a new method to solve stochastic programming problems under probabilistic constraint with discrete random variables

A NEW METHOD TO SOLVE STOCHASTIC PROGRAMMING PROBLEMS UNDER PROBABILISTIC CONSTRAINT WITH DISCRETE RANDOM VARIABLES BY TONGYIN LIU A dissertation submitted to the Graduate School—New

A NEW METHOD TO SOLVE STOCHASTIC PROGRAMMING PROBLEMS UNDER

PROBABILISTIC CONSTRAINT WITH DISCRETE

RANDOM VARIABLES

BY TONGYIN LIU

A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey

in partial fulfillment of the requitements

for the degree of Doctor of Philosophy Graduate Program in Operations Research Written under the direction of Professor Andras Prékopa and approved by

UMI Number: 3203387

Copyright 2006 by Liu, Tongyin

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ABSTRACT OF THE DISSERTATION

A New Method to Solve Stochastic Programming

Problems Under Probabilistic Constraint with Discrete

Random Variables

by Tongyin Liu Dissertation Director: Professor Andras Prékopa

In this dissertation, probabilistic constrained stochastic programming problems are con- sidered with discrete random variables on the r.h.s in the stochastic constraints In Chapter 2 and 3, it is assumed that the random vector has multivariate Poisson, bino- mial or geometric distribution We prove a general theorem that implies that in each

of the above cases the c.d.f majorizes the product of the univariate marginal c.d.f’s and then use the latter one in the probabilistic constraints The new problem is solved

in two steps: (1) first we replace the c.d.f’s in the probabilistic constraint by smooth logconcave functions and solve the continuous problem; (2) search for the optimal so- lution for the case of the discrete random variables In Chapter 4, numerical examples are presented and comparison is made with the solution of a problem taken from the literature In Chapter 5, some properties of p level efficient points of a random vari- able are studied, and a new algorithm to enumerate all the p level efficient points is developed In Chapter 6, p level efficient points in linear systems are studied

il

Acknowledgements

I would like to take this opportunity to express my deepest gratitude and appreciation

to my advisor Professor Andrd4s Prékopa for the opportunity working with him on this topic, and for his advice and guidance

I have very much enjoyed courses taught by Professors Andrés Prékopa, Endre Boros, and David Shanno I appreciate the Hungarian brains I am lucky to have taken the Case Study course from Professor Michael Rothkopf, which provided me some opportunities to practice my OR skills I am glad to audit Professor Andrzej Ruszczynski’s Operations Research Models in Finance, and benefit from his ideas of the finanical risk modeling I am happy and thankful to have attended Dr Hammer’s class and RUTCOR’s seminars Also I am grateful for the friendly RUTCOR staff Clare and Terry

My colleagues and friends at RUTCOR will be at the heart of my fondest memories: Igor Zverovich, Sandor Szedmak, James Wojtowicz and Lijie Shi The leisure time with Martin Milanic, Noam Goldberg and Gabor Rudolf gives me so much fun

I thank my parents for so many years supports, their kindness and hard work char- acters are the utmost examples to follow During the visit of my parents in-law, they gave us so much help to take care of my son I also would like to take this opportunity

to express my thanks to my wife Xiaoling and our son Jialiang for everything

Finally, I would like to thank DIMACS for their generous financial support, which makes life easier

iii

Dedication

To Xiaoling, Leon, and my family

iv

Table of Contents

Acknowledgements .Ặ Ặ Q Q Q Q HQ Q Q ee iii

List of Abbreviations QC Q Q Q Q Q Q v gà va kia ix

1.1 Some definitions 20.00.00 00 0 ee 3

2 The case of independent Poisson, binomial and geometric random variables Q Q Q LH HQ ng cà n ngà kg k KV kg ki ki ko ki kia ĩ 2.1 Independent Poisson random variables 8 2.2 Independent binomial random variables 9 2.3 Independent geometric random variables 12 2.4 Relations between the feasible sets of the convex programming problems and the discrete cases Q Q Q Q LH ng ng gà gà kàng 13

3 Inequalities for the joint probability distribution of partial sums of independent random variables .-.2.0+000.- 20

4 Numerical examples 00 pepe eee eee ens 24 4.1 AÀ vehicle routing example ee 25 4.2 A stochastic network design problem 29

5 Methods of enumerating all PLEP’s of a random vector 35 5.1 An algorithm to enumerate all PLEP’s 36

vi

Complulting T@SUÏS Q Q LH Q ng và kg ga 45

vii

An illustration of PVB algorithm 0-.20050 42

An illustration of the new algorithm .2005 43

vill

PLEP

List of Abbreviations

p level efficient point

set of real numbers

set of nonnegative real numbers

set of integers

set of nonnegative integers

ix

Chapter 1

Introduction

Stochastic programming is the science that provides us with tools to design and control stochastic systems with the aid of mathematical programming techniques [27] Proba- bilistic constraints are one of the main challenges of modern stochastic programming The motivation is as follows: if in the linear program

min c

subject to Tx > €

z > 0, where A is an m x n matrix, T is an s X n matrix, and € = (&, ,€ )? is a random vector with values in Rể c, z € R” and b € R*”, we require that Tx > € shall hold at least with some specified probability p € (0,1), rather than all for all possible realizations of the right hand side Then it can be written in the following problem

formulation:

min cr

where P denotes probability For furhter reference, we can also write (1.1) in the following equivalent form:

min c's subject to P(Eé<y)>p

max P(Tr > €)

z>0

For detailed presentation of these results, the reader is referred to (27] and [29]

Much is known about the problem (1.1) in the case where € has a continuous prob- ability distribution (e.g., see [27], [29] and the references therein) However, only a few papers handle the case of a discrete distribution But the number of potential appli- cations is large Before a brief literature review for the discrete case, we first present

some primary definitions for the stochastic programming

1.1 Some definitions

Logeoncave measures have been introduced in the stochastic programming framework

in [20], [22] and [21] but they became widely used also in statistics, convex geometry, mathematical analysis, economics, etc

Definition 1.1 A nonnegative function f defined on a conver subset A of the space R™ is said to be logarithmically concave (logconcave) if for every pair x,y € A and O<A<1 we have the inequality

f(Ar + (1— Ady) > [F(@)P LAY) (1.4)

If f is positive valued, then log f is a concave function on A If the equality holds strictly for « # y, then f is said to be strictly logconcave

Definition 1.2 A probability measure defined on the Borel sets of R” is said to be logconcave if for any conver subsets of R": A, B andQ <A <1 we have the inequality

PA4+(1-—A)) > [P(4)]`IP(8)]'~^,

tphere ÀA + (1— À)}B = {z = Àz+(1—À)w|zc€ A,ue BỊ

For the case of a discrete €, the concept of p level efficient point(PLEP) has been introduced in [25] Below, we recall this definition Let F(z) designate the probability distribution function of €, ie., F(z) = P(E < z), ze R’

Definition 1.3 Let Z be the set of possible values of £ A vector z € Z is said to be ap level efficient point or PLEP of the probability distribution of € if F(z) =P(€<z)>p and there is no y € Z such that F(y) >p,y<z andy#z

Dentcheva, Prékopa and Ruszezyriski (2000) remarked that, by a classical theorem

of Dickson [2] on partially ordered sets (posets), the number of PLEP’s is finite even if

Z is not a finite set Let v), 7 © J be the set of PLEP’s Since

JET

a further equivalent form of problem (1.1) is the following:

of the convex hull of the given disjunctive set implied by the probabilistic constraint in (1.2) Prékopa, Vizvári and Badics (1998) relaxed problem (1.5) in the following way:

min c!z subject to Tx > yy, 0

S2; =1, >0, ý€{1, ,|Z|} (1.6)

Ar>b z>0, gave an algorithm to find all the PLEP’s and a cutting plane method to solve problem (1.6) In general, however, the number of p level efficient points for € is very large

To avoid the enumeration of all PLEP’s, Dentcheva, Prékopa and Ruszezyriski (2000) presented a cone generation method to solve problem (1.6) Vizvdri (2002) further analyzed the above solution technique with emphasis on the choices of Lagrange mul- tipliers and the solution of the knapsack problem that comes up as a PLEP generation technique in case of independent random variables

Discrete random vectors in the contexts of (1.1) and (1.3) come up in many practical problems Singh et al [40] present and solve a chip fabrication problem, where the components of the random vector designate the number of chip sites in a wafer that

produce good chips of given types These authors solve a similar problem as (1.3) rather than problem (1.1), where the objective function in (1.3) is 1 — P(T'x > €) Dentcheva

et al [7] present and solve a traffic assignment problem in telecommunication, where the problem is of type (1.1) and the random variables are demands for transmission In the design of a stochastic transportation network in power systems, Prékopa and Boros [26] present a method to find the probability of the existence of a feasible flow problem, where the demands for power at the nodes of the network are integer valued random

variables

1.3 Outline of the thesis

In this dissertation, we present a new method to solve stochastic programming prob- lems under probabilistic constraints with discrete random variables In Chapter 2, we assume that the components of the random vector € have all Poisson, all binomial or all geometric distributions We use incomplete gamma, complementary of incomplete beta and exponential functions, respectively, to convexify the problem by replacing smooth distribution functions for the discrete ones such that they coincide at lattice points Then the optimal solutions to the convex problems are used to find the optimal solu- tion of problem (1.1) This is carried out by the use of a modified Hooke and Jeeves direct search method Also we prove that the complementary of the incomplete beta function is logconcave as well as incomplete beta function itself In Chapter 2, we solve problem (1.1) under the assumption that the components of € are independent and the solution of (1.3) is discussed for the case of discrete random vector € In Chapter 3 we look at the case where the components of € are partial sums of independent random variables We prove a general theorem that implies that in our cases the multivariate c.d.f majorizes the product of the univariate marginal c.d.f’s In Chapter 4 numerical results, along with new applications are presented [16] Comparison is made between the solution of a telecommunication problem presents in [7] and our solution to the same problem In Chapter 5, some properties of PLEP are studied, and a new algorithm of enumerating all PELP’s of a random vector € is presented In Chapter 6, we further study the properties of p level efficient points in the linear systems, which is essential

to solve the problem of designing a stochastic network with discrete random variables.

Chapter 2 The case of independent Poisson, binomial and geometric

Tins Fi(ys) 2 v-

Note that the inequality

P(Œz > &) > P(Tz >€) >p, (0<p< 1)

implies 7¿z > 0, ¿ = 1, ,r Thus if £ is a discrete random vector, the above problem

is equivalent to the following:

min cÝz subjectto T7+z >, €c Z2"

2.1 Independent Poisson random variables

Let € be a random variable that has Poisson distribution with parameter \ > 0 The values of its c.d.f at nonnegative integers are

k=0 `

Let

an Find) = | Tø+D° đz, where

For the log concavity of P,,, we recall the following theorem

Theorem 2.1 (/2/, [27]) For any fired \ > 0, the function F(n;) is logconcave on the entire real line, strictly logconcave on {n | n > —-1} and P, = F(n,A) for any nonnegative integer n

Suppose that €,, ,€,- are independent Poisson random variables with parameters Ai;++* Ap, respectively To solve (2.2), first we consider the following problem:

By Theorem 2.1, (2.4) is a convex nonlinear programming problem

2.2 Independent binomial random variables

Suppose € has binomial distribution with parameter 0 < p < 1 Let z be a nonnegative integer It is known (see, e.g., Singh et al., 1980; Prekopa 1995) that

For fixed a > 0 define G(a, x), as a function of the continuous variable z, by the equation (2.6) for z > 0 and let G(a, z) = 0 for ¢ < a, We have the following Theorem Theorem 2.2 (/27/,/40]) Let a > 0 be a fired number Then G(a, x) is strictly increasing and strictly logconcave for x > a

If x is an integer then G(a,r) = 1 — F(a —1) where F is the c.d.f of the binomial distribution with parameters x and p While Theorem 2.2 provides us with a useful tool in some applications (c.f [40]), we need a smooth logconcave extension of F and

it can not be derived from G(a,z)

Let X be a binomial random variable with parameter n and p Then, by (2.6), we have for every x =0,1, ,n —1:

_ J4)” ”'dụ

=———

Js 2 — 0)*—*—~Ìdụ The function of the variable x, on the right hand side of (2.7), is defined for every

We have the following:

Theorem 2.3 (/16/) The function F(x;n,p) satisfies the relations

10

it is strictly increasing in the interval (-1, n), has continuous derivative and is logcon- cave on TRÀ

Proof We skip the proof of the relations because it requires any standard reasoning

To prove the other assertions first transform the integral in (2.8) by the introduction

of the new variable -— + =t, We obtain

mô PŒUP) — pm Tứin,P) = 0

It is enough to prove the logconcavity of F(2;n,p) for the case of -1 < # < n, because the logconcavity of the function on R! easily follows from it We have the

g(t) = froe (z+1)u }“tranmrrdu (re) , 00 <t <0, (2.13)

where z is a fixed number satisfying -1 < x <n The function g(t) is exactly logconcave

on the entire real line Let X be a random variable that has p.d.f equal to (2.13) Then (2.12) can be rewritten as

Burridge (1982) has shown that if a random variable X has a logconcave p.d.f., then

E(X? | X >u) — E°(X |X >)

is a decreasing function (a proof of this fact is given in Prékopa 1995 pp.118-119) If

we apply this in connection with the function (2.13), then we can see that the value in

12

Suppose ¢1, £, ,& are independent binomial random variables with parameters (n1,pi), -, (mr, pr), respectively To solve problem (2.2), we first solve the following problem:

min cz

subject to Tr=y

Az >ob Viet (in by £⁄(1 — £)P+~W—1đ—

In fy t#(1 — tute) > Inp

(2.15)

œ>9

This is again a convex programming problem from Theorem 2.3

2.3 Independent geometric random variables

Let € be a random variable with geometric distribution € has probability function

For the c.d.f F(z) = 1 —e~*, we have the following theorem:

Theorem 2.4 Let \ > 0 be a fixed number Then F(x) = 1—e~™ is strictly increasing and strictly logconcave for x > 0

Proof Since F’ = e~** > 0, F(z) is strictly increasing for z > 0 Rewrite F(x) as follows:

Ax —

cÀz

13

Then In F(x) = In(e** — 1) — Az Take the second derivative of In F(x), we get

— A2 ert

Suppose the components €1, ,&- of random vector € are independent geometric variables with parameters p,, ,p,, respectively In this case, to solve problem (2.2),

we first solve the following convex programming problem:

min cs subject to Tx =y

to solve the problems in the numerical example chapter

2.4 Relations between the feasible sets of the convex programming problems and the discrete cases

First, we have the following theorem:

Theorem 2.5 (/16]) Let € be a discrete random variable, F(z) the c.d.f of §, z€ Z+, and P the set of all PLEP’s of € Let F(x) be a smooth function, x € Ri, such that F(z) = F(z) when2 € Z, Then

Proof Let Zp = {y € Zs | P(€ < y) > p}, where P is the distribution function of

€ Then P © Zy Since the values of F(z) and F(a) coincide at the lattice points,

Corollary 2.1 (a) If all components of € have independent Poisson distribution with parameters 1,A2, ,Ar, respectively, then P C 2z;

(b) Tƒ all components oƒ € haue independent binomial distribution with parameters (m,1), (nạ, Ð>), , (n„, py), respectively Then P C 29;

(c) If all components of € have independent geometric distribution with parameters D1, P2, ++,Pr, respectively Then P © Ze

Proof The proof can he directly derived from Theorem 2.19 O

So for problem (2.2), if all components of € have independent Poisson, or binomial

or geometric distribution, we can get the corresponding relaxed convex programming problem as shown in (2.5), (2.15) and (2.18), respectively

2.5 Local search method

From the optimal solutions of the relaxed problems, we use a direct search method to find the optimal solutions of the discrete optimization problems The method is based

on Hooke and Jeeves search method (Hook and Jeeves, 1961) and in each step, we have

to check the feasibility of the new trial point To state the method, without loss of generality, we simplify problem (2.2) in the following way:

min clr

subject to 7z >1, t¿¡ € 2

[h-¡ Fis) = z€2+

Let z be the optimal solution of problem (2.1) and z* = |z] Let ƒ(z) = eŸz and

,

D={z| || (+) >p, Ar >b, xe Z.}

¡=1 where T; is the ?-th row vector of 7T,

16

The modified Hooke and Jeeves direct search method is as follows In each searching step, it comprises two kinds of moves: Exploratory and Pattern Let Aa, be the step length in each of the directions e;, i = 1,2, ,7

The Method

Exploratory move

Step 0: Set i = 1 and compute F = f(x*) where z* = |z] =

(11,23, , #z)

Step 1: Set ø := (Z1,Z2, ,¿ + A1, , đr)

Step 2: If f(z) < F and z € D then set F = f(x), i:=i+1; Goto

Step 1

Step 3: If f(z) > F and x € D then set x := (21,2%9, ,2; —

2Aqg;, ,0,) If f(x) < F and x € D, the new trial point is retained

Set F = ƒ(z), ¿:= ¿+ 1, and goto Step 1

If f(z) > F then the move is rejected, +; remains unchanged Set

i:=%+1 and goto Step 1

Pattern move

Step 1: Set « = 2? + (x? — @8), where x? is the point arrived

by the Exploratory moves, and #° is a point which is also arrived

by exploratory move in previous step where x? is obtained from the

exploratory move starting from 7°

Step 2: Starts the Exploratory move If for the point z obtained by

the Exploratory moves f(x) < f(x?) and x € D, then the pattern

move is recommended Otherwise x? is the starting point and the

process restarts from 2

Remark When we consider the discrete random variables which have Poisson, binomial

or geometric distributions, we set Az; = 1

A simple two-dimensional Hooke and Jeeves search method is illustrated in the

In figure 2.1, the points labeled numbers according to the sequence are selected x!

is a starting base After x® failed, and #2 and x4 are successes, then the new base x4 is

obtained by the Exploratory moves, where f(r*) < f(x') z® is obtained from Pattern move From 2°, 2° is a base if f(x2°) < f(x*) after the Exploratory moves We repeat these steps, finally we reach zø!3, If ƒ(z!3) > ƒ(zŠ), we have to return to #Š, which is a starting base and performance this procedure with reduced step lengths

18

2.6 Probability maximization under constraints

Now we consider problem (1.1) and the following problem together:

max P(Tr > &) subject to cax<K

(2.21)

Ar >b z>0, where € is a random vector and K is fixed number In [27], the relations between problem (1.1) and (2.21) are discussed

Suppose the components of random vector € are independent, then the objective

function of problem (2.21) is h(x) = []j_, Fi(yi), where Tx = y Since F;(y;) > 0, we

take natural logarithm of h(x) and problem (2.21) can be written in the following form:

From Theorem 2.1, the objective function of problem (2.23) is concave Let z be the optimal solution of problem (2.22) and x* = |x] Then we apply the modified Hooke and Jeeves search method to search the optimal solution of problem (2.21) around 2*

as described above, and D is replaced by

D={x|c?<K, Av>b, rE Zs}

20

Chapter 3 Inequalities for the joint probability distribution of partial

sums of independent random variables

In this chapter, we consider the joint probability distribution of partial sums of inde pendent random variables We assume that the r-h.s random variables of problem (1.1) are partial sums of independent ones where all of them are either Poisson or binomial

or geometric with arbitrary parameters The probability that a joint constraint of one

of these types is satisfied is shown to be bounded from below by the product of the probabilities of the individual constraints The probabilistic constraint is imposed on the lower bound Then smooth logconcave c.d.f’s are fitted to the univariate discrete c.d.f’s and the continuous problem can be solved numerically

For the proof of our main theorems in this chapter, we need the following

Lemma 3.1 (/16]/) LetO0<p, <1, g=1-p, a > a1, bạ >bị, , zọ > 24, then

we have the inequality

paobo - - - Z0 + gatbg - - -Z\ > (pao + gai)(pbo + ghi) (p20 + gz1)

Proof We prove the assertion by induction For the case of ao > aj, bo > bi, the assertion is

pagby + qaib, > (pag + ga1)(pbo + gb1)

This is easily seen to be the same as

pq(ao — a1)(bạ — bị) > 0.

21

which holds true, by assumption Looking at the general case, we can write

pao (bo - - - zo) + ga (b4 - - - Z1)

> (pag + gay) (pbo - - zo + đÙI - - - Z1)

> (pao + qga1)(pbo + qbì) (pco - - - Z0 + đề - - - 21)

> (pag + qaz)(pbo + gbi) (pzo + 921)

Let A = (a; 4) #0 be an mxr matrix with 0—1 entries and X;, ,X; independent,

0 - 1 valued not necessarily identically distributed random variables Consider the transformed random variables

r

=1

We prove the following

Theorem 3.1 (/16/) For any nonnegative integers y1, ,Ym we have the inequality

m™m

P( <Sựi,' .Ym < m) > |][ PƠi < 1ì (3.2)

¿=1 Proof Let Ï = {¿ | œ\ = 1}, Ï = {1, ,m}\ T, pịì = P(X: = 0), g =1— pị Then

we have the relation

Theorem 3.2 (/16]) Let X1, ,X, be independent, binomially distributed random variables with parameter (n1,pi),. , (Mr, Pr), respectively Then for the random vari- ables (3.1), then inequality (3.2) holds true

Proof The assertion is an immediate consequence of Theorem 3.1 O

Proof If in Theorem 3.2, we let n; — oo, py; — 00 such that njpj — À¿, ý = 1, ,7,

In case of Theorem 3.3, the random variables Y;, 1 = 1, ,m have Poisson distri- bution with parameter )7j_, ai,nA;, i = 1, m, respectively

Theorem 3.3 obviously remain true if X;, ,X; are independent random variables such that the c.d.f of X; can be obtained as a suitable limit of the c.d.f of binomial distri- butions, i = 1, ,r Since m-variate normal distribution with nonnegative correlations can be obtained this way as the joint distribution of the random variables Y1, ,Ym

In this case, inequality (3.2) is a special case of the well-known Slepian-inequality (see Slepian, 1962).