EXPLICIT CHARACTERIZATION OF A CLASS OF PARAMETRIC SOLUTION SETS

Transcription

1 Доклади на Българската академия на науките Comptes rendus de l Académie bulgare des Sciences Tome 6, No 10, 009 MATHEMATIQUES Algèbre EXPLICIT CHARACTERIZATION OF A CLASS OF PARAMETRIC SOLUTION SETS Evgenija D. Popova (Submitted by Corresponding Member V. Drensky on June, 009 Abstract Consider a linear system A(p x = b(p, where the elements of the matrix and the right-hand side vector depend affine-linearly on a m-tuple of parameters p = (p 1,...,p m varying within given intervals. It is a fundamental problem how to describe the parametric solution set Σ (A(p, b(p, [p] := {x R n p [p], A(px = b(p}. So far, the solution set description can be obtained by a lengthy (and not unique parameter elimination process. In this paper we introduce a new classification of the parameters with respect to the way they participate in the equations of the system and give numerical characterization for each class of parameters. For a class of parametric linear systems, where each uncertain parameter occurs in only one equation of the system and does not matter how many times within that equation, an explicit characterization of the parametric solution set is derived which generalizes the famous Oettly-Prager theorem for non-parametric linear systems. Key words: linear systems, solution set, interval parameters 000 Mathematics Subject Classification: 65F05, 65G99 1. Introduction. Consider the linear algebraic system (1 ( A(p x = b(p, where a ij (p := a ij,0 + a ij,µ p µ, b i (p := b i,0 + b i,µ p µ, =1 a ij,µ,b i,µ R, µ = 0,...,m, i,j = 1,...,n This work is supported by the Bulgarian National Science Fund under grant No DO 0 359/

2 and the parameters are considered to be uncertain, varying within given intervals (3 p [p] = ([p 1 ],...,[p m ]. Such systems are common in many engineering analysis or design problems, control engineering, robust Monte Carlo simulations, etc., where there are complicated dependencies between the coefficients of the system. The set of solutions to (1 (3, called parametric solution set, is (4 Σ p = Σ (A(p,b(p,[p] := {x R n p [p],a(px = b(p}. Denote by R n, R n m the set of real vectors with n components and the set of real n m matrices, respectively. A real compact interval is [a] = [a,a + ] := {a R a a a + }. By IR n, IR n m we denote the sets of interval n-vectors and interval n m matrices, respectively. For [a] = [a,a + ], define mid-point ǎ := (a + a + / and radius a := (a + a /. The end-point functionals (,( +, as well as the mid-point and radius functionals are applied to interval vectors and matrices componentwise. The well-known non-parametric interval linear system [A]x = [b], which is the most studied in the interval literature can be considered as a special case of the parametric linear system with n + n independent parameters a ij [a ij ], b i [b i ], i,j = 1,...,n, where [a ij ] = a ij ([p] = a ij,0 + a ij,µ [p µ ], [b i ] = b i ([p] = b i,0 + b i,µ [p µ ]. The non-parametric solution set, called also united solution set, is defined as (5 Σ ([A],[b] := {x R n A [A], b [b],a x = b}. The solution set (5 is well studied with a lot of results concerning its characterization and properties, see e.g. [ 1 ]. In particular, the famous Oettly-Prager theorem [ ] characterizes the non-parametric solution set by the inequalities (6 A(ˇpx b(ˇp A ([p] x + b ([p]. It is a fundamental problem of considerable practical importance how to describe the parametric solution set (4 by a logical combination of inequalities. Such a description is useful for visualizing the parametric solution set exploring its properties and for computing componentwise boundaries. So far the solution set description can be obtained by a lengthy (and not unique parameter elimination process [ 3 ] shortly recalled in Section. Explicit descriptions of the symmetric and skew-symmetric solution sets are given in [ 4 ]. The goal of this paper is to give an explicit characterization of the parametric solution set for another class 108 E. D. Popova

3 of parametric systems. To this end we introduce in Section 3 a new classification of the parameters with respect to the way they participate in the equations of the system. In Section 4 the parameter elimination process is studied in details for a class of parametric linear systems, where each uncertain parameter occurs in only one equation of the system and does not matter how many times the parameter occurs within that equation. For such systems we give a simple explicit characterization of the parametric solution set which generalizes the famous Oettly-Prager theorem. Some important properties regarding the elimination of zero and 1st class parameters are explored. The explicit parametric solution set characterization is illustrated in Section 5 on some numerical examples and its advantages are compared to other approaches.. Fourier-Motzkin like elimination of parameters. The solution set (4 is characterized by a trivial set of inequalities as follows: Σ p = {x R n p µ R,µ = 1,...,m : (7 (8 hold}, where for i = 1,...,n (7 (8 a ij,0 + a ij,0 + a ij,µ p µ x j b i,0 + a ij,µ p µ x j, p µ p µ p + µ, b i,µ p µ µ = 1,...,m. Starting from such a description, Theorem.1 below shows how the parameters in this set can be eliminated successively in order to obtain a new description not involving p µ, µ = 1,...,m. Theorem.1 (Alefeld et al. [ 3 ]. Let f λµ, g λ, λ = 1,...,k (, µ = 1,...,m, be real-valued functions of x = (x 1,...,x n on some subset D R n. Assume that there is a positive integer k 1 < k such that: f λ1 (x 0 for all λ {1,...,k}; f λ1 (x 0 for all x D and all λ {1,...,k}; for each x D there is an index β = β (x {1,...,k 1 } with f β 1(x > 0 and an index γ = γ (x {k 1 +1,...,k} with f γ 1(x > 0. For m parameters p 1,...,p m varying in R and for x varying in D define the sets S 1,S by S 1 S := {x D p µ R,µ = 1,...,m : (9,(10 hold}, := {x D p µ R,µ =,...,m : (11 holds}, Compt. rend. Acad. bulg. Sci., 6, No 10,

4 where inequalities (9, (10 and (11 respectively, are given by (9 (10 g β (x + f βµ (xp µ f β1 (xp 1, β = 1,...,k 1, µ= f γ1 (xp 1 g γ (x + f γµ (xp µ, γ = k 1 + 1,...,k µ= and for β = 1,...,k 1, γ = k 1 + 1,...,k (11 g β (xf γ1 (x + f βµ (xf γ1 (xp µ g γ (xf β1 (x + f γµ (xf β1 (xp µ. µ= µ= (Trivial inequalities such as 0 0 can be omitted. Then S 1 = S. Theorem.1 defines the transition from inequalities (9, (10 to inequalities (11, where the parameter p 1 does not occur. The parameter elimination process resembles the so-called Fourier-Motzkin elimination of variables, see e.g. [ 5 ]. As it is demonstrated below, it is a lengthy and not unique process. Therefore explicit parametric solution set characterizations are of particular interest. 3. Classification of the parameters. Here we classify the parameters involved in the system into three classes with respect to the way they participate in the parametric system. Our goal is to reveal the specific way by which the elimination of each class of parameters updates the set of characterizing inequalities. With the notations A µ := (a ij,µ R n n b µ ( := (b i,µ R n, µ = 0,...,m the system (1 can be rewritten equivalently as A 0 + m p µa µ x = b 0 + m p µb µ. For a matrix A R n n, A m denotes the m-th row of A. Definition 3.1. A parameter p µ, 1 µ m, is of class zero if it is involved only in the right-hand side and only in one equation of system (1. A parameter p µ is of class zero iff A µ = 0 R n n and only one component of the numerical vector b µ is nonzero (b µ i 0 for exactly one i, 1 i n. For example, the parameter p 3 involved in the system from Example 5.1 is of class zero. It is obvious that the elimination of each parameter of class zero, by applying Theorem.1, removes one couple of parameter inequalities (8 and updates the inequalities (7 without changing their number. Definition 3.. A parameter p µ, 1 µ m, is of 1st class if it is not of class zero and occurs in only one equation of system (1 and does not matter how many times it appears within that equation. A parameter p µ is of 1st class iff A µ 0 and b µ A µ x has only one nonzero component (that is b µ i Aµ i x 0 for exactly one i, 1 i n. For example, the parameters p 1 and p involved in the system from Example 5.1 are parameters of 1st class. All parameters, except p n, involved in the system from Example 5. are parameters of 1st class. The elimination of every parameter of 1st class removes 110 E. D. Popova

5 one couple of parameter inequalities (8. We demonstrate in the next section that eliminating the parameters of 1st class by applying Theorem.1 to the end-point parameter inequalities (8 updates the inequalities (7 expanding exponentially their number. An efficient elimination procedure will be defined for parameters of zero and 1st class and an explicit characterization of the solution set to a special class of parametric linear systems will be derived in Section 4. Definition 3.3. A parameter p µ, 1 µ m, is of nd class if it is involved in more than one equation of system (1. A parameter p µ is of nd class iff the vector b µ A µ x has more than one nonzero components. During the elimination procedure by Theorem.1 the elimination of each parameter of nd class removes one couple of parameter inequalities (8, updates those inequalities (7 corresponding to the nonzero components of b µ A µ x and expands the total number of characterizing inequalities. The elimination process involving nd class parameters will be studied in another article. 4. A special class of parametric systems. For λ R, define sign(λ := {+ if λ 0, if λ < 0} and apply the sign functional to vectors and matrices componentwise. Denote by U(s := { sign(u u R s, u = (1,...,1 } the set of all s-dimensional sign vectors, where the absolute value u is understood componentwise. The set U(s consists of Card(U(s = s elements. For σ {+, } and {, }, denote a σ 0 to be equivalent to a 0 if σ = + and to be equivalent to a 0 if σ =. These relations are applied to vectors componentwise. Let all the uncertain parameters participating in system (1 3 be of zero and 1st class and l n equations involve all m uncertain parameters. Denote by i j, 1 j l, the indexes of the equations involving uncertain parameters. Let us suppose that (m i1 + r i1 + (m i + r i + + (m il + r il = m, where n ij := m ij + r ij, 1 j l, is the number of the parameters participating in the i j -th equation and 0 r ij n ij of these parameters occur only in the right-hand side of this equation. Denote by K(r ij := {k 1,...,k rij } the set of indexes of the parameters involved only in the r.h.s. of the equation i j and K(m ij := {k 1,...,k mij } be the set of indexes of the other parameters involved in equation i j, such that K(m ij K(r ij =, K ij := K(m ij K(r ij, and Card ( K ij = nij 1. Let fix a i j, for which m ij 1, r ij 1. For a fixed µ K ij denote c(µ,x := b µ A µ x and its i j -th component is c ij (µ,x := b µ i j A µ i j x. Then for a fixed µ from p µ p µ p + µ we obtain (1 c ij (µ,xp σ i j µ c ij (µ,xp µ c ij (µ,xp σ i j µ, where σ ij = sign(c ij (µ,x. Compt. rend. Acad. bulg. Sci., 6, No 10,

6 Denote K := {1,...,m} \ K ij, A := b 0 A 0 x + (b µ K ij Ξ := ( σ ij (µ µ K(m ij, µ A µ x and where σ i j (µ = sign(c ij (µ,x {+, }. Inequalities (7 can be rewritten equivalently as (13 0 b i,0 b i,0 a ij,0 x j + a ij,0 x j + p µ b i,µ b i,µ a ij,0 x j p µ a ij,0 x j 0, i = 1,...,n. Then the application of Theorem.1 to the i j -th couple of inequalities (13 and the inequalities (1 for µ K ij removes the parameters p µ, µ K ij. Thus we obtain the following equivalent representation for the parametric solution set: Σ p = x Rn p µ R,µ = 1,...,m,µ K ij : Ξ U(m ij 0 A ij + A ij + c ij (,xp σ i j ( + c ij (,xp σ i j ( + K(r ij K(r ij p σ(b i j p σ(b i j 0 c ij (,x σ i ( j n 0 A s = 0 p µ p µ p + µ s=1,s =i j µ K ij. Expanding this result over all parameters in the system we obtain the following explicit characterization of the parametric solution set: 11 E. D. Popova

7 (14 (15 Σ p = x R n l 0 A ij + Ξ U( È l m i j A ij + c ij (,xp σ i j ( + c ij (,xp σ i j ( + c ij (,x σ i ( j 0 K(r ij K(r ij n p σ(b i j p σ(b i j A s = 0 s=1 s l K i j 0. Thus applying the elimination process, as described in [ 3 ], we obtain an explicit characterization of the solution set to a parametricsystem involving only zero È l l and 1st class parameters that consist of m i j l + inequalities and n l equalities. The number of characterizing inequalities can be reduced È l l if we replace in (15 the set of m i j m ij inequalities c ij (,x σ i j ( 0 by the set consisting of l m ij m ij inequalities l µ K(m ij c(µ,i j 0. Thus the last characterization of the considered parametric solution set involves È l m i j l+ l m i j inequalities and n l equalities. As it is seen, the number of characterizing inequalities grows exponentially with the number of 1st class parameters. In order to reduce further the number of inequalities characterizing the solution set of a linear system involving only parameters of zero and 1st class, instead of parameter inequalities p µ p µ p + µ, we shall use the equivalent inequalities ˇp µ p µ p µ ˇp µ + p µ for the parameters of 1st class. Thus for λ R we have (16 λˇp µ λ p µ λp µ λˇp µ + λ p µ. Then the elimination process, based on relations (16 instead of relations (1, leads to the following inequalities: Compt. rend. Acad. bulg. Sci., 6, No 10,

8 l (17 A ij + K(r ij 0 A ij + n K(r ij A s = 0 s=1 s l K i j p σ(b i j p σ(b i j + + c ij (,xˇp c ij (,xˇp + c ij (,x p 0 c ij (,x p which can be rewritten in the following more general but equivalent forms or l b ij (ˇp A ij (ˇpx b µ i j A µ i j x p µ 0 µ K ij 0 b ij (ˇp A ij (ˇpx + b µ i j A µ i j x p µ b 0 s A 0 s x = 0, µ K ij s l K i j (18 b(ˇp A(ˇpx m b µ A µ x p µ 0 b(ˇp A(ˇpx+ b µ A µ x p µ. In this way we have proven the following theorem which generalizes the famous Oettly-Prager solution set characterization (6 for parametric systems involving only parameters of zero and 1st class. Theorem 4.1. The solution set of system (1 (3 involving only uncertain parameters of zero and 1st class has the following explicit characterization: Σ p = {x R n A(ˇpx b(ˇp b µ A µ x p µ }. The solution set characterization (18, respectively Theorem 4.1, can be obtained by a parameter elimination process specified by the following proposition. Proposition 4.1. Let f µ, g 1,g, µ = 1,...,m, be real-valued functions of x = (x 1,...,x n on some subset D R n. For m parameters p 1,...,p m varying in R, where the -th parameter is of zero or 1st class, f (x 0, {1,...,m}, 114 E. D. Popova

9 and for x varying in D, it holds S 1 = S where the sets S 1, S are defined by S 1 := {x D p µ R,µ = 1,...,m : p p p + g 1 (x f µ (xp µ f (xp g (x f µ (xp µ }, µ S := {x D p µ R,µ = 1,...,m, µ : g 1 (x f µ (xp µ f (x ˇp + f (x p µ f (x pˇ f (x p g (x µ f µ (xp µ }. µ Proof. The proposition follows from Theorem.1 and the equivalence of the relations p p p + and ˇp p p ˇp + p. Being specialized for parameters of zero and 1st class, Proposition 4.1 removes the restriction of Theorem.1 for positiveness of the coefficient functions f λ1, λ = 1,..., k in (9, (10. Furthermore, utilizing the midpoint-radius representation of intervals, Proposition 4.1 allows updating the inequalities (7 without changing their number. Thus Proposition 4.1 defines a more efficient elimination process for parameters of zero and 1st class than Theorem.1. Theorem 4.1 characterizes the considered special class of parametric solution sets by n inequalities and this number does not depend on the number of the involved uncertain parameters. As being evident from the characterizations (14 (15 or (17, the inequalities characterizing this special class of parametric solution sets are linear. Furthermore, the order of elimination of the parameters is not significant for the characterization of this class parametric solution sets. 5. Numerical examples. Example 5.1. Consider a simple parametric system ( p1 1 + p p p x = ( p3 1 3p, where p 1,p [0,1],p 3 [ 1,1]. Applying Theorem 4.1, the parametric solution set is characterized by x 1 x 3 x 1 + x 1 x + x 1 + x 1 + x 1 3 x 1 + x x 1 x x 1 + x. For comparison, the Mathematica R function Resolve, doing quantifier elimination, characterizes the solution set by 181 inequalities. Compt. rend. Acad. bulg. Sci., 6, No 10,

10 Example 5.. Consider the parametric system A (px = b(p, where A(p R n n is the Milnes matrix [ 6 ] { 1 j i, A(p = (a ij (p, a ij (p = p j j < i, b(p = (p 1,...,p n, ˇp i = 1/(i + 1, p i = % ˇp i = 1/(50i According to Theorem 4.1 the solution set of this problem is characterized by the following system of n inequalities i (i x j x j /(i x j /(50i n i=1 0 (i j=i+1 i x j j=i+1 j=i+1 x j /(i j=i+1 x j /(50i For n = 0, e.g., the solution set characterization based on Theorem.1 and the end-point parameter inequalities (8 would require inequalities. The execution of the Mathematica R function Resolve on the same problem for n = 5 took seconds which was about 490 times slower than an interpreted implementation of Theorem 4.1. REFERENCES [ 1 ] Alefeld G., V. Kreinovich, G. Mayer. Computing Suppl., 16, 00, 1. [ ] Oettli W., W. Prager. Numer. Math., 6, 1964, [ 3 ] Alefeld G., V. Kreinovich, G. Mayer. J. Comput. Appl. Math., 15, 003, [ 4 ] Hladík M. Description of symmetric and skew-symmetric solution set, KAM-DIMATIA Series , 007. [ 5 ] Schrijver A. Theory of Linear and Integer Programming, New York, Wiley, [ 6 ] Milnes H. W. Math. Comput.,, 1968, No 104, Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev Str., Bl Sofia, Bulgaria epopova@bio.bas.bg 116 E. D. Popova