# Online Supplement for The Robust Network Loading Problem under Hose Demand Uncertainty: Formulation, Polyhedral Analysis, and Computations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Onine Suppement for The Robust Network Loading Probem under Hose Demand Uncertaint: Formuation, Pohedra Anasis, and Computations Aşegü Atın Department of Industria Engineering, TOBB Universit of Economics and Technoog, Sögütözü Ankara, Turke Hande Yaman, Mustafa Ç. Pınar Department of Industria Engineering, Bikent Universit, Ankara, Turke For i W and e E, et u e i be a a unit vector of dimension W E where the entr corresponding to e and i is 1 and other entries are zero. For e E and L, et ve be a unit vector of dimension E L where the entr corresponding to e and is one and other entries are zero. Let M be a ver arge integer. Proof of Proposition 3.1 The dimension of P and P is at most the number of variabes, i.e., ( W + L ) E. Suppose that a points in P (resp. P ) satisf αλ + β = γ. Let e E, L and (λ, ) P (resp. (λ, ) P ). Observe that the vector (λ, + v e) is aso in P (resp. in P ). Hence βe = 0. Let i W and e E. Consider the vectors (λ, ) where λ = i W e E ue i ue i and = L e E Mv and (λ + e ue i, ). Since these two vectors are in P (resp. in P ), we have αi e = 0. As there is no equait that is satisfied b a points in P (resp. in P ), pohedron P (resp. P ) is fu-dimensiona, i.e., its dimension is ( W + L ) E. Next we give two emmas which are often used in the seque. Lemma 1. Let F (resp. F ) be a projection of F (resp. of F ) into the subspace of a subset of variabes and et αλ + β γ be a facet defining inequait for P (resp. for P ). If the coefficients of the variabes that are projected out are zero in the inequait αλ + β γ, then the inequait aso defines a facet of conv(f ) (resp. of conv(f )). Proof. Foows from Coroar 3.6 in Baas and Oosten (1998). Lemma 2. Suppose that a points in P (resp. in P ) such that αλ + β = γ aso satisf αλ + β = γ. Let e E and L be such that βe = 0. Then β e = 0. 1

2 Proof. Let (λ, ) P (resp. in P ) such that αλ + β = γ and = + v e. The vector (λ, ) is aso in P (resp. in P ) and satisfies αλ + β = γ since β e = 0. Hence β e = 0. Proof of Proposition 3.2 B Lemma 1, if inequait σλ σ 0 is facet defining for P (resp. for P ) then it is facet defining for F λ (resp. for F λ ). Suppose that inequait σλ σ 0 is facet defining for F λ (resp. for F λ ) and a points in P (resp. in P ) such that σλ = σ 0 aso satisf αλ + β = γ. Then b Lemma 2, β = 0. As σλ σ 0 is facet defining for F λ (resp. for F λ ) and F λ = P roj λ (F ) (resp. F λ = P roj λ(f )), (α, γ) is a mutipe of (σ, σ 0 ). Thus inequait σλ σ 0 is facet defining for P (resp. for P ). Proof of Theorem 3.1 Let e E be such that δ(s) \ {e} for ever S V such that there exists q Q with s(q) S and t(q) V \ S. It foows from Lemma 1 that if inequait αλ e + β e γ is facet defining for P (resp. for P ) then it defines a facet of P e. Suppose that inequait αλ e + β e γ is facet defining for P e (resp. for P e). Suppose aso that a points in P (resp. in P ) such that αλ e + β e = γ aso satisf αλ + β = γ. B Lemma 2, β e = 0 for a e E \ {e} and L. Let i W and e E \ {e}. Let (λ e, e ) be a vector in P e (resp. in P e) such that αλ e + β e = γ. Consider (λ, ) where λ = i W λe i u e i + i W e E\{e} ue i u e and i = L e E\{e} Mv + e L eve. Since δ(s) \ {e} = for ever S V such that q Q with s(q) S and t(q) V \ S, this vector is in P (resp. in P ) and satisfies αλ e + β e = γ. Consider aso (ˆλ, ŷ) where ˆλ = λ + u e i and ŷ =. This vector is aso in P (resp. in P ) and satisfies αˆλ e + βŷ e = γ. Hence α e i Now we can concude that αλ + β = γ is a mutipe of αλ e + β e = γ as F e = P roj (λ e, e )(F ) (resp. F e = P roj (λ e, e )(F )). Proof of Proposition 3.3 Suppose that S V such that the subgraphs induced b S and V \ S are both connected and B(S) > 0. Let δ(s) F (S) and assume that B(S) = b(s). The proof for the other case is simiar. Define (ˆλ, ŷ) as foows: ˆλ e i = 1 for e E \ δ(s) and i W, ˆλ e i = 0 for e δ(s) and i W \ S, ŷ = L e δ(s) eve + L e E\δ(S) Mv e. Notice that since the subgraphs induced b S and V \ S are both connected, for S V 2

3 such that S S, S V \ S, and δ(s ) \ δ(s). Hence, if (ˆλ, ŷ) satisfies ˆλ e i = 1 i S W (1) e δ(s) ˆλe i L C ŷ e e δ(s) (2) ˆλ e i 0 i S W, e δ(s) (3) then (ˆλ, ŷ) is in F and F. B Farkas Lemma, this sstem is feasibe if and on if α i + C ŷeβ e 0 (4) for a (α, β) such that e δ(s) L α i + β e 0 i S W, e δ(s) β e 0 e δ(s). Since L C ŷe 0 for a e δ(s), we can imit ourseves to (α, β) such that β e = ( ) + α max i for a e δ(s). Now, we can rewrite the eft hand side of (4) as α i + ( ) + C ŷe αi max. (5) e δ(s) L ( ) + α If max i = 0, then αi 0 for a i S W, and (5) is nonnegative. Suppose now that max ( α i ) + = α i Now as e δ(s) > 0 for some i S W. Then (5) is α i α i L C ŷ e b(s) and α i α i α i b(s) = e δ(s) L C ŷe. (6) > 0, (6) is greater than or equa to (α i α i ). As α i α i for a i S W, b (α i α i b b i) 0. So the sstem (1)-(3) awas has a i i soution and F (S) P roj δ(s) (F ) and F (S) P roj δ(s) (F ). As L e δ(s) C e B(S) is a vaid inequait for F and F, P roj δ(s) (F ) F (S) and P roj δ(s) (F ) F (S). Proof of Theorem 3.2 Let S V be such that the subgraphs induced b S and V \ S are 3

4 both connected and suppose that inequait L e δ(s) β ee β 0 is facet defining for P (S) and for each e δ(s) there exists a vector δ(s) F (S) such that L e δ(s) β ee = β 0 and L C > B(S). Assume that B(S) = b(s). The proof for the other case is simiar. e Suppose that a points in P such that L e δ(s) β e e = β 0 aso satisf αλ + β = γ. B Lemma 2, β e = 0 for a e E \ δ(s) and L. Define λ δ(s) S W to be the restriction of the vector λ to its entries with edges in δ(s) and vertices in S W. Let δ(s) F (S) be such that L e δ(s) β ee = β 0 and λδ(s) S W be a soution to the sstem (1)-(3) with δ(s). Let i W and e E \ δ(s). Let λ = i W e E\δ(S) ue i u e + i e δ(s) λe i u e i and = e δ(s) eve + e E\δ(S) Mv e. The vector (λ, ) is in P and satisfies L e δ(s) β e e = β 0. Consider aso (ˆλ, ) where ˆλ = λ + u e. This atter vector is aso in P and satisfies i L e δ(s) β e e = β 0. Hence α e = 0. i Let i W, e δ(s) and δ(s) F (S) be such that L e δ(s) β ee = β 0 and L C > b(s). Let λs W e δ(s) be a soution to the sstem (1)-(3) with δ(s). Consider the vector (λ, ) where λ = i W e E\δ(S) ui e + e δ(s) λi eu i e and = e δ(s) eve + e E\δ(S) Mv e. This vector is in P and satisfies L e δ(s) β e e = β 0. Now et ɛ > 0 be ver sma and consider aso the vector (ˆλ, ) where ˆλ = λ + ɛu e. This vector is aso in P i since L C > b(s) and satisfies e L e δ(s) β e e = β 0. Thus α e = 0. i As F (S) = P roj δ(s) (F ) and as L e δ(s) β ee β 0 is facet defining for P (S), αλ + β = γ is a mutipe of L e δ(s) β ee = β 0. Proof of Proposition 3.5 Suppose that S V such that the subgraphs induced b S and V \ S are both connected, L such that R (S) > 0 and C 1 = 1. The cutset inequait (23) is facet defining for P (S) (Yaman 2007). To show that it aso defines a facet of P, we need to prove, for each e δ(s), the existence of a vector δ(s) F (S) which satisfies the inequait (23) with equait and L C e > B(S). For edge e δ(s), consider the vector δ(s) = ve. This vector satisfies (23) with equait, and we have L C e > B(S) B(S) C since R (S) > 0. Proof of Proposition 3.6 Inequait i S λ e i L C e is a vaid inequait for P. We substitute λ e i = 1 λ e i for i S and obtain b(s) L C e + i S λ e i. Let L and divide the ast inequait with C. This ieds b(s) C L C C e + i S C λe i. Let 4

5 L u = { L : g(, ) > r (S)} and L d = L \ L u. The ast inequait impies b(s) C C C e + g(, ) C e + C C e + λ e C i. L d L d L u i S Here C L d C e + C L u C e is integer and g(, ) L d C e + i S C λe i is a nonnegative rea. Hence the MIR inequait g(, ) C e + ( b(s) ) λ e C i r (S) Ld C e Lu C C C C C e L d i S is vaid for P. Mutiping both sides with C and organizing the terms, we obtain ( ) C g(, ) + r (S) C e + C r (S) C e + λ e b(s) i r (S) C L u i S L d which is the same as ( C r (S) + min { g(, ), r (S) }) C e + i S L λ e b(s) i r (S) C. Substituting λ e i = 1 λ e i for i S ieds inequait (24). Experimenta Resuts Beow, we provide detaied resuts of our experimenta tests. The entries in the tabes contain the foowing information: the instance characteristics, i.e., the name of the instance as we as the numbers of nodes V, inks E, and terminas W (the number of commodities is W ( W 1)), faciit capacities, i.e., C for the singe faciit case, C 1 and C 2 for the two-faciit case, the best upper bound z cp, the gap at termination g cp, the soution time t cp in CPU seconds, and the number of B&C nodes # cp for Cpex, the best upper bound z B&C, the gap at termination g B&C, the soution time t B&C in CPU seconds, and the number of B&C nodes # B&C for our B&C agorithm, where gap at termination is the gap between the best upper and ower bounds at termination defined as (upper ower)/ower 100 for each method. Given the difficut of the probem, it is not surprising that some of the instances are not soved to optimait within two hours 5

6 time imit. We indicate such cases with a in the corresponding z coumn. Moreover, the gaps at termination (g) for them are provided under the corresponding time (t) coumns in parenthesis. Besides, we mark those cases for which soving the NLP hose GD or NLP hose modes did not ied even a feasibe soution within the time imit with NoI whereas MA indicates a termination due to insufficient memor aocation. The gap at termination for such cases are noted as INF in a resut tabes. Instance ( V, E, W,C) z cp t cp (g cp ) # cp z B&C t B&C (g B&C ) # B&C metro (11,42,5,24) nsf1b (14,21,10,24) 97, , at-cep1 (15,22,6,24) 51, , pacbe (15,21,7,24) 11, , bhv6c (27,39,15,24) 840, , 251 (1.22%) 37,702 bhvdc (29,36,13,24) 1.1e e pdh (11,34,6,480) 1.1e e poska (12,18,12,155) 44, 253 (1.16%) 10,793 44, 287 (0.42%) 46,133 poska (12,18,12,100) ,719 dfn (11,47,11,155) 52, 380 (6.34%) , 416 (3.74%) 6624 nework (16,49,16,1000) 1.499e + 6 (56.74%) e + 6 (45.4%) 21 france (25,45,14,2500) 17, 400 (3.61%) , 400 (5.45%) 304 n-cep2 (16,49,9,24) (3.8%) (2.52%) 7998 atanta (15,22,15,1000) 4.7e + 8 (0.2%) e + 8 (0.54%) 19,135 tai (24,51,19,504k) 1.6e + 12 (99.7%) e + 7 (19.94%) 13 janos (26,42,26,64) 1.29e + 9 (99.7%) e nobe-eu (28,41,28,20) 1.47e + 10 (99.7%) e + 6 (2.01%) 8 sun (27,51,24,40) 6.3e + 10 (99.7%) (20.5%) 6 Tabe 1: Resuts for the singe faciit probem. In Tabe 1, we provide the resuts for the singe faciit case. In Tabe 2, for each of the four settings, F, F &D, F &R, and a, we report the percentage gap between the optima vaue and the ower bound at the root node under the gap coumns as we as the corresponding soution times under the t coumns. Fina, in Tabe 3, we provide the detaied resuts for the two-faciit case. References Baas, E., Oosten, M On the dimension of the projected pohedra. Discrete App. Math

7 Instance t F gap F t F +D gap F +D t F +R gap F +R t a gap a metro % % % % nsf1b % % % % at-cep % % % % pacbe % % % % bhvdc % % % % pdh % % % % Tabe 2: Resuts with different cuts for the singe faciit case. Instance ( V, E, W, C 1, C 2 ) z cp t cp # cp z B&C t B&C # B&C metro (11, 42, 5, 1, 24) nsf1b (14, 21, 10, 1, 24) 96, , at-cep1 (15, 22, 6, 1, 24) 50, , pacbe (15, 21, 7, 1, 24) 11, , bhv6c (27, 39, 15, 1, 24) 826, ,548* (0.5%) 36,108 bhvdc (29, 36, 13, 1, 24) 1.09e e pdh (11, 34, 6, 30, 480) 8.18e e poska (12, 18, 12, 155, 622) 34,006* (2.18%) ,348* (3.11%) 50,403 dfn (11, 47, 11, 155, 622) 44,352* (21.37%) ,316* (12.43%) 21,736 nework (16, 49, 16, 1k, 4k) NoI MA-INF 2 1.6e+6* (70.4%) 30 atanta (15, 22, 15, 1k, 4k) 1.376e+8* (1.15%) e+8* (1.44%) 14,598 tai (24, 51, 19, 504k, 1008k) NoI INF e+7* (10.48%) 8702 nobe-eu (28, 41, 28, 20, 40) NoI INF e+6* (1.6%) 690 pioro (40, 89, 40, 155, 622) NoI INF e+6 (2.47%) 2 norwa (27, 51, 27, 1k, 4k) NoI INF 2 NoI INF 1 cost266 (37, 57, 37, 7560, 30240) NoI INF e+7* (29.75%) 2 gui39 (39, 86, 39, 160, 320) NoI INF 2 NoI INF 1 Tabe 3: Resuts for the two faciit probem. 7

### Simultaneous Routing and Power Allocation in CDMA Wireless Data Networks

Simutaneous Routing and Power Aocation in CDMA Wireess Data Networks Mikae Johansson *,LinXiao and Stephen Boyd * Department of Signas, Sensors and Systems Roya Institute of Technoogy, SE 00 Stockhom,

α α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =

### Secure Network Coding with a Cost Criterion

Secure Network Coding with a Cost Criterion Jianong Tan, Murie Médard Laboratory for Information and Decision Systems Massachusetts Institute of Technoogy Cambridge, MA 0239, USA E-mai: {jianong, medard}@mit.edu

### Proximal mapping via network optimization

L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

### Lecture 4: BK inequality 27th August and 6th September, 2007

CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

### ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007

ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007 FRANÇOIS SIMENHAUS Université Paris 7, Mathématiques, case 7012, 2, pace Jussieu, 75251 Paris, France

### P. Jeyanthi and N. Angel Benseera

Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

### Polynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will

### Finance 360 Problem Set #6 Solutions

Finance 360 Probem Set #6 Soutions 1) Suppose that you are the manager of an opera house. You have a constant margina cost of production equa to \$50 (i.e. each additiona person in the theatre raises your

### Face Hallucination and Recognition

Face Haucination and Recognition Xiaogang Wang and Xiaoou Tang Department of Information Engineering, The Chinese University of Hong Kong {xgwang1, xtang}@ie.cuhk.edu.hk http://mmab.ie.cuhk.edu.hk Abstract.

### Normalization of Database Tables. Functional Dependency. Examples of Functional Dependencies: So Now what is Normalization? Transitive Dependencies

ISM 602 Dr. Hamid Nemati Objectives The idea Dependencies Attributes and Design Understand concepts normaization (Higher-Leve Norma Forms) Learn how to normaize tabes Understand normaization and database

Proc of IEEE Internationa Conference on Robotics and Automation, Karsruhe, Germany, 013 Muti-Robot Tas Scheduing Yu Zhang and Lynne E Parer Abstract The scheduing probem has been studied extensivey in

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### Logics preserving degrees of truth from varieties of residuated lattices

Corrigendum Logics preserving degrees of truth from varieties of residuated attices FÉLIX BOU and FRANCESC ESTEVA, Artificia Inteigence Research Institute IIIA - CSIC), Beaterra, Spain. E-mai: fbou@iiia.csic.es;

### Risk Margin for a Non-Life Insurance Run-Off

Risk Margin for a Non-Life Insurance Run-Off Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas February 2, 2011 Abstract For sovency purposes insurance companies need to cacuate so-caed best-estimate

### Several Views of Support Vector Machines

Several Views of Support Vector Machines Ryan M. Rifkin Honda Research Institute USA, Inc. Human Intention Understanding Group 2007 Tikhonov Regularization We are considering algorithms of the form min

### Key Features of Life Insurance

Key Features of Life Insurance Life Insurance Key Features The Financia Conduct Authority is a financia services reguator. It requires us, Aviva, to give you this important information to hep you to decide

### IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 12, DECEMBER 2013 1

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 12, DECEMBER 2013 1 Scaabe Muti-Cass Traffic Management in Data Center Backbone Networks Amitabha Ghosh, Sangtae Ha, Edward Crabbe, and Jennifer

### A Branch-and-Price Algorithm for Parallel Machine Scheduling with Time Windows and Job Priorities

A Branch-and-Price Agorithm for Parae Machine Scheduing with Time Windows and Job Priorities Jonathan F. Bard, 1 Siwate Rojanasoonthon 2 1 Graduate Program in Operations Research and Industria Engineering,

### On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

### 500 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013

500 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 3, NO. 3, MARCH 203 Cognitive Radio Network Duaity Agorithms for Utiity Maximization Liang Zheng Chee Wei Tan, Senior Member, IEEE Abstract We

### THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

### The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method

The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem

### Negative Binomial. Paul Johnson and Andrea Vieux. June 10, 2013

Negative Binomial Paul Johnson and Andrea Vieux June, 23 Introduction The Negative Binomial is discrete probability distribution, one that can be used for counts or other integervalued variables. It gives

### Split Nonthreshold Laplacian Integral Graphs

Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br

### Cyber-Security Analysis of State Estimators in Power Systems

Cyber-Security Analysis of State Estimators in Electric Power Systems André Teixeira 1, Saurabh Amin 2, Henrik Sandberg 1, Karl H. Johansson 1, and Shankar Sastry 2 ACCESS Linnaeus Centre, KTH-Royal Institute

### Minimally Infeasible Set Partitioning Problems with Balanced Constraints

Minimally Infeasible Set Partitioning Problems with alanced Constraints Michele Conforti, Marco Di Summa, Giacomo Zambelli January, 2005 Revised February, 2006 Abstract We study properties of systems of

### SAT Math Must-Know Facts & Formulas

SAT Mat Must-Know Facts & Formuas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas

### Web-based Supplementary Materials for Bayesian Effect Estimation. Accounting for Adjustment Uncertainty by Chi Wang, Giovanni

1 Web-based Supplementary Materials for Bayesian Effect Estimation Accounting for Adjustment Uncertainty by Chi Wang, Giovanni Parmigiani, and Francesca Dominici In Web Appendix A, we provide detailed

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### Minimizing the Total Weighted Completion Time of Coflows in Datacenter Networks

Minimizing the Tota Weighted Competion Time of Cofows in Datacenter Networks Zhen Qiu Ciff Stein and Yuan Zhong ABSTRACT Communications in datacenter jobs (such as the shuffe operations in MapReduce appications

### Fast Robust Hashing. ) [7] will be re-mapped (and therefore discarded), due to the load-balancing property of hashing.

Fast Robust Hashing Manue Urueña, David Larrabeiti and Pabo Serrano Universidad Caros III de Madrid E-89 Leganés (Madrid), Spain Emai: {muruenya,darra,pabo}@it.uc3m.es Abstract As statefu fow-aware services

### Pacific Journal of Mathematics

Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000

### Pricing and Revenue Sharing Strategies for Internet Service Providers

Pricing and Revenue Sharing Strategies for Internet Service Providers Linhai He and Jean Warand Department of Eectrica Engineering and Computer Sciences University of Caifornia at Berkeey {inhai,wr}@eecs.berkeey.edu

### Bargaining Solutions in a Social Network

Bargaining Solutions in a Social Network Tanmoy Chakraborty and Michael Kearns Department of Computer and Information Science University of Pennsylvania Abstract. We study the concept of bargaining solutions,

### Statistical Machine Learning

Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

### Load Balancing in Distributed Web Server Systems with Partial Document Replication *

Load Baancing in Distributed Web Server Systems with Partia Document Repication * Ling Zhuo Cho-Li Wang Francis C. M. Lau Department of Computer Science and Information Systems The University of Hong Kong

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### Random graphs with a given degree sequence

Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

### 5 Directed acyclic graphs

5 Directed acyclic graphs (5.1) Introduction In many statistical studies we have prior knowledge about a temporal or causal ordering of the variables. In this chapter we will use directed graphs to incorporate

### SAT Math Facts & Formulas

Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### Discounted Cash Flow Analysis (aka Engineering Economy)

Discounted Cash Fow Anaysis (aka Engineering Economy) Objective: To provide economic comparison of benefits and costs that occur over time Assumptions: Future benefits and costs can be predicted A Benefits,

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### A train dispatching model based on fuzzy passenger demand forecasting during holidays

Journa of Industria Engineering and Management JIEM, 2013 6(1):320-335 Onine ISSN: 2013-0953 Print ISSN: 2013-8423 http://dx.doi.org/10.3926/jiem.699 A train dispatching mode based on fuzzy passenger demand

### Leakage detection in water pipe networks using a Bayesian probabilistic framework

Probabiistic Engineering Mechanics 18 (2003) 315 327 www.esevier.com/ocate/probengmech Leakage detection in water pipe networks using a Bayesian probabiistic framework Z. Pouakis, D. Vaougeorgis, C. Papadimitriou*

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

### Steiner Tree Approximation via IRR. Randomized Rounding

Steiner Tree Approximation via Iterative Randomized Rounding Graduate Program in Logic, Algorithms and Computation μπλ Network Algorithms and Complexity June 18, 2013 Overview 1 Introduction Scope Related

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### Documents de Travail du Centre d Economie de la Sorbonne

Documents de Travai du Centre d Economie de a Sorbonne Rationaizabiity and Efficiency in an Asymmetric Cournot Oigopoy Gabrie DESGRANGES, Stéphane GAUTHIER 2014.28 Maison des Sciences Économiques, 106-112

### (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties

Lecture 1 Convex Sets (Basic definitions and properties; Separation theorems; Characterizations) 1.1 Definition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### 0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup

456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

### On the Unique Games Conjecture

On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary

### Dynamic TCP Acknowledgement: Penalizing Long Delays

Dynamic TCP Acknowledgement: Penalizing Long Delays Karousatou Christina Network Algorithms June 8, 2010 Karousatou Christina (Network Algorithms) Dynamic TCP Acknowledgement June 8, 2010 1 / 63 Layout

### Capacity of Multi-service Cellular Networks with Transmission-Rate Control: A Queueing Analysis

Capacity of Muti-service Ceuar Networs with Transmission-Rate Contro: A Queueing Anaysis Eitan Atman INRIA, BP93, 2004 Route des Lucioes, 06902 Sophia-Antipois, France aso CESIMO, Facutad de Ingeniería,

### A Simple Model of Price Dispersion *

Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion

### POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS

POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS N. ROBIDOUX Abstract. We show that, given a histogram with n bins possibly non-contiguous or consisting

### Optimal Online Assignment with Forecasts

Optimal Online Assignment with Forecasts Erik Vee Yahoo! Research Santa Clara, CA erikvee@yahoo-inc.com Sergei Vassilvitskii Yahoo! Research New York, NY sergei@yahoo-inc.com Jayavel Shanmugasundaram Yahoo!

### The Equivalence of Linear Programs and Zero-Sum Games

The Equivalence of Linear Programs and Zero-Sum Games Ilan Adler, IEOR Dep, UC Berkeley adler@ieor.berkeley.edu Abstract In 1951, Dantzig showed the equivalence of linear programming problems and two-person

### Risk Margin for a Non-Life Insurance Run-Off

Risk Margin for a Non-Life Insurance Run-Off Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas August 15, 2011 Abstract For sovency purposes insurance companies need to cacuate so-caed best-estimate reserves

### Multiplier-accelerator Models on Time Scales

International Journal of Statistics and Economics; [Formerly known as the Bulletin of Statistics & Economics ISSN 0973-7022)]; ISSN 0975-556X,; Spring 2010, Volume 4, Number S10; Copyright 2010 by CESER

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### WIRELESS Mesh Networks (WMNs) have recently attracted

3968 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 8, AUGUST 23 A New MPLS-Based Forwarding Paradigm for Muti-Radio Wireess Mesh Networks Stefano Avaone and Giovanni Di Stasi Abstract Routing

### Powers of Two in Generalized Fibonacci Sequences

Revista Colombiana de Matemáticas Volumen 462012)1, páginas 67-79 Powers of Two in Generalized Fibonacci Sequences Potencias de dos en sucesiones generalizadas de Fibonacci Jhon J. Bravo 1,a,B, Florian

### 9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

### Regular Induced Subgraphs of a Random Graph*

Regular Induced Subgraphs of a Random Graph* Michael Krivelevich, 1 Benny Sudaov, Nicholas Wormald 3 1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; e-mail: rivelev@post.tau.ac.il

### The Envelope Theorem 1

John Nachbar Washington University April 2, 2015 1 Introduction. The Envelope Theorem 1 The Envelope theorem is a corollary of the Karush-Kuhn-Tucker theorem (KKT) that characterizes changes in the value

### A Similarity Search Scheme over Encrypted Cloud Images based on Secure Transformation

A Simiarity Search Scheme over Encrypted Coud Images based on Secure Transormation Zhihua Xia, Yi Zhu, Xingming Sun, and Jin Wang Jiangsu Engineering Center o Network Monitoring, Nanjing University o Inormation

### Date: April 12, 2001. Contents

2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

### On an anti-ramsey type result

On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### LECTURE 15: AMERICAN OPTIONS

LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These

### Optimal File Sharing in Distributed Networks

Optimal File Sharing in Distributed Networks Moni Naor Ron M. Roth Abstract The following file distribution problem is considered: Given a network of processors represented by an undirected graph G = (V,

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### On Minimal Valid Inequalities for Mixed Integer Conic Programs

On Minimal Valid Inequalities for Mixed Integer Conic Programs Fatma Kılınç Karzan June 27, 2013 Abstract We study mixed integer conic sets involving a general regular (closed, convex, full dimensional,

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### TOPOLOGICAL DESIGN OF MULTIPLE VPNS OVER MPLS NETWORK Anotai Srikitja David Tipper

TOPOLOGICAL DESIGN OF MULTIPLE VPNS OVER MPLS NETWORK Anotai Sriitja David Tier Det. of Information Science and Teecommunications University of Pittsburgh N. Beefied Avenue, Pittsburgh, PA 60 ABSTRACT

### Cooperative Content Distribution and Traffic Engineering in an ISP Network

Cooperative Content Distribution and Traffic Engineering in an ISP Network Wenjie Jiang, Rui Zhang-Shen, Jennifer Rexford, Mung Chiang Department of Computer Science, and Department of Eectrica Engineering

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

### Adverse Selection and the Market for Health Insurance in the U.S. James Marton

Preliminary and Incomplete Please do not Quote Adverse Selection and the Market for Health Insurance in the U.S. James Marton Washington University, Department of Economics Date: 4/24/01 Abstract Several

### Link Dimensioning and LSP Optimization for MPLS Networks Supporting DiffServ EF and BE traffic classes

Link Dimensioning and LSP Optimization for MPLS Networks Supporting DiffServ EF and BE traffic casses Kehang Wu and Dougas S. Reeves Departments of Eectrica and Computer Engineering and Computer Science

### The Whys of the LOIS: Credit Risk and Refinancing Rate Volatility

The Whys of the LOIS: Credit Risk and Refinancing Rate Voatiity Stéphane Crépey 1, and Raphaë Douady 2 1 Laboratoire Anayse et Probabiités Université d Évry Va d Essonne 9137 Évry, France 2 Centre d économie

### A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,

### Betting Strategies, Market Selection, and the Wisdom of Crowds

Betting Strategies, Market Seection, and the Wisdom of Crowds Wiemien Kets Northwestern University w-kets@keogg.northwestern.edu David M. Pennock Microsoft Research New York City dpennock@microsoft.com

### Notes on the Negative Binomial Distribution

Notes on the Negative Binomial Distribution John D. Cook October 28, 2009 Abstract These notes give several properties of the negative binomial distribution. 1. Parameterizations 2. The connection between

### A Latent Variable Pairwise Classification Model of a Clustering Ensemble

A atent Variabe Pairwise Cassification Mode of a Custering Ensembe Vadimir Berikov Soboev Institute of mathematics, Novosibirsk State University, Russia berikov@math.nsc.ru http://www.math.nsc.ru Abstract.

### Market Design & Analysis for a P2P Backup System

Market Design & Anaysis for a P2P Backup System Sven Seuken Schoo of Engineering & Appied Sciences Harvard University, Cambridge, MA seuken@eecs.harvard.edu Denis Chares, Max Chickering, Sidd Puri Microsoft

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong