On closed-form solutions of a resource allocation problem in parallel funding of R&D projects

Save this PDF as:

Size: px
Start display at page:

Download "On closed-form solutions of a resource allocation problem in parallel funding of R&D projects"

Transcription

2 230 U. Gurler et al. / Operations Research Letters 27 (2000) [4] tackle a similar problem using game theoretic approaches. In the present paper, we identify conditions under which the nonlinear and nonconvex optimization problem introduced by Gerchak and Kilgour [2] admits closed-form solutions. For a general and detailed exposition of resource allocation problems the interested reader is directed to the book [6]. 2. The optimization problem and main results Gerchak and Kilgour [2] modeled the resource allocation problem as follows. They assume that (1) the future achievement of a research team is a continuous nonnegative random variable, (2) the research teams are independent, (3) the achievements of the research teams have identical probability distributions, given that their funding is done on an equal basis, and (4) the achievements of the funded teams are exponentially distributed. Suppose that the total available budget is B, and there are M potential research activities. For 1; 2;:::;M, let B be the budget allocated to activity and n be the number of parallel research teams to work on activity, each of which receives equal funding which amounts to B n. The achievement of each team within activity has an identical distribution function given by F (x; B n ), where B n is related to the parameter of the distribution. The optimization problem consists in imizing a weighted sum of the probabilities that the most successful team within each activity exceeds a specic threshold value. Since the probability that the best team in activity exceeds the threshold T can be written as 1 [F (T ; B n )] n, the optimization problem is formulated as follows (see [2]): B 1;:::; B M ;n 1;:::;n M B B; p {1 [F (T ; B n )] n }; where p denotes the positive weight attached to activity. Gerchak and Kilgour [2] set the parameter of the exponential distribution to L(nB), where (0; 1] reects the sensitivity of a research team to the budget allocation, and L is an appropriate constant. The case 1 indicates a high sensitivity where larger budget allocations would induce higher achievements. With multiple activities this problem translates into B 1:::; B M ;n 1;:::;n M B B: p {1 [1 e L(nB) T ] n }; Since the threshold values T s and the parameters L s are xed, we can absorb L into T in the analysis without loss of generality. Therefore, we assume L to be equal to one. We also suppose that the research teams achievement is very sensitive to resource allocation. In other words, we set 1;; 2;:::;M. Therefore, the goal of imizing the weighted sum of probabilities that the most successful team exceeds an activity-specic threshold value T translates into min B;n M f(b; n; p) p [1 e (nb)t ] n ; B B; where B(B 1 ;:::;B M ); n and p are dened similarly. Before proceeding to the results, we present below an example with two activities. Let B10; T 1 T 2 8. We plot in Fig. 1 the obective function for several choices of (n 1 ;n 2 ) as a function of B 1, when L 1 L 2 1. We observe that the curve corresponding to n 1 n 2 1 lies below all the others. This observation is formally proved in Theorem 1. Furthermore, the function for n 1 n 2 1 is neither convex nor concave but has three local minima, two of which are both global minima, namely B 1 0, and B In the rest of the paper, we use 1 to indicate a vector with all elements equal to one. For ease of notation, we dene r T B which can be interpreted as a scaled measure of the success threshold for activity per unit budget. The higher this ratio gets, the smaller is the probability that an activity is considered successful. Hence, larger values of r indicate higher levels of ambition for that activity.

3 U. Gurler et al. / Operations Research Letters 27 (2000) Fig. 1. The plot of the function f on a two activity problem for dierent values of n 1 ;n 2. Theorem 1. If r ln 2 for 1;:::;M;f(B; n; p) is minimized at n 1, for all B. Proof. We rst show that when r ln 2 for 1; 2;:::;M; f(b; 1; p) is smaller than or equal to f(b; n; p) for n 1 (that is, each component is greater than or equal to one), for all values of B. Since f(b; 1; p) p i (1 e TiBi ); i then, for any n 1, the dierence f(b; n; p) f(b; 1; p) is given by p i [(1 e nitibi ) ni (1 e TiBi )]: i1 Note that if n i 1 for some i, the dierences corresponding to those i values will be zero and they will not contribute to the above sum. Letting T i x i ; B i and omitting p i which is a positive multiplicative constant, the ith term in the previous expression is written as (1 e nixi ) ni (1 e xi ): This expression is zero when n i 1, and otherwise its derivative with respect to n i is { } ni xe nixi (1 e nixi ) ni +ln(1 e nixi ) : 1 e nixi This derivative vanishes when n i x i ln 2, it is negative if n i x i ln 2, and positive if n i x i ln 2. Then, (1 e nixi ) ni (1 e xi ) is always nonnegative for all values of n i 1 if x i ln 2. Thus, for all B;f(B; 1; p)6f(b; n; p) for n 1 if T i ln 2: B i Since 0 B i B, a sucient condition is given as r i ln 2: Now suppose for some i 1 ;i 2 ;:::;i k ;n i 0 and the rest of the n s are 1. For simplicity, let i 1 M k + 1; i 2 M k +2;:::;i k M, so that the rst M k elements of n are 1 and the rest are zero and denote the corresponding n vector by n k. Then f(b; n k ; p) f(b; 1; p) ( p i (1 e n it ib i ) ni (1 e TiBi ) ) i1

4 232 U. Gurler et al. / Operations Research Letters 27 (2000) p i (1 e TiBi )+ M k i1 im k+1 p i e TiBi 0: im k+1 p i p i (1 e TiBi ) i1 Hence, a sucient condition for the theorem to hold is that r ln 2 min r ln 2 We conclude from the above theorem that when r ln 2 for 1;:::;M, at most one research team in each activity can be funded. This result which may not seem intuitive at rst sight can be attributed to the form of the obective function since the success of a team is measured by the tail probability of an exponential distribution. If the desired achievement rate per unit budget T B is high the policy avoids parallel funding in order to maintain the success level expressed by this small probability. Hence, other obective functions can result in dierent policies which may encourage more parallel funding. The following theorem complements the above result by providing the optimal budget allocation. De- ne c p for 1;:::;M where a r ln 2. Theorem 2. Let c {c } and J{ 166M and c c }. Then for r ln 2 for 1;:::;M; the following holds: 1. If J is a singleton; then f(b; 1; p) is minimized at B i B; and B 0; for all i; for which c i c. 2. If J{i 1 ;i 2 ;:::;i r } and if p i1 p i2 p ir p; then f(b; 1; p) is minimized at B ik B; and B 0; for all i k ; for k 1; 2;:::;r. Furthermore; if it also holds that c p (1r) 1r 1 ; then f(b; 1; p) is also minimized at B ik Br, for k 1; 2;:::;r and B 0; for all i k. Proof. The original problem of minimizing f(b; 1; p) can be transformed into B 1;:::; B M p e TB (1) B B: Since r ln 2, we can write r a ln 2 for some a 1 which implies T Ba ln 2 and (1) can be written as X f(p; X) X ( ) 1x 1 p ; (2) where X (x 1 ;x 2 ;:::;x M ) and 06x i 61. First, observe that lim f(p; X) p i c i : x i 1 x 0 i 2 ai Then it is sucient to show that the imum in (2)6 i {c i } c. To see this we write f(p; X) p ( 1 ) 1x p ( ) (1 x)x 1 M ( ) (1 x)x 1 6 c (3) 6c M ( ) (1 x)x 1 : (4) 2 Now using the method of Lagrange multipliers we can easily see that the RHS of (4) is imized at x 1M for ; 2;:::;M and at this point its value is c M(0:5) M 1 6c for all values of M 1. This proves part 1. For part 2, suppose c c i1 c i2 c ir and p i1 p i2 p ir p. These imply also that a i1 a i2 a ir a. For ease of notation let i 1 1;i 2 2;:::;i 2 r, for r6m. It is sucient to check whether f(p 1 ; X 1 )f(p 1 ; X 2 ), with p 1 (p;p;:::;p;p r+1 ;:::;p M ); X 1 (1r; 1r;:::;1r; 0;:::;0); X 2 (0;:::;1;:::;0;:::;0);

5 U. Gurler et al. / Operations Research Letters 27 (2000) where in X 2, the single entry 1 occurs anywhere in the rst r entries. Then r ( ) r 1 f(p 1 ; X 1 ) p ( ) r 1 pr 2 a ; (5) ( ) 1 f(p 1 ; X 2 )p 2 a : (6) Then we see that (5) and (6) are equal if 12 a c p (1r) 1(r 1). Having established in Theorem 1 that only a single team within each activity should be funded when T B ln 2; 1;:::;M, Theorem 2 states the following: 1. If there is a single activity that dominates in terms of its weight (c ) then that activity receives all the available budget. 2. If several activities simultaneously dominate in terms of their weights, there are multiple optima where one of the dominating activities gets all the funding. Furthermore, if a certain condition on the imum weight c holds, there is yet another optimum funding scheme where all activities equally share the available budget. Note that part 2 of Theorem 2 always holds without the condition on c p for M 2. Also, it is enough to compute the c i values and select the imum to nd the activity that receives all the funding. As a simple illustration of the above theorem, consider the example in Fig. 1. Here p 1 p 2 1, and we compute c 1 c 2 0: Hence, clause 2 of the theorem applies and we have two global minima, (B 1 10;B 2 0) and (B 1 0;B 2 10) Maximizing the expected number of teams achieving a threshold In the foregoing discussion, we considered the probability that the most successful team within each activity exceeds a threshold and a weighted sum of these probabilities is aimed to be imized. As an alternative obective, Gerchak [3] considers the problem of imizing the expected number of teams attaining a pre-specied threshold (Problem 3 in the above reference). Since, it is assumed that the teams within each activity work independently with an identical achievement distribution, the number of teams achieving a certain threshold will have a binomial distribution with parameters n i and p i [1 F (T ; B n )], where F ( ) corresponds to the achievement distribution of a team in activity ; 1;:::;M. If activities have dierent priorities reected by p s as before, the obective function for M 2; i 1;i1; 2 becomes [pn 1 e T1(n1B1) +(1 p)n 2 e T2(n2(1 B1)) ] B 1;n 1;n 2 f(b 1 ;n 1 ;n 2 ); B 1;n 1;n 2 where B is taken is to be one without loss of generality. We briey indicate below that the results obtained above also extend to this problem. In particular, we observe that f(b 1 ;n 1 ;n 2 ) f(b 1 ;n 1 ; 0) for all n 1 1 and f(b 1 ;n 1 ;n 2 ) f(b 1 ; 0;n 2 ) for all n 2 1. Further, f(b 1 ; 1; 1) f(b 1 ; 1;n 2 ) (1 p)[e T2(1 B1) n 2 e T2n2(1 B1) ]: For n 2 2, the above expression is always positive if ln 26T 2, which follows from the arguments presented in the previous section. Similarly, we can show that f(b 1 ; 1; 1) f(b 1 ;n 1 ; 1) is always nonnegative if ln 26T 1 holds. Therefore, the sucient condition for (n 1 ;n 2 )(1; 1) to be the optimal number of teams turns out to be that ln 26T i ;i1; 2, similar to the previous problem. The problem of nding the optimal value of B 1 that imizes f(b 1 ; 1; 1) also reduces to the previous problem since {pe T1B1 +(1 p)e T1(1 B1) } B 1 min{p[1 e T1(1B1) ] B 1 +(1 p)[1 e T1(1(1 B1)) ]}: This seemingly counterintuitive result can again be explained by the tail probability expression involved in the obective function. On the other hand, the sucient conditions of the above results may be stronger than necessary. That is, if this condition is not satised the optimal policy may still advocate funding a single team as exemplied in [3, Table 8].

6 234 U. Gurler et al. / Operations Research Letters 27 (2000) Conclusion In this paper, we obtained a sucient condition under which a nonlinear resource allocation problem introduced by Gerchak and Kilgour [2] admits closed-form solutions. The cases where the sucient condition fails to hold are deemed less interesting in applications by Gerchak and Kilgour [2]. Closed-form solutions in such cases are not possible as observed by Gerchak and Kilgour [2] through numerical experimentation since the optimal decision diers from one case to another. [2] Y. Gerchak, D.M. Kilgour, Optimal parallel funding of R&D proects, IIE Trans. 31 (2) (1999) [3] Y. Gerchak, On allocating R&D budgets among and within proects, R&D Management 28 (4) (1998) [4] Y. Gerchak, Parlar, Allocating resources to R&D proects in a competitive environment, IIE Trans. 31 (1999) [5] Y. Gerchak, Budget Allocation Among R&D teams with interdependent uncertain achievement levels and a common goal. Department of Management Sciences, University of Waterloo, [6] T. Ibaraki, N. Katoh, Resource Allocation Problems: Algorithmic Approaches, MIT Press, Cambridge, MA, May References [1] W.J. Abernathy, R.S. Rosenbloom, Parallel strategies in development proects, Management Sci. 15 (10) (1969)

constraint. Let us penalize ourselves for making the constraint too big. We end up with a

Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the

Data analysis in supersaturated designs

Statistics & Probability Letters 59 (2002) 35 44 Data analysis in supersaturated designs Runze Li a;b;, Dennis K.J. Lin a;b a Department of Statistics, The Pennsylvania State University, University Park,

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

Introduction to Logistic Regression

OpenStax-CNX module: m42090 1 Introduction to Logistic Regression Dan Calderon This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 Abstract Gives introduction

Prot Maximization and Cost Minimization

Simon Fraser University Prof. Karaivanov Department of Economics Econ 0 COST MINIMIZATION Prot Maximization and Cost Minimization Remember that the rm's problem is maximizing prots by choosing the optimal

4.1 Introduction - Online Learning Model

Computational Learning Foundations Fall semester, 2010 Lecture 4: November 7, 2010 Lecturer: Yishay Mansour Scribes: Elad Liebman, Yuval Rochman & Allon Wagner 1 4.1 Introduction - Online Learning Model

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

Portfolio selection based on upper and lower exponential possibility distributions

European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.

Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =

Topology-based network security

Topology-based network security Tiit Pikma Supervised by Vitaly Skachek Research Seminar in Cryptography University of Tartu, Spring 2013 1 Introduction In both wired and wireless networks, there is the

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

Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

Chapter 7. Sealed-bid Auctions

Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

Fuzzy regression model with fuzzy input and output data for manpower forecasting

Fuzzy Sets and Systems 9 (200) 205 23 www.elsevier.com/locate/fss Fuzzy regression model with fuzzy input and output data for manpower forecasting Hong Tau Lee, Sheu Hua Chen Department of Industrial Engineering

Abstract. this constraint. A series of examples is presented which demonstrate greater protability

Flexible Retrot Design of Multiproduct Batch Plants R. Fletcher 1, J.A.J. Hall 2, and W.R. Johns 3 Abstract We treat the addition of new equipment to an existing multiproduct batch plant for which new

Online Ad Auctions. By Hal R. Varian. Draft: February 16, 2009

Online Ad Auctions By Hal R. Varian Draft: February 16, 2009 I describe how search engines sell ad space using an auction. I analyze advertiser behavior in this context using elementary price theory and

Game Theory: Supermodular Games 1

Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution

A note on optimal policies for a periodic inventory system with emergency orders

Computers & Operations Research 28 (2001) 93}103 A note on optimal policies for a periodic inventory system with emergency orders Chi Chiang* Department of Management Science, National Chiao Tung University,

Procedia Computer Science 00 (2012) 1 21. Trieu Minh Nhut Le, Jinli Cao, and Zhen He. trieule@sgu.edu.vn, j.cao@latrobe.edu.au, z.he@latrobe.edu.

Procedia Computer Science 00 (2012) 1 21 Procedia Computer Science Top-k best probability queries and semantics ranking properties on probabilistic databases Trieu Minh Nhut Le, Jinli Cao, and Zhen He

Single-Period Balancing of Pay Per-Click and Pay-Per-View Online Display Advertisements Changhyun Kwon Department of Industrial and Systems Engineering University at Buffalo, the State University of New

Decision-making with the AHP: Why is the principal eigenvector necessary

European Journal of Operational Research 145 (2003) 85 91 Decision Aiding Decision-making with the AHP: Why is the principal eigenvector necessary Thomas L. Saaty * University of Pittsburgh, Pittsburgh,

Using Generalized Forecasts for Online Currency Conversion

Using Generalized Forecasts for Online Currency Conversion Kazuo Iwama and Kouki Yonezawa School of Informatics Kyoto University Kyoto 606-8501, Japan {iwama,yonezawa}@kuis.kyoto-u.ac.jp Abstract. El-Yaniv

Name. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.

Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price

2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

Algorithmic Mechanism Design for Load Balancing in Distributed Systems

In Proc. of the 4th IEEE International Conference on Cluster Computing (CLUSTER 2002), September 24 26, 2002, Chicago, Illinois, USA, IEEE Computer Society Press, pp. 445 450. Algorithmic Mechanism Design

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

Offline sorting buffers on Line

Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

Availability of a system with gamma life and exponential repair time under a perfect repair policy

Statistics & Probability Letters 43 (1999) 189 196 Availability of a system with gamma life and exponential repair time under a perfect repair policy Jyotirmoy Sarkar, Gopal Chaudhuri 1 Department of Mathematical

On the Interaction and Competition among Internet Service Providers

On the Interaction and Competition among Internet Service Providers Sam C.M. Lee John C.S. Lui + Abstract The current Internet architecture comprises of different privately owned Internet service providers

3 Does the Simplex Algorithm Work?

Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

Moral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania

Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically

Week 1: Introduction to Online Learning

Week 1: Introduction to Online Learning 1 Introduction This is written based on Prediction, Learning, and Games (ISBN: 2184189 / -21-8418-9 Cesa-Bianchi, Nicolo; Lugosi, Gabor 1.1 A Gentle Start Consider

Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

Alternative proof for claim in [1]

Alternative proof for claim in [1] Ritesh Kolte and Ayfer Özgür Aydin The problem addressed in [1] is described in Section 1 and the solution is given in Section. In the proof in [1], it seems that obtaining

Chapter 21: The Discounted Utility Model

Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal

9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

Hacking-proofness and Stability in a Model of Information Security Networks

Hacking-proofness and Stability in a Model of Information Security Networks Sunghoon Hong Preliminary draft, not for citation. March 1, 2008 Abstract We introduce a model of information security networks.

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

Institute for Empirical Research in Economics University of Zurich. Working Paper Series ISSN 1424-0459. Working Paper No. 229

Institute for Empirical Research in Economics University of Zurich Working Paper Series ISSN 1424-0459 Working Paper No. 229 On the Notion of the First Best in Standard Hidden Action Problems Christian

Chapter 7. Continuity

Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

Stochastic Inventory Control

Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

Application of Game Theory in Inventory Management

Application of Game Theory in Inventory Management Rodrigo Tranamil-Vidal Universidad de Chile, Santiago de Chile, Chile Rodrigo.tranamil@ug.udechile.cl Abstract. Game theory has been successfully applied

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...........

Worked examples Basic Concepts of Probability Theory

Worked examples Basic Concepts of Probability Theory Example 1 A regular tetrahedron is a body that has four faces and, if is tossed, the probability that it lands on any face is 1/4. Suppose that one

DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

How I won the Chess Ratings: Elo vs the rest of the world Competition

How I won the Chess Ratings: Elo vs the rest of the world Competition Yannis Sismanis November 2010 Abstract This article discusses in detail the rating system that won the kaggle competition Chess Ratings:

CHAPTER 1 Splines and B-splines an Introduction

CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the

6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation

6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitely-repeated prisoner s dilemma

IMPLICIT COLLUSION IN DEALER MARKETS WITH DIFFERENT COSTS OF MARKET MAKING ANDREAS KRAUSE Abstract. This paper introduces dierent costs into the Dutta-Madhavan model of implicit collusion between market

School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213

,, 1{8 () c A Note on Learning from Multiple-Instance Examples AVRIM BLUM avrim+@cs.cmu.edu ADAM KALAI akalai+@cs.cmu.edu School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 1513 Abstract.

capital accumulation and growth. Luisa Fuster y January 1998

Eects of uncertain lifetime and annuity insurance on capital accumulation and growth. Luisa Fuster y January 1998 I am indebted to Jordi Caballe for his advice and encouragement. I am also very grateful

A simple criterion on degree sequences of graphs

Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

OPTIMAL DESIGN OF A MULTITIER REWARD SCHEME. Amir Gandomi *, Saeed Zolfaghari **

OPTIMAL DESIGN OF A MULTITIER REWARD SCHEME Amir Gandomi *, Saeed Zolfaghari ** Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Ontario * Tel.: + 46 979 5000x7702, Email:

In this section, we will consider techniques for solving problems of this type.

Constrained optimisation roblems in economics typically involve maximising some quantity, such as utility or profit, subject to a constraint for example income. We shall therefore need techniques for solving

Lecture 13: Martingales

Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x

Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written

Secretary Problems. October 21, 2010. José Soto SPAMS

Secretary Problems October 21, 2010 José Soto SPAMS A little history 50 s: Problem appeared. 60 s: Simple solutions: Lindley, Dynkin. 70-80: Generalizations. It became a field. ( Toy problem for Theory

Pricing the Cloud: Resource Allocations, Fairness, and Revenue

Pricing the Cloud: Resource Allocations, Fairness, and Revenue Carlee Joe-Wong and Soumya Sen Princeton University and University of Minnesota Emails: coe@princeton.edu, ssen@umn.edu Abstract As more businesses

5.1 Derivatives and Graphs

5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

Outsourcing Prioritized Warranty Repairs

Outsourcing Prioritized Warranty Repairs Peter S. Buczkowski University of North Carolina at Chapel Hill Department of Statistics and Operations Research Mark E. Hartmann University of North Carolina at

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

Notes from Week 1: Algorithms for sequential prediction

CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 22-26 Jan 2007 1 Introduction In this course we will be looking

Optimal shift scheduling with a global service level constraint

Optimal shift scheduling with a global service level constraint Ger Koole & Erik van der Sluis Vrije Universiteit Division of Mathematics and Computer Science De Boelelaan 1081a, 1081 HV Amsterdam The

A Spectral Clustering Approach to Validating Sensors via Their Peers in Distributed Sensor Networks

A Spectral Clustering Approach to Validating Sensors via Their Peers in Distributed Sensor Networks H. T. Kung Dario Vlah {htk, dario}@eecs.harvard.edu Harvard School of Engineering and Applied Sciences

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

A Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems

myjournal manuscript No. (will be inserted by the editor) A Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems Eduardo Álvarez-Miranda Ivana Ljubić Paolo Toth Received:

Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

The Conference Call Search Problem in Wireless Networks

The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,

A Systematic Approach. to Parallel Program Verication. Tadao TAKAOKA. Department of Computer Science. Ibaraki University. Hitachi, Ibaraki 316, JAPAN

A Systematic Approach to Parallel Program Verication Tadao TAKAOKA Department of Computer Science Ibaraki University Hitachi, Ibaraki 316, JAPAN E-mail: takaoka@cis.ibaraki.ac.jp Phone: +81 94 38 5130

A Stochastic Inventory Placement Model for a Multi-echelon Seasonal Product Supply Chain with Multiple Retailers

The Sixth International Symposium on Operations Research and Its Applications (ISORA 06) Xinjiang, China, August 8 12, 2006 Copyright 2006 ORSC & APORC pp. 247 257 A Stochastic Inventory Placement Model

A Detailed Price Discrimination Example

A Detailed Price Discrimination Example Suppose that there are two different types of customers for a monopolist s product. Customers of type 1 have demand curves as follows. These demand curves include

10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

Corollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality

Corollary For equidistant knots, i.e., u i = a + i (b-a)/n, we obtain with (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality 120202: ESM4A - Numerical Methods

ECON 305 Tutorial 7 (Week 9)

H. K. Chen (SFU) ECON 305 Tutorial 7 (Week 9) July 2,3, 2014 1 / 24 ECON 305 Tutorial 7 (Week 9) Questions for today: Ch.9 Problems 15, 7, 11, 12 MC113 Tutorial slides will be posted Thursday after 10:30am,

Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis

Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................

Revenue Management with Correlated Demand Forecasting

Revenue Management with Correlated Demand Forecasting Catalina Stefanescu Victor DeMiguel Kristin Fridgeirsdottir Stefanos Zenios 1 Introduction Many airlines are struggling to survive in today's economy.

Two-step competition process leads to quasi power-law income distributions Application to scientic publication and citation distributions

Physica A 298 (21) 53 536 www.elsevier.com/locate/physa Two-step competition process leads to quasi power-law income distributions Application to scientic publication and citation distributions Anthony

14.451 Lecture Notes 10

14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

6.231 Dynamic Programming and Stochastic Control Fall 2008

MIT OpenCourseWare http://ocw.mit.edu 6.231 Dynamic Programming and Stochastic Control Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 6.231

Invertible elements in associates and semigroups. 1

Quasigroups and Related Systems 5 (1998), 53 68 Invertible elements in associates and semigroups. 1 Fedir Sokhatsky Abstract Some invertibility criteria of an element in associates, in particular in n-ary

BX in ( u, v) basis in two ways. On the one hand, AN = u+

1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

Simultaneous or Sequential? Search Strategies in the U.S. Auto. Insurance Industry. Elisabeth Honka 1. Pradeep Chintagunta 2

Simultaneous or Sequential? Search Strategies in the U.S. Auto Insurance Industry Elisabeth Honka 1 University of Texas at Dallas Pradeep Chintagunta 2 University of Chicago Booth School of Business October

Buered Probability of Exceedance: Mathematical Properties and Optimization Algorithms

Buered Probability of Exceedance: Mathematical Properties and Optimization Algorithms Alexander Mafusalov, Stan Uryasev RESEARCH REPORT 2014-1 Risk Management and Financial Engineering Lab Department of

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

1 Example of Time Series Analysis by SSA 1

1 Example of Time Series Analysis by SSA 1 Let us illustrate the 'Caterpillar'-SSA technique [1] by the example of time series analysis. Consider the time series FORT (monthly volumes of fortied wine sales

Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

Pricing of Limit Orders in the Electronic Security Trading System Xetra

Pricing of Limit Orders in the Electronic Security Trading System Xetra Li Xihao Bielefeld Graduate School of Economics and Management Bielefeld University, P.O. Box 100 131 D-33501 Bielefeld, Germany

1 Another method of estimation: least squares

1 Another method of estimation: least squares erm: -estim.tex, Dec8, 009: 6 p.m. (draft - typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i

Removing Partial Inconsistency in Valuation- Based Systems*

Removing Partial Inconsistency in Valuation- Based Systems* Luis M. de Campos and Serafín Moral Departamento de Ciencias de la Computación e I.A., Universidad de Granada, 18071 Granada, Spain This paper

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

Notes on Symmetric Matrices

CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

Math 315: Linear Algebra Solutions to Midterm Exam I

Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =