Adjacent basis based algorithm for multiparametric linear programming

23 3.3 Determine the critical region and the optimal value for a given point 25 3.4 Validity test of the adjacent basis based algorithm.. This thesis presents a novel algorithm, namely a

ADJACENT BASIS BASED ALGORITHM FOR MULTIPARAMETRIC LINEAR

PROGRAMMING

CHEN FEI

(B.Sc., QFNU)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2009

First of all, I am grateful to Professor Zhao Gongyun, my thesis supervisor Hiskind and patient guidance has provided the most impetus for my study and re-search, without which this thesis would not have been completed I’m particularlyindebted to his advice in how to fit a research project in a big picture and deducealgorithms from theories

In addition, I deeply appreciate the encouragement and help from my friends inNUS, Zhao Xinyuan, Gao Yan, and Li Lu, just to name a few My thanks also go

to the staff of the Department of Mathematics for their support and help

Last but not least, the support and love from my husband and other familymembers should not be ignored It is their encouragement and warm care for both

my study and daily life that make me still energetic

Chen FeiJuly 2009

ii

3 Adjacent Basis Based Algorithm for Multiparametric Linear

3.1 Formulas of p & d boundaries and the cutting hyperplane 163.1.1 Deduce p & d boundary 16

iii

Contents iv

3.1.2 Deduce the cutting hyperplane 20

3.2 Algorithm based on basis partition of the SLP 21

3.2.1 Sub-algorithms 22

3.2.2 Main algorithm–Adjacent basis based algorithm 23

3.3 Determine the critical region and the optimal value for a given point 25 3.4 Validity test of the adjacent basis based algorithm 26

3.5 Numerical experiments 26

3.6 Comparison between the adjacent basis based algorithm and the geometric algorithm 36

3.6.1 The geometric algorithm 36

3.6.2 Comparison between the adjacent basis based algorithm and the geometric algorithm 38

4 Further Research on Adjacent Basis 40 4.1 Determine p boundary related feasible adjacent bases 40

4.2 Determine d boundary related feasible adjacent bases 46

This thesis presents a novel algorithm, namely adjacent basis based algorithm, tosystematically identify all the critical regions in a parameter space for solving mul-tiparametric linear programming (mpLP) problems, based on the studies of Zhaoabout basis partition of the space of linear programs (SLP)

In the algorithm, a cutting theorem is proposed to efficiently find a feasible point

in the parameter space The feasible basis of the feasible point can be determined

and the corresponding critical region can be represented by relevant p & d

bound-aries From the feasible basis, we can generate all the relevant adjacent bases.Then, we identify all the feasible bases out of the adjacent bases and determinetheir corresponding critical regions in the parameter space

An introduction is given in Chapter 1 to provide an overview of the subject inquestion Chapter 2 briefly introduces the relevant definitions and how to relatethe SLP to Grassmann Manifold We also describe the characterization and con-struction of critical regions and their boundaries Then, we propose the cutting

v

Summary vi

theorem which gives a method to remove part of the region which is not strictly

feasible In Chapter 3, formulas are developed to represent the p & d boundaries

and the cutting hyperplane Then, two sub-algorithms are given before the mainalgorithm is proposed to make the latter clear The method for determining thecritical region of a fixed point and the method for testing the validity of the adja-cent basis based algorithm are also discussed Furthermore, numerical experimentsare shown to test the feasibility of the proposed algorithms Finally, a comparisonbetween the adjacent bases based algorithm and the geometric algorithm is carriedout In Chapter 4, further research is done to filter feasible bases from all adjacentbases Conclusions are given in Chapter 5

List of Tables

3.1 Optimal solution for Example 2 29

3.2 Boundaries for Example 3 29

3.3 Represent of critical regions for Example 3 30

3.4 Optimal solution for Example 3 31

3.5 Results for different cases in Example 4 32

3.6 Results for case 3 with two parameters tt = 2 32

3.7 Results for case 3 with four parameters tt = 4 33

vii

mpLP is an important research topic in operational research It was first discussed

by Gass and Saaty [1], a few years after the simplex method was developed byDantzig Since then, extensive research has been done on the topic [3-14] Forinstance, the book by Gal [3] contains hundreds of references related to sensitivityand parametric analysis Generally, the study of mpLP problems has focused ontwo levels: sensitivity analysis and parametric programming Sensitivity analysischaracterizes the change of the solution with respect to a small perturbation of theparameters and provides a solution in the neighborhood of the nominal value of theparameters Parametric programming is based on the sensitivity analysis theory

1

and aims to identify subregions in a parameter space In each of the subregions,the optimal solution is an affine function of the parameters

As far as we know, there are two types of methods for solving mpLP problems

on parametric programming level The first method was presented by Gal andNedoma [2] It enumerates all optimal bases of the associated LP tableau using amethod derived from the simplex algorithm Subsequently, similar methods wereproposed by Yu, Zeleny [12] and Schechter [5] However, these methods are verysensitive to the number of parameters As it works with polyhedral in the parame-ter space, the amount of computing required by the method increases exponentiallywith the increasing number of parameters Only recently, an essentially differentmethod named geometric algorithm was proposed by Borrelli and Bemporad et

al [9, 10, 11], which uses the geometric properties of the problem to explore theparameter space It directly partitions the parameter space into subsets to ob-tain critical regions However, it cannot solve problems with high dimensionalparameter space, because the number of new sub-regions defined by the parti-tioning strategy increases exponentially as the dimension of the parameter spaceincreases Furthermore, this method introduces a large number of artificial cuts inthe parameter space

Fortunately, the study of the space of linear programs (SLP) provides a new spective on understanding essential relationships among critical regions and thenature of LP Works of Zhao [16, 17, 18] introduces a representation and somebasic structures of SLP, including detailed geometric structures of the critical re-gions and their boundaries in the SLP Characterization of the basis partition canpotentially lead to new methods for solving parametric LP This motivates us tostudy the mpLP problems based on the basis partitions of the SLP

In this thesis, we consider a class of non-degenerate mpLP problems with linear

parameters in the cost c and the right-hand side b simultaneously The problem

can be reformed as:

The rest of the thesis is organized as follows:

In Chapter 2, we consider SLP as a collection of all LPs with the same number ofdecision variables and constrains We briefly introduce the relationship betweenSLP and Grassmann Manifold which has rich geometric and algebraic structures.Then, we describe the characterization and construction of critical regions andtheir boundaries Finally, the cutting theorem is presented as a method to removesome of the infeasible points from the parameter space

Chapter 3 aims to solve the mpLP problems based on theories presented in the

previous chapter In section 3.1, we deduce formulas of p & d boundaries and

a cutting hyperplane In section 3.2, two sub-algorithms are given to check thefeasibility of a basis and remove redundant linear constraints Then, the mainalgorithm, namely adjacent basis based algorithm, is proposed In the algorithm,the cutting theorem is used to efficiently find a feasible point in the parameterspace The feasible basis of the point can be determined and the corresponding

critical region can be represented by relevant p & d boundaries From the feasible

basis, all the adjacent bases can be generated We can then identify all the feasible

bases out of the adjacent bases and determine their corresponding critical regions

in the parameter space In section 3.3, a method for determining the critical gion of a fixed point is given Once the partitioning of the parametric space iscompleted, the optimal solution of the point can be obtained without solving thecorresponding LP problem In section 3.4, the method for testing the validity ofthe adjacent basis based algorithm is discussed Numerical experiments are shown

re-to implement and test the algorithms in section 3.5 Furthermore, a comparisonbetween the adjacent bases based algorithm and the geometric algorithm is carriedout in section 3.6

However, checking all the relevant adjacent bases increases the cost of calculation,

as not all the adjacent bases are feasible bases Therefore, further research to testthe feasibility of adjacent bases is given in Chapter 4

In Chapter 5, conclusions include the characters of the adjacent basis based rithm and an important topic for future research

algo-Chapter 2

Preliminaries

This chapter briefly introduces relevant definitions and theories of basis partition

of the space of liner programs based on works of Zhao[16, 17, 18] We introducethe relationships between linear programs and Grassmann Manifold which hasrich geometric and algebraic structures Furthermore, the characterization andconstruction of critical regions and their boundaries are presented Finally, thecutting theorem is given It provides a method to remove part of the region which

is not strictly feasible

Manifold

Consider the linear program:

min c T x s.t Ax = b (2.1)

x ≥ 0

5

2.1 Relating linear programming to Grassmann Manifold 6its dual is:

min b T y s.t A T y + s = c (2.2)

s ≥ 0

where A ∈ R m×n is of full row rank, b ∈ R m , x, c, s ∈ R n

Definition 1 If the primal and dual problems (2.1) and (2.2) have strictly feasiblesolutions, i.e both primal and dual feasible regions have relative interior points, the

coefficients (A, b, c) is considered as a strictly feasible instance (in short, instance)

of linear programming The set of all strictly feasible instances of dimension (n, m)

is denoted by SLP (n, m) We call SLP (n, m) the space of linear programs.

Definition 2 An index set B ⊂ {1, · · · , n}, |B| = m is said to be the optimal basis

of (A, b, c), if for each i ∈ B the dual constraint a T

i y ≤ c i is binding, i.e a T

i y = c i

for every dual optimal solution y and for each i ∈ N the primal constraint x i ≥ 0 is

satisfied at equality for every primal optima solution x, where a i is the i th column

of A and c i the i th component of c.

Note that an instance need not be non-degenerate A basis can be any subset, even

an empty set or a full set

It is known that for any feasible instance (A, b, c), there exists a unique optimal basis (B, N) Furthermore, there exists an optimal solution (x, y) such that

2.1 Relating linear programming to Grassmann Manifold 7

Since each instance (A, b, c) possesses a unique basis B, we can partition SLP (n, m) into {SLP (B) : B ∈ {1, · · · , n}}, where SLP (B) is the set of all (A, b, c) whose basis is B This partition is referred to as the basis partition of SLP (n, m).

Definition 3 A critical region CR B is a set of parameters t such that all the instances (A(t), b(t), c(t)) possess the same optimal basis B If CR B 6= Ø, the

critical region is considered feasible The corresponding basis B is a feasible basis Definition 4 Two critical regions CR B1, CR B2 are called adjacent if their inter- section CR B1 ∩ CR B2 is of dimension m − 1 The two optimal bases B1, B2 are

called adjacent.

In this work, we consider two kinds of boundaries which are formed by adjacentcritical regions

The boundary between adjacent critical regions CR B∪{i} and CR B∪{j} for i, j ∈

N, i 6= j is called p boundary, where |B| = m − 1.

The boundary between adjacent critical regions CR B\{i} and CR B\{j} for i, j ∈

B, i 6= j is called d boundary, where |B| = m + 1.

We denote by N the complementary index set of B, i.e N = {1, · · · , n}/B If we

write ¯B or ˆ B, then we will write the corresponding complementary index set as ¯ N

or ˆN Let J, K be any index subsets For a vector x, x J stands for the subvector of

x consisting of all components of x with indices in J For a matrix Q ∈ R m×n , Q JK stands for the submatrix of Q consisting of all entries of Q with row indices in J and column indices in K Q K consists of the columns indexed by K Q j is the j th column of Q Q −j consists of all the columns of Q except the j th column For any

partition (B, N) of {1, , n}, we follow the convention to write A = (A B , A N)

Notice that this does not mean that the columns in A B come before the columns

in A N In fact, it should be written as A = (A B , A N)Π for certain permutation

2.1 Relating linear programming to Grassmann Manifold 8matrix Π But we omit Π for simplicity.

For any strictly feasible instance, we consider the perturbed KKT system:

The KKT system has a unique solution to any t ∈ R and the solution (x(t), s(t)), t ≥

0, is known as the central path of the LP instance It is known that the centralpath converges to a pair of strictly complementary primal and dual optimal solu-

tions When t = 0, (x(0), s(0)) is considered as the center of the LP instance.

Based on the central path, we define the projection matrix :

2.1 Relating linear programming to Grassmann Manifold 9

A novel tool we use to characterize the basis partition is a differential equationwhich is defined on the space of projection matrices

G(n, m) := {M ∈ S n : MM = M, rank(M) = m}.

where S n is the set of all symmetric n × n matrices The space of projection

matrices is also known as the Grassmann manifold The differential equation weuse to characterize the basis partition is,

M 0 = h(M),

where the derivative 0 is taken with respect to t ∈ (−∞, +∞) and

h(M) = MbM1e + bM1eM − 2MbM1eM.

A close relationship between SLP (n, m) and G(n, m) and the basis partitions on

them was shown in [16] Here, we briefly summarize the relationship Throughout

the paper, we define the map Υ : R m×n → G(n, m) by

Υ(A) = A T (AA T)−1 A,

and the map Γ : SLP (n, m) → G(n, m) by

Γ(A, b, c) = Υ(Abxe), where (x, s, y) is the analytic center of (A, b, c), i.e the unique solution of the

system

Ax = b, A T y + s = c, x ◦ s = 1, x > 0, s > 0

Conversely, for any M ∈ G(n, m) we can construct an instance (A, b, c) such that Γ(A, b, c) = M, Thus, the map Γ is surjective but not injective For any strictly feasible instance (A, b, c) ∈ SLP (n, m), both the map from (A, b, c) to the analytic center (x, s, y) and the map Υ are smooth Therefore, the map Γ is smooth.

2.2 The characterization and construction of critical regions and their

The map Γ : SLP (n, m) → G(n, m) is surjective but not injective By the following

lemma we can determine the set:

Γ−1 (M) = {(A, b, c) ∈ SLP (n, m) : Γ(A, b, c) = M}.

Lemma 2.1 Let M ∈ G(n, m) and (A, b, c) ∈ Γ −1 (M) Then ( ¯ A, ¯b, ¯c) ∈ Γ −1 (M)

if and only if there exist a positive vector 0 < z ∈ R n and a nonsingular matrix

Q ∈ R m×m such that ¯ A = QAbze, ¯b = Qb, ¯c = bzec + bzeA T u for some u ∈ R m

Furthermore, let x(t), x ∗ and ¯ x(t), ¯ x ∗ be the central paths and optimal solutions of

(A, b, c) and ( ¯ A, ¯b, ¯c) respectively Then they are related as follows: ¯ x(t) = bze −1 x(t) for all t ∈ R and ¯ x ∗ = bze −1 x ∗

Since z > 0, ¯ x ∗ = bze −1 x ∗ , we see that (A, b, c) and ( ¯ A, ¯b, ¯c) have the same basis.

This shows that, for a projection matrix M, all instances which are mapped to M

by Γ have the same basis B Thus, we also refer to this B as the basis of M.

crit-ical regions and their boundaries

Here, three remarks are given to describe the characters of critical regions and theirboundaries

Remark 1 There are

2.2 The characterization and construction of critical regions and their

for i ∈ ¯ B There are m such B.

Each d boundary of CR B¯ is determined by

for j ∈ ¯ N There are n − m such B.

Remark 3 Each piece of p(d) boundary intersects all pieces of d(p) boundaries.

Stationary points of the form M1 :=

c which is the boundary of many pairs of critical regions

For each such point, we can determine a pair of critical regions between which

this point lies And from theorems proposed by Zhao [18] in chapter 3, we can

construct special instance (A, b, c) for M i such that Γ(A, b, c) = M i , i = 1, 2.

Theorem 2.2 If u p and u q are the largest and the smallest components of u, then

2.2 The characterization and construction of critical regions and their

Then, (A, b, c) ∈ Γ −1 (M) for any (A, b, c) ∈ L p (B, i),

Next, we consider the d boundary based on

Next, we present the cutting theorem which provides a method to construct a

cutting hyperplane The hyperplane can partition part of the region which is not

strictly feasible

Theorem 2.5 Cutting Theorem1

For t0 ∈ T , if the instance (A, b(t0), c(t0)) is infeasible, there exist u ∗ , z ∗ such that

a cutting hyperplane b(t) T u ∗ − c(t) T z ∗ = 0 can be constructed The hyperplane

1 The author contributes to the presentation of the theorem and proof based on ideas provided

by Zhao.

2.2 The characterization and construction of critical regions and their

can remove set {t ∈ T |b(t) T u ∗ − c(t) T z ∗ > 0} which is not strictly feasible Here,

(u ∗ , z ∗ ) can be obtained by solving the following LP problem,

Therefore, (A, b, c) is strictly feasible if and only if (2.6) has solutions.

Otherwise, if the instance (A, b, c) is not strictly feasible, then by Faskas’ theorem [15], there exist u, v, W , such that the following inequation system has solutions,

2.2 The characterization and construction of critical regions and their

The instance is not strictly feasible if and only if (2.7) has solutions

For a parameter t0 ∈ T , the instance (A, b(t0), c(t0)) is infeasible We can find

(u ∗ , z ∗ ) by solving the LP problem (2.4) and construct a cutting hyperplane b(t) T u ∗ − c(t) T z ∗ = 0 It partitions set {t ∈ T |b(t) T u ∗ − c(t) T z ∗ > 0} which is not strictly

feasible Next, we prove the statement

Let ˜T := {t ∈ T |b(t) T u ∗ − c(t) T z ∗ > 0} For any t ∈ ˜ T , the instance (A, b(t), c(t))

is not strictly feasible if and only if (2.7) has solutions Obviously, (u ∗ , z ∗)

sat-isfies A T u ≤ 0, Az = 0, z ≥ 0 since it is feasible to LP (2.4) For any t ∈ ˜ T , b(t) T u ∗ − c(t) T z ∗ > 0 Thus, (u ∗ , z ∗) is a solution to (2.7) Therefore, instance

(A, b(t), c(t)) is not strictly feasible.

Note that t0 is also cut off, i.e t0 ∈ ˜ T

There exist (u 0 , z 0 ) such that b(t0)T u 0 − c(t0)T z 0 > 0, since (2.7) has solutions for

infeasible point t0 In addition, since (u ∗ , z ∗ ) is the solution to LP (2.4), b(t0)T u ∗ − c(t0)T z ∗ ≥ b(t0)T u 0 −c(t0)T z 0 Thus, b(t0)T u ∗ −c(t0)T z ∗ > 0 Therefore, t0 ∈ ˜ T

Chapter 3

Adjacent Basis Based Algorithm for

Multiparametric Linear Programming

In this chapter, we deduce formulas of p & d boundaries and the cutting hyperplane

based on theories presented in the previous chapter Next, two sub-algorithms aregiven to check the feasibility of a basis and remove redundant linear constraints.Then, the main algorithm, namely adjacent basis based algorithm, is proposed.Then, the method for determining the critical region of a fixed point and themethod for testing the validity of the adjacent basis based algorithm are alsodiscussed Furthermore, numerical experiments are shown to test the feasibility ofthe proposed algorithms Finally, a comparison between the adjacent bases basedalgorithm and the geometric algorithm is carried out

15

3.1 Formulas of p & d boundaries and the cutting hyperplane 16

3.1 Formulas of p & d boundaries and the cutting

hyperplane

Here, formulas to represent p & d boundaries are given and the formula of the

cutting hyperplane which can remove part of the infeasible region is obtained

3.1.1 Deduce p & d boundary

The formula to represent a p boundary is presented in Lemma 3.1 As shown in

the proof, the formula is deduced from Lemma 2.1 and Lemma 2.3

Lemma 3.1 For an instance (A, b(t), c(t)), an optimal basis ¯ B, | ¯ B| = m, i ∈ ¯ B

and B = ¯ B \ {i}, the p boundary related to the optimal basis ¯ B and i is determined

 According to Lemma 2.1, the relation

between (A, b, c) and ( ¯ A, ¯b, ¯c) is

A = Q ¯ Abze, b = Q¯b, c = bze¯c + bze ¯ A T µ

where, 0 < z ∈ R n , nonsingular matrix Q ∈ R m×m , µ ∈ R m , and A ∈ R m×n

3.1 Formulas of p & d boundaries and the cutting hyperplane 17Then we have:

Next, the formula to represent a d boundary is presented in Lemma 3.2 As shown

in the proof, the formula is deduced from Lemma 2.1 and Lemma 2.4

Lemma 3.2 For an instance (A, b(t), c(t)), an optimal basis ¯ B, | ¯ B| = m, i ∈ ¯ N and N = ¯ N \ {i}, the d boundary related to the optimal basis ¯ B and i is determined

3.1 Formulas of p & d boundaries and the cutting hyperplane 18

Proof For simplicity, let (A, b, c) := (A, b(t), c(t)) | ¯ B| = m and i ∈ ¯ N, B =

According to Lemma 2.1, the relationship between (A, b, c) and ( ¯ A, ¯b, ¯c) is

A = Q ¯ Abze, b = Q¯b, c = bze¯c + bze ¯ A T µ

where, 0 < z ∈ R n , nonsingular matrix Q ∈ R m×m , µ ∈ R m , and A ∈ R m×n

3.1 Formulas of p & d boundaries and the cutting hyperplane 19

From Remark 2 of section 2.2, we know that each critical region CR B¯ has m pieces

of p boundaries, with ¯ B = B ∪ {i} for i ∈ ¯ B and n − m pieces of d boundaries,

3.1 Formulas of p & d boundaries and the cutting hyperplane 20

with ¯B = B \ {j} for j ∈ ¯ N Thus, the critical region can be represented by n

pieces of p & d boundaries as follows:

3.1.2 Deduce the cutting hyperplane

From the Cutting Theorem 2.5, we construct the cutting hyperplane for an sible point

infea-For t0 ∈ T , if (A, b(t0), c(t0)) is infeasible, (u ∗ , z ∗) can be obtained by solving thefollowing LP problem:

3.2 Algorithm based on basis partition of the SLP 21

The optimal solution of the LP problem is

parti-tions the infeasible part {t ∈ T |b(t) T u ∗ − c(t) T z ∗ > 0}.

Refer to Theorem 1 (p 178) and Theorem 2 (p 179) in book by Gal [3], we know

that there are either finite critical regions or no feasible point in T , and the set of all critical regions is a closed polyhedral set in T Without loss of generality, we assume that T has finite critical regions.

Next, two sub-algorithms are given to check the feasibility of a basis and removeredundant linear constraints Then, the main algorithm for solving the mpLPproblem, named adjacent basis based algorithm, is proposed

3.2 Algorithm based on basis partition of the SLP 22

3.2.1 Sub-algorithms

In this section, two sub-algorithms are given

• Algorithm 1: Check if basis ¯ B is feasible or not.

• Algorithm 2: Remove redundant linear constraints.

Algorithm 1: Check if basis ¯B is feasible or not

There are two cases where a basis is infeasible Case 1: If A B¯ is a singular matrix

the basis is infeasible Case 2: If the related region CR B¯ does not have an interiorpoint, the basis is infeasible

The set of linear constrains in (3.3) can be reformed as:

Where, t ∈ T , G with appropriate dimension.

CR B¯ has no interior point if the result of the following LP problem is ’no feasiblepoint found’; otherwise it has interior points

min t1s.t Gt < 0

t ∈ T

The algorithm can be summarized as follows:

Step 1, if A B¯ is singular, the basis is infeasible

Step 2, elseif calculate CR B¯ from (3.3) if CR B¯ has no interior point, the basis

is infeasible, elseif it is feasible.

Algorithm 2: Remove redundant linear constraints

To obtain a concise representation of critical region CR B¯, we need to remove

3.2 Algorithm based on basis partition of the SLP 23

redundant constraints from the set of linear constraints We briefly introduce themethod to check if a constrain is redundant or not For details, please refer to

book [3] Let J be the index set of linear constrains in (3.4) Constrain r ∈ J (J ⊆ {1, , n} & |J| = j) is redundant if there is a solution to the following

problem:

min ² r s.t G J t + ² = 0

² ∈ R j×1

such that ² k ≥ 0.

The method for removing redundant constraints can be summarized as follows:

Step 1 : Set J := {1, , n}.

Step 2 : if constrain r ∈ J is redundant, let J ← J \ {r}, elseif J ← J

Step 3 : Repeat step 2 until there is no redundant constrain Then, we get a concise

representation of the critical region G J t ≤ 0.

3.2.2 Main algorithm–Adjacent basis based algorithm

For an instance (A, b(t), c(t)), since b(t) & c(t) is the linear combination of t, let

3.2 Algorithm based on basis partition of the SLP 24

input: instance (A, b 0 , c 0)

output: Baseset, CRset, N CR

Step 1, Set Baseset ← ∅, CRset ← ∅, N CR ← ∅, Bbset ← ∅, Bbrest ← ∅.

Step 2, Find a feasible point and determine the corresponding feasible basis.

For any t0 ∈ T , set t = t0 in mpLP (1.2)

if (A, b(t0), c(t0)) is infeasible, determine a cutting hyperplane by (3.1.2) to removepart of the region Then, choose another point from the rest of the region and test

if the corresponding instance is feasible The process should be repeated until a

feasible point is found elseif (A, b(t0), c(t0)) is feasible, solve the LP problem to

obtain the optimizer x ∗ (t0) The feasible basis ¯B := {j ∈ {1, , n}|x ∗

j > 0} and

let Bbset ← Bbset ∪ { ¯ B}.

Step 3, Construct the critical region for a feasible basis ¯ B.

For ¯B ∈ Bbset, calculate CR0 :=

, where T is the set

of linear inequalities which define the region of parameters Next, execute

Algo-rithm 2 to get nonredundant critical region CR B¯, record the index of nonredundant

constraints into J and record the number of the nonredundant constrains into N CR

Then, let CRset ← CRset ∪ {CR B¯}.

Step 4, Determine adjacent bases of a feasible basis ¯ B.

First, determine the bases of relevant p & d boundaries from the index set J which

is generated in step 3 For the p boundary related to index i, B = ¯ B \{i}, there are

(n − m) adjacent bases of ¯ B, { ¯ B adj := BS{j}, j ∈ ¯ N} For the d boundary related

3.3 Determine the critical region and the optimal value for a given point25

to index i, B = ¯ BS{i}, there are m adjacent bases of ¯ B, { ¯ B adj := B \{j}, j ∈ ¯ B}.

Then, put all the bases which are not in Bbset and Bbrest into Bbset.

Step 5, Pick one basis ¯ B from Bbset, then execute Algorithm 1 if ¯ B is infeasible,

let Bbset ← Bbset \ { ¯ B}, Bbrest ← Bbrest ∪ { ¯ B} Then pick another basis from Bbset to repeat the process until one feasible basis is found elseif ¯ B is feasible,

let Baseset ← Baseset ∪ { ¯ B} and go to Step 3 The process will be finished until

there is no element in Bbset.

opti-mal value for a given point

Once the partitioning of the parametric space is completed, we want to find a

method to determine which critical region a given point t0 belongs to Therefore,the basis and the optimal solution of the point can be obtained without solvingthe corresponding LP problem

In the main algorithm, we store all the feasible bases in Baseset, the corresponding critical regions in CRset and the number of boundaries in each critical region in set N CR Since each critical region is represented as a subset of CRset, we can check if t0 satisfies one of the subsets If t0 satisfies one subset CR B¯, t0 belongs tothe region Then, the basis ¯B is obtained and the optimal solution can be obtained

3.4 Validity test of the adjacent basis based algorithm 26

al-gorithm

This section discusses how to test the validity of the main algorithm For any

t0 ∈ T , we set t = t0 in the mpLP (1.1) Then, the optimal solution x ∗ (t0) and

z ∗

min (t0) can be obtained by solving the LP problem If the solution is the same asthe solution determined by the algorithm shown in section 3.3, the adjacent basisbased algorithm is valid Otherwise, it is invalid

In this section, numerical experiments are given to test the algorithms Examples

1, 2, 3 aim to test the main algorithm with respect to three scenarios, where the

right-hand side b have parameters, the cost c have parameters and both b and c

have parameters separately Example 4 is constructed to test which factors affectthe amount of labor required in the main algorithm Example 5 aims to illustratehow the algorithm obtains the critical region and optimal value for a given pointand test the validity of the main algorithm

Example 1, Consider the mpLP problem with the right-hand side b that has two parameters t1, t2

3.5 Numerical experiments 27

min z = −x1− x2s.t −x1− x2 ≤ −t1− 3t2

Three feasible bases are obtained by solving this example The corresponding

crit-ical regions are constructed by d boundaries since only the cost c have parameters.

The solution is shown as follows

Bases ¯B1 = {1 5}, ¯ B2 = {2 5} and ¯ B3 = {3 5} The corresponding critical regions are CR1, CR2 and CR3

The corresponding optimizers and optimal values are shown in Table 3.1

Example 3, Consider the following mpLP problem with both the right-hand side

Table 3.2: Boundaries for Example 3

b and the cost c having parameters.

By solving this problem, we obtain boundaries {L1, L2, , L13}, which are listed

in Table 3.2 The critical regions CR1, CR2, , CR8 are shown in the Table 3.3which are represented by boundaries The optimal solutions are shown in Table3.4

Table 3.3: Represent of critical regions for Example 3

Example 4, This example aims to test which factors affect the amount of

la-bor required We call (tt, m, n) the size of the problem, where tt = dim(T ), size (A mn )=(m, n) Now, we use MatLab language to construct examples in three cases.

Case 1, only b(t) with parameters, the input (A, b 0 , c 0) is as follows:

A = [randn(m, n−m), eye(m)]; b 0 = rand(m, tt+1); c 0 = [rand(n−m, 1), zeros(n−

m, tt); zeros(m, tt + 1)].

Case 2, only c(t) with parameters, the input (A, b 0 , c 0) is as follows:

A = [randn(m, n − m), eye(m)]; b 0 = [rand(m, 1), zeros(m, tt)]; c 0 = [rand(n −

m, tt + 1); zeros(m, tt)].

Case 3, both b(t) and c(t) with parameters, the input (A, b 0 , c 0) is as follows:

A = [randn(m, n − m), eye(m)]; b 0 = rand(m, tt + 1); c 0 = [rand(n − m, tt + 1); zeros(m, tt + 1)].