationDiscrete Optimization. Lecture Notes 2
Discrete Optimization. Lecture Notes 2 Disjunctive Constraints Defining variables and formulating linear constraints can be straightforward or more sophisticated, depending on the problem structure. The
More informationFrom the Separation to the Intersection Sub-problem in Benders Decomposition Models with Prohibitively-Many Constraints
From the Separation to the Intersection Sub-problem in Benders Decomposition Models with Prohibitively-Many Constraints Daniel Porumbel CEDRIC CS Lab, CNAM, 292 rue Saint-Martin, F-75141 Paris, France
More informationMarcia Fampa Universidade Federal do Rio de Janeiro Rio de Janeiro, RJ, Brazil
A specialized branch-and-bound algorithm for the Euclidean Steiner tree problem in n-space Marcia Fampa Universidade Federal do Rio de Janeiro Rio de Janeiro, RJ, Brazil fampa@cos.ufrj.br Jon Lee University
More informationConvex Optimization. 2. Convex Sets. Prof. Ying Cui. Department of Electrical Engineering Shanghai Jiao Tong University. SJTU Ying Cui 1 / 33
Convex Optimization 2. Convex Sets Prof. Ying Cui Department of Electrical Engineering Shanghai Jiao Tong University 2018 SJTU Ying Cui 1 / 33 Outline Affine and convex sets Some important examples Operations
More informationTarget Cuts from Relaxed Decision Diagrams
Target Cuts from Relaxed Decision Diagrams Christian Tjandraatmadja 1, Willem-Jan van Hoeve 1 1 Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA {ctjandra,vanhoeve}@andrew.cmu.edu
More informationConvex sets and convex functions
Convex sets and convex functions Convex optimization problems Convex sets and their examples Separating and supporting hyperplanes Projections on convex sets Convex functions, conjugate functions ECE 602,
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 6
EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 6 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 19, 2012 Andre Tkacenko
More informationLinear programming and duality theory
Linear programming and duality theory Complements of Operations Research Giovanni Righini Linear Programming (LP) A linear program is defined by linear constraints, a linear objective function. Its variables
More informationConvex Optimization. Convex Sets. ENSAE: Optimisation 1/24
Convex Optimization Convex Sets ENSAE: Optimisation 1/24 Today affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes
More informationInteger Programming Theory
Integer Programming Theory Laura Galli October 24, 2016 In the following we assume all functions are linear, hence we often drop the term linear. In discrete optimization, we seek to find a solution x
More informationCS675: Convex and Combinatorial Optimization Spring 2018 The Simplex Algorithm. Instructor: Shaddin Dughmi
CS675: Convex and Combinatorial Optimization Spring 2018 The Simplex Algorithm Instructor: Shaddin Dughmi Algorithms for Convex Optimization We will look at 2 algorithms in detail: Simplex and Ellipsoid.
More informationInteger Programming Chapter 9
Integer Programming Chapter 9 University of Chicago Booth School of Business Kipp Martin October 25, 2017 1 / 40 Outline Key Concepts MILP Set Monoids LP set Relaxation of MILP Set Formulation Quality
More informationLP Geometry: outline. A general LP. minimize x c T x s.t. a T i. x b i, i 2 M 1 a T i x = b i, i 2 M 3 x j 0, j 2 N 1. where
LP Geometry: outline I Polyhedra I Extreme points, vertices, basic feasible solutions I Degeneracy I Existence of extreme points I Optimality of extreme points IOE 610: LP II, Fall 2013 Geometry of Linear
More informationProbabilistic Graphical Models
School of Computer Science Probabilistic Graphical Models Theory of Variational Inference: Inner and Outer Approximation Eric Xing Lecture 14, February 29, 2016 Reading: W & J Book Chapters Eric Xing @
More informationPolytopes Course Notes
Polytopes Course Notes Carl W. Lee Department of Mathematics University of Kentucky Lexington, KY 40506 lee@ms.uky.edu Fall 2013 i Contents 1 Polytopes 1 1.1 Convex Combinations and V-Polytopes.....................
More informationShiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 2. Convex Optimization
Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 2 Convex Optimization Shiqian Ma, MAT-258A: Numerical Optimization 2 2.1. Convex Optimization General optimization problem: min f 0 (x) s.t., f i
More informationAdvanced Operations Research Techniques IE316. Quiz 2 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 2 Review Dr. Ted Ralphs IE316 Quiz 2 Review 1 Reading for The Quiz Material covered in detail in lecture Bertsimas 4.1-4.5, 4.8, 5.1-5.5, 6.1-6.3 Material
More information5.3 Cutting plane methods and Gomory fractional cuts
5.3 Cutting plane methods and Gomory fractional cuts (ILP) min c T x s.t. Ax b x 0integer feasible region X Assumption: a ij, c j and b i integer. Observation: The feasible region of an ILP can be described
More informationUnconstrained Optimization Principles of Unconstrained Optimization Search Methods
1 Nonlinear Programming Types of Nonlinear Programs (NLP) Convexity and Convex Programs NLP Solutions Unconstrained Optimization Principles of Unconstrained Optimization Search Methods Constrained Optimization
More informationStable sets, corner polyhedra and the Chvátal closure
Stable sets, corner polyhedra and the Chvátal closure Manoel Campêlo Departamento de Estatística e Matemática Aplicada, Universidade Federal do Ceará, Brazil, mcampelo@lia.ufc.br. Gérard Cornuéjols Tepper
More informationCutting Planes for Some Nonconvex Combinatorial Optimization Problems
Cutting Planes for Some Nonconvex Combinatorial Optimization Problems Ismael Regis de Farias Jr. Department of Industrial Engineering Texas Tech Summary Problem definition Solution strategy Multiple-choice
More informationConvex Optimization MLSS 2015
Convex Optimization MLSS 2015 Constantine Caramanis The University of Texas at Austin The Optimization Problem minimize : f (x) subject to : x X. The Optimization Problem minimize : f (x) subject to :
More informationDistributed Mixed-Integer Linear Programming via Cut Generation and Constraint Exchange
Distributed Mixed-Integer Linear Programming via Cut Generation and Constraint Exchange Andrea Testa, Alessandro Rucco, and Giuseppe Notarstefano 1 arxiv:1804.03391v1 [math.oc] 10 Apr 2018 Abstract Many
More informationOpen problems in convex geometry
Open problems in convex geometry 10 March 2017, Monash University Seminar talk Vera Roshchina, RMIT University Based on joint work with Tian Sang (RMIT University), Levent Tunçel (University of Waterloo)
More informationFundamentals of Integer Programming
Fundamentals of Integer Programming Di Yuan Department of Information Technology, Uppsala University January 2018 Outline Definition of integer programming Formulating some classical problems with integer
More informationSome Advanced Topics in Linear Programming
Some Advanced Topics in Linear Programming Matthew J. Saltzman July 2, 995 Connections with Algebra and Geometry In this section, we will explore how some of the ideas in linear programming, duality theory,
More informationLinear Programming in Small Dimensions
Linear Programming in Small Dimensions Lekcija 7 sergio.cabello@fmf.uni-lj.si FMF Univerza v Ljubljani Edited from slides by Antoine Vigneron Outline linear programming, motivation and definition one dimensional
More information15.082J and 6.855J. Lagrangian Relaxation 2 Algorithms Application to LPs
15.082J and 6.855J Lagrangian Relaxation 2 Algorithms Application to LPs 1 The Constrained Shortest Path Problem (1,10) 2 (1,1) 4 (2,3) (1,7) 1 (10,3) (1,2) (10,1) (5,7) 3 (12,3) 5 (2,2) 6 Find the shortest
More informationA Center-Cut Algorithm for Quickly Obtaining Feasible Solutions and Solving Convex MINLP Problems
A Center-Cut Algorithm for Quickly Obtaining Feasible Solutions and Solving Convex MINLP Problems Jan Kronqvist a, David E. Bernal b, Andreas Lundell a, and Tapio Westerlund a a Faculty of Science and
More information3 INTEGER LINEAR PROGRAMMING
3 INTEGER LINEAR PROGRAMMING PROBLEM DEFINITION Integer linear programming problem (ILP) of the decision variables x 1,..,x n : (ILP) subject to minimize c x j j n j= 1 a ij x j x j 0 x j integer n j=
More informationLecture 2: August 31
10-725/36-725: Convex Optimization Fall 2016 Lecture 2: August 31 Lecturer: Lecturer: Ryan Tibshirani Scribes: Scribes: Lidan Mu, Simon Du, Binxuan Huang 2.1 Review A convex optimization problem is of
More informationExploring Domains of Approximation in R 2 : Expository Essay
Exploring Domains of Approximation in R 2 : Expository Essay Nicolay Postarnakevich August 12, 2013 1 Introduction In this paper I explore the concept of the Domains of Best Approximations. These structures
More informationOn the selection of Benders cuts
Mathematical Programming manuscript No. (will be inserted by the editor) On the selection of Benders cuts Matteo Fischetti Domenico Salvagnin Arrigo Zanette Received: date / Revised 23 February 2010 /Accepted:
More informationarxiv: v1 [math.co] 15 Dec 2009
ANOTHER PROOF OF THE FACT THAT POLYHEDRAL CONES ARE FINITELY GENERATED arxiv:092.2927v [math.co] 5 Dec 2009 VOLKER KAIBEL Abstract. In this note, we work out a simple inductive proof showing that every
More information