Adjacent basis based algorithm for multiparametric linear programming

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 Acknowledgements First of all, I am grateful to Professor Zhao Gongyun, my thesis supervisor. His kind and patient guidance has provided the most impetus for my study and research, without which this thesis would not have been completed. I’m particularly indebted to his advice in how to fit a research project in a big picture and deduce algorithms from theories. In addition, I deeply appreciate the encouragement and help from my friends in NUS, 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 family members should not be ignored. It is their encouragement and warm care for both my study and daily life that make me still energetic. Chen Fei July 2009 ii Contents Acknowledgements ii Summary v List of Tables vii Introduction Preliminaries 2.1 Relating linear programming to Grassmann Manifold . . . . . . . . 2.2 The characterization and construction of critical regions and their boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Adjacent Basis Based Algorithm for Multiparametric Linear Programming 15 3.1 Formulas of p & d boundaries and the cutting hyperplane . . . . . . 16 3.1.1 16 Deduce p & d boundary . . . . . . . . . . . . . . . . . . . . iii Contents 3.1.2 3.2 iv Deduce the cutting hyperplane . . . . . . . . . . . . . . . . 20 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 . . . . . . . . . . . . . . . . . . . . Further Research on Adjacent Basis 38 40 4.1 Determine p boundary related feasible adjacent bases . . . . . . . . 40 4.2 Determine d boundary related feasible adjacent bases . . . . . . . . 46 Conclusions 50 A Source of Program X 52 Summary This thesis presents a novel algorithm, namely adjacent basis based algorithm, to systematically identify all the critical regions in a parameter space for solving multiparametric linear programming (mpLP) problems, based on the studies of Zhao about 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 boundaries. 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 determine their corresponding critical regions in the parameter space. An introduction is given in Chapter to provide an overview of the subject in question. Chapter briefly introduces the relevant definitions and how to relate the SLP to Grassmann Manifold. We also describe the characterization and construction of critical regions and their boundaries. Then, we propose the cutting v Summary 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 main algorithm is proposed to make the latter clear. The method for determining the critical region of a fixed point and the method for testing the validity of the adjacent basis based algorithm are also discussed. Furthermore, numerical experiments are shown to test the feasibility of the proposed algorithms. Finally, a comparison between the adjacent bases based algorithm and the geometric algorithm is carried out. In Chapter 4, further research is done to filter feasible bases from all adjacent bases. Conclusions are given in Chapter 5. vi List of Tables 3.1 Optimal solution for Example . . . . . . . . . . . . . . . . . . . . 29 3.2 Boundaries for Example . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Represent of critical regions for Example . . . . . . . . . . . . . . 30 3.4 Optimal solution for Example . . . . . . . . . . . . . . . . . . . . 31 3.5 Results for different cases in Example . . . . . . . . . . . . . . . . 32 3.6 Results for case with two parameters tt = . . . . . . . . . . . . 32 3.7 Results for case with four parameters tt = . . . . . . . . . . . . 33 vii Chapter Introduction The multiparametric linear programming (mpLP) problem is as follows: z(t) = cT (t)x s.t. A(t)x ≤ b(t) (1.1) x≥0 where A ∈ Rm×(n−m) , b ∈ Rm , x ∈ Rn−m , c ∈ Rn−m , z ∈ R and t ∈ T is the vector of parameters. 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 by Dantzig. Since then, extensive research has been done on the topic [3-14]. For instance, the book by Gal [3] contains hundreds of references related to sensitivity and parametric analysis. Generally, the study of mpLP problems has focused on two levels: sensitivity analysis and parametric programming. Sensitivity analysis characterizes the change of the solution with respect to a small perturbation of the parameters and provides a solution in the neighborhood of the nominal value of the parameters. Parametric programming is based on the sensitivity analysis theory 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 and Nedoma [2]. It enumerates all optimal bases of the associated LP tableau using a method derived from the simplex algorithm. Subsequently, similar methods were proposed by Yu, Zeleny [12] and Schechter [5]. However, these methods are very sensitive to the number of parameters. As it works with polyhedral in the parameter space, the amount of computing required by the method increases exponentially with the increasing number of parameters. Only recently, an essentially different method named geometric algorithm was proposed by Borrelli and Bemporad et al. [9, 10, 11], which uses the geometric properties of the problem to explore the parameter space. It directly partitions the parameter space into subsets to obtain critical regions. However, it cannot solve problems with high dimensional parameter space, because the number of new sub-regions defined by the partitioning strategy increases exponentially as the dimension of the parameter space increases. Furthermore, this method introduces a large number of artificial cuts in the parameter space. Fortunately, the study of the space of linear programs (SLP) provides a new perspective on understanding essential relationships among critical regions and the nature of LP. Works of Zhao [16, 17, 18] introduces a representation and some basic structures of SLP, including detailed geometric structures of the critical regions and their boundaries in the SLP. Characterization of the basis partition can potentially lead to new methods for solving parametric LP. This motivates us to study 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: z(t) = cT (t)x s.t. Ax = b(t) (1.2) x≥0 where A ∈ Rm×n is of full row rank, b ∈ Rm , c ∈ Rn , x ∈ Rn and t ∈ T is the vector of parameters. 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 of decision variables and constrains. We briefly introduce the relationship between SLP and Grassmann Manifold which has rich geometric and algebraic structures. Then, we describe the characterization and construction of critical regions and their boundaries. Finally, the cutting theorem is presented as a method to remove some of the infeasible points from the parameter space. Chapter 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 the feasibility of a basis and remove redundant linear constraints. Then, the main algorithm, namely adjacent basis based algorithm, is proposed. In the algorithm, the cutting theorem is used to efficiently find a feasible point in the parameter space. 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 Mickey 55 %%%%%%%%%%move the same boundary %%%%%%%%%%%%%%%% [nc1,nc2]=size(nrCr); if length(Crcheck)==0 for k=1:nc1 nrCrt(k,:)=nrCr(k,:)/norm(nrCr(k,:)); end Crcheck=[Crcheck;nrCrt]; else temp=[]; for k=1:nc1 nrCrt(k,:)=nrCr(k,:)/norm(nrCr(k,:)); rt=1-max(abs(Crcheck*nrCrt(k,:)’)); if rt[...]... Next, two sub-algorithms are given to check the feasibility of a basis and remove redundant linear constraints Then, the main algorithm for solving the mpLP problem, named adjacent basis based algorithm, is proposed 3.2 Algorithm based on basis partition of the SLP 3.2.1 22 Sub-algorithms In this section, two sub-algorithms are given ¯ • Algorithm 1: Check if basis B is feasible or not • Algorithm 2:... presented in the previous chapter Next, two sub-algorithms are given 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 the method for testing the validity of the adjacent basis based algorithm are also discussed Furthermore, numerical... 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 −c(t0 )T z Thus, b(t0 )T u∗ −c(t0 )T z ∗ > 0 Therefore, t0 ∈ T 14 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... the basis B is obtained and the optimal solution can be obtained ∗ x∗¯ (t0 ) = A−1 b(t0 ) x∗¯ = 0 and zmin (t0 ) = cT (t0 )A−1 b(t0 ) Otherwise, it belongs to ¯ ¯ ¯ B N B Bi Bi no region, i.e the point is infeasible 3.4 Validity test of the adjacent basis based algorithm 3.4 Validity test of the adjacent basis based algorithm This section discusses how to test the validity of the main algorithm For. .. adjacent basis based algorithm is discussed Numerical experiments are shown to implement and test the algorithms in section 3.5 Furthermore, a comparison between the adjacent bases based algorithm and the geometric algorithm is carried out 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... feasibility of adjacent bases is given in Chapter 4 In Chapter 5, conclusions include the characters of the adjacent basis based algorithm and an important topic for future research 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 introduce the relationships between linear programs... shown to test the feasibility of the proposed algorithms Finally, a comparison between the adjacent bases based algorithm and the geometric algorithm is carried out 15 3.1 Formulas of p & d boundaries and the cutting hyperplane 3.1 16 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... 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 AB is a singular matrix ¯ the basis is infeasible Case 2: If the related region CRB does not have an interior ¯ point, the basis is infeasible The set of linear constrains in (3.3) can be reformed as: Gt ≤ 0 (3.4) Where, t ∈ T , G... s.t Gt < 0 t∈T The algorithm can be summarized as follows: Step 1, if AB is singular, the basis is infeasible ¯ Step 2, elseif calculate CRB from (3.3) if CRB 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 CRB , we need to remove ¯ 3.2 Algorithm based on basis partition of... optimal basis B If CRB = Ø, the critical region is considered feasible The corresponding basis B is a feasible basis Definition 4 Two critical regions CRB1 , CRB2 are called adjacent if their intersection CRB1 ∩ CRB2 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 adjacent critical regions The boundary between adjacent . ADJACENT BASIS BASED ALGORITHM FOR MULTIPARAMETRIC LINEAR PROGRAMMING CHEN FEI (B.Sc., QFNU) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT. 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. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Adjacent Basis Based Algorithm for Multiparametric Linear Pro- gramming 15 3.1 Formulas of p & d boundaries and the cutting hyperplane