AE solutions and AE solvability to general interval linear systems
aa r X i v : . [ m a t h . NA ] M a y AE solutions and AE solvability to general interval linearsystems
Milan Hlad´ık ∗ September 3, 2018
Abstract
We consider linear systems of equations and inequalities with coefficients varying insidegiven intervals. We define their solutions (so called AE solutions) and solvability (so called AEsolvability) by using forall-exists quantification of interval parameters. We present an explicitdescription of the AE solutions, and discuss complexity issues as well. For AE solvability,we propose a sufficient condition only, but for a specific sub-class of problems, a completecharacterization is developed. Moreover, we investigate inequality systems for which AEsolvability is equivalent to existence of an AE solution.
Interval linear systems appear in many situations. The basic problem of solving interval linearequations [3, 18, 19, 28, 31] is important in solving and verifying real-valued linear and nonlinearsystems, and in solving engineering problems with uncertain data. Interval linear inequalitiesemerge in global optimization when linearization techniques are used [10] and in mathematicalprogramming when dealing with uncertainty [2, 6, 8, 9].Traditionally, a solution of an interval system is defined as a solution for some realizationof intervals. In order to model robustness in interval equations solving, generalized concepts ofsolutions using quantifications appeared. The commonly used one is an AE solution [4, 5, 21, 22,31], characterized by ∀∃ -quantification of interval parameters.To the best of our knowledge, the concept of AE solutions has not been utilized for intervalinequalities yet. There are, however, many special cases studied. In interval linear programming,for example, the concept of quantified solutions was recently introduced in [14, 15, 16].In this paper, we study AE solutions for general interval linear systems of equations, inequal-ities or both. Its direct applicability is in interval linear programming to characterize variousmodels of robust solutions. Notation.
The sign of a real r is defined as sgn( r ) = 1 if r ≥ r ) = − s ) stands for the diagonal matrix with entriesgiven by s .An interval matrix is defined as A := { A ∈ R m × n ; A ≤ A ≤ A } , where A and A , A ≤ A , are given matrices, and the inequality between matrices is understoodcomponentwise. The midpoint and radius matrices are defined as A c := 12 ( A + A ) , A ∆ := 12 ( A − A ) . ∗ Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostransk´en´am. 25, 11800, Prague, Czech Republic, e-mail: [email protected] A ) := 12 ( A + A ) , rad( A ) := 12 ( A − A ) . The set of all m × n interval matrices is denoted by IR m × n . Naturally, intervals and interval vectorsare considered as special cases of interval matrices. For interval arithmetic see, e.g., [1, 18, 19].Given A ∈ IR m × n and b ∈ IR m , the corresponding interval linear system of equations is thefamily of systems Ax = b, A ∈ A , b ∈ b . (1) Solution concepts.
There are different definitions of a solution of the interval system (1); cf.[3]. We say that x ∈ R n is • a (weak) solution if ∃ A ∈ A , ∃ b ∈ b : Ax = b , • a strong solution if ∀ A ∈ A , ∀ b ∈ b : Ax = b , • a tolerable solution if ∀ A ∈ A , ∃ b ∈ b : Ax = b , • a controllable solution if ∀ b ∈ b , ∃ A ∈ A : Ax = b .Similarly, we define analogous solutions for other types of linear systems (inequalities, or mixedequations and inequalities).Weak solutions are the most commonly used ones [18, 19]. Strong solutions are more ap-propriate in the context of interval inequalities [3, 7]. Tolerable solutions were studied, e.g., by[22, 24, 31, 32], and controllable solutions in [22, 30, 31].The above solution types were generalized to the so called AE solutions [4, 5, 22, 31]. Eachinterval is associated either with the universal, or with the existential quantifier. Thus, we cansplit the interval matrix as A = A ∀ + A ∃ , where A ∀ is the interval matrix comprising univer-sally quantified coefficients, and A ∃ concerns existentially quantified coefficients. Similarly, wedecompose the right-hand side vector b = b ∀ + b ∃ . Now, x ∈ R n is an AE solution if ∀ A ∀ ∈ A ∀ , ∀ b ∀ ∈ b ∀ , ∃ A ∃ ∈ A ∃ , ∃ b ∃ ∈ b ∃ : ( A ∀ + A ∃ ) x = b ∀ + b ∃ . In the same manner we define AE solutions for interval inequalities and other interval linearsystems.The following characterization of AE solutions is from [31].
Theorem 1.
A vector x ∈ R n is an AE-solution to interval equations A x = b if and only if | A c x − b c | ≤ (cid:0) A ∃ ∆ − A ∀ ∆ (cid:1) | x | + b ∃ ∆ − b ∀ ∆ . Goal.
The purpose of this paper is to generalize the above characterization of AE solutions togeneral interval systems, including inequalities or mixed systems of equations and inequalities(Section 2). The second focus is on the related problem of AE solvability (Section 3), which is,however, a more difficult problem.
Before we state a characterization of AE solutions for the general case, we state a specific caseof inequalities first. 2 roposition 1.
A vector x ∈ R n is an AE-solution to interval inequalities A x ≤ b if and onlyif A c x − b c ≤ (cid:0) A ∃ ∆ − A ∀ ∆ (cid:1) | x | + b ∃ ∆ − b ∀ ∆ . (2) Proof.
An AE solution must satisfy ∀ A ∀ ∈ A ∀ ∀ b ∀ ∈ b ∀ ∃ A ∃ ∈ A ∃ ∃ b ∃ ∈ b ∃ : A ∀ x − b ∀ ≤ b ∃ − A ∃ x. By eliminating the existential quantifiers, we equivalently get ∀ A ∀ ∈ A ∀ ∀ b ∀ ∈ b ∀ : A ∀ x − b ∀ ≤ b ∃ − A ∃ x, and by eliminating the universal quantifiers, we arrive at A ∀ x − b ∀ ≤ b ∃ − A ∃ x. This condition can be formulated as A ∀ c x + A ∀ ∆ | x | − b ∀ ≤ b ∃ − A ∃ c x + A ∃ ∆ | x | , which is equivalent to the form (2).Now, we extend the above results to an interval system in a general form. Consider a linearsystem Ax + By = a, Cx + Dy ≤ b, x ≥ , (3)where the constraint matrices and right-hand side vectors vary in given interval matrices A ∈ IR m × n , B ∈ IR m × n ′ , C ∈ IR m ′ × n , D ∈ IR m ′ × n ′ , and interval vectors a ∈ IR m , and b ∈ IR m ′ .We briefly denote this interval system as A x + B y = a , C x + D y ≤ b , x ≥ . (4)Each interval linear system can be transformed to this formulation [6], so it serves as a generalform of interval linear systems. Proposition 2.
A pair of vectors ( x, y ) ∈ R n + n ′ is an AE-solution to (4) if and only if | A c x + B c y − a c | ≤ (cid:0) A ∃ ∆ − A ∀ ∆ (cid:1) x + (cid:0) B ∃ ∆ − B ∀ ∆ (cid:1) | y | + a ∃ ∆ − a ∀ ∆ , (5a) C c x + D c y − b c ≤ (cid:0) C ∃ ∆ − C ∀ ∆ (cid:1) x + (cid:0) D ∃ ∆ − D ∀ ∆ (cid:1) | y | + b ∃ ∆ − b ∀ ∆ , (5b) x ≥ . (5c) Proof. (5 a ) follows from Theorem 1 applied on A x + B y = b and utilizing nonnegativity of x .Similarly, (5 b ) follows from Proposition 1 applied on C x + D y ≤ a .In Table 1, we list some special cases of AE solutions and feasibility. For equations, we getalmost the same results as in [3]; the only difference is for strong solutions. However, in view of b ≤ b the condition Ax ≥ b, Ax ≤ b, x ≥ Ax = b = Ax = b, x ≥
0, or A c x = b c , A ∆ x = b ∆ = 0, which is the characterization from [3]. For weak and strong solutionsof inequalities (both cases), the characterizations also coincide to known results [3, 7, 27]. Theother cases (tolerable and controllable solutions of inequalities) seem not to be published yet. Proposition 3.
The set of AE solutions is a union of at most m ′ convex polyhedral sets. Ax = b Ax = b, x ≥ Ax ≤ b Ax ≤ b, x ≥ | A c x − b c | ≤ A ∆ | x | + b ∆ Ax ≤ b, Ax ≥ b, x ≥ A c x ≤ A ∆ | x | + b Ax ≤ b, x ≥ | A c x − b c | ≤ − A ∆ | x | − b ∆ Ax ≥ b, Ax ≤ b, x ≥ A c x + A ∆ | x | ≤ b Ax ≤ b, x ≥ | A c x − b c | ≤ − A ∆ | x | + b ∆ Ax ≤ b, Ax ≥ b, x ≥ A c x + A ∆ | x | ≤ b Ax ≤ b, x ≥ | A c x − b c | ≤ A ∆ | x | − b ∆ Ax ≤ b, Ax ≥ b, x ≥ A c x − A ∆ | x | ≤ b Ax ≤ b, x ≥ Proof.
Let s ∈ {± } n ′ . The AE solution set in the orthant diag( s ) y ≥ A c x + B c y − a c ≤ (cid:0) A ∃ ∆ − A ∀ ∆ (cid:1) x + (cid:0) B ∃ ∆ − B ∀ ∆ (cid:1) diag( s ) y + a ∃ ∆ − a ∀ ∆ , − A c x − B c y + a c ≤ (cid:0) A ∃ ∆ − A ∀ ∆ (cid:1) x + (cid:0) B ∃ ∆ − B ∀ ∆ (cid:1) diag( s ) y + a ∃ ∆ − a ∀ ∆ ,C c x + D c y − b c ≤ (cid:0) C ∃ ∆ − C ∀ ∆ (cid:1) x + (cid:0) D ∃ ∆ − D ∀ ∆ (cid:1) diag( s ) y + b ∃ ∆ − b ∀ ∆ ,x ≥ A ∃ + A ∀ ) x + (cid:0) B c − B ∃ ∆ diag( s ) + B ∀ ∆ diag( s ) (cid:1) y ≤ a ∃ + a ∀ , (6a) − ( A ∃ + A ∀ ) x + (cid:0) − B c − B ∃ ∆ diag( s ) + B ∀ ∆ diag( s ) (cid:1) y ≤ − a ∃ − a ∀ , (6b)( C ∃ + C ∀ ) x + (cid:0) D c − D ∃ ∆ diag( s ) + D ∀ ∆ diag( s ) (cid:1) y ≤ b ∃ + b ∀ , (6c) x ≥ s ∈{± } n ′ .As a consequence, we obtain the following method for finding an AE solution. Corollary 1.
A pair of vectors ( x, y ) ∈ R n + n ′ is an AE-solution to (4) if and inly if they satisfy (6) for some s ∈ {± } n ′ . Given an AE solution ( x ∗ , y ∗ ), the following natural question arises: For a realization of ∀ -parameters, what are the values of ∃ -parameters, for which ( x ∗ , y ∗ ) remains to be a solution? Proposition 4.
Let A ∀ ∈ A ∀ , B ∀ ∈ B ∀ , C ∀ ∈ C ∀ , D ∀ ∈ D ∀ , a ∀ ∈ a ∀ , and b ∀ ∈ b ∀ . Then ( x ∗ , y ∗ ) solves (3) for the setting A ∃ = A ∃ c − diag ( u ) A ∃ ∆ , (7a) B ∃ = B ∃ c − diag ( u ) B ∃ ∆ diag (sgn( y ∗ )) , (7b) a ∃ = a ∃ c + diag ( u ) b ∃ ∆ , (7c) C ∃ = C ∃ , (7d) D ∃ = D ∃ c − D ∃ ∆ diag (sgn( y )) , (7e) b ∃ = b ∃ , (7f)4 here u ∈ [ − , m is defined entrywise as u i = ( A c x + B c y − a c ) i (cid:0) A ∃ ∆ x + B ∃ ∆ | y | + a ∃ ∆ (cid:1) i if (cid:0) A ∃ ∆ x + B ∃ ∆ | y | + a ∃ ∆ (cid:1) i > , otherwise . Proof.
Since ( x ∗ , y ∗ ) is an AE solution, it must be a weak solution to( A ∃ + A ∀ ) x + ( B ∃ + B ∀ ) y = a ∃ + a ∀ , ( C ∃ + C ∀ ) x + ( D ∃ + D ∀ ) y ≤ b ∃ + b ∀ , x ≥ . By Hlad´ık [7], ( x ∗ , y ∗ ) solves the constraints for (7). Checking whether a given pair ( x, y ) is an AE solution is easy by checking (5). On the otherhand, computing an AE solution, or just checking whether there exists any AE solution, may bea computationally hard problem.Some special cases of (5) are polynomially solvable by reducing to linear system of equationsand inequalities and utilizing polynomiality of linear programming [29].
Proposition 5. If B ∃ ∆ = 0 and D ∃ ∆ = 0 , then computing an AE solution or checking its existenceis a polynomial time problem.Proof. Under the assumption, the AE solution set is described by | A c x + B c y − a c | ≤ (cid:0) A ∃ ∆ − A ∀ ∆ (cid:1) x − B ∀ ∆ | y | + a ∃ ∆ − a ∀ ∆ ,C c x + D c y − b c ≤ (cid:0) C ∃ ∆ − C ∀ ∆ (cid:1) x − D ∀ ∆ | y | + b ∃ ∆ − b ∀ ∆ , x ≥ . Equivalently, it is the projection to the ( x, y )-subspace of the convex polyhedral set described in( x, y, z )-space as A c x + B c y − a c + B ∀ ∆ z ≤ (cid:0) A ∃ ∆ − A ∀ ∆ (cid:1) x + a ∃ ∆ − a ∀ ∆ , − A c x − B c y + a c + B ∀ ∆ z ≤ (cid:0) A ∃ ∆ − A ∀ ∆ (cid:1) x + a ∃ ∆ − a ∀ ∆ ,C c x + D c y − b c + D ∀ ∆ z ≤ (cid:0) C ∃ ∆ − C ∀ ∆ (cid:1) x + b ∃ ∆ − b ∀ ∆ ,x ≥ , y ≤ z, − y ≤ z, or, in an alternative form( A ∃ + A ∀ ) x + B c y + B ∀ ∆ z ≤ a ∃ + a ∀ , (8a) − ( A ∃ + A ∀ ) x − B c y + B ∀ ∆ z ≤ − a ∃ − a ∀ , (8b)( C ∃ + C ∀ ) x + D c y + D ∀ ∆ z ≤ b ∃ + b ∀ , (8c) x ≥ , y ≤ z, − y ≤ z. (8d)In general, however, the problem of finding an AE solution is NP-hard. It remains NP-hardeven on the following sub-cases: • weak solutions to A x = b ; see [3, 11, 12] • weak solutions to A x ≤ b ; see [3] • controllable solutions to A x = b ; see [3, 12]5 AE solvability
The interval systems (4) is called
AE solvable if for each realization of ∀ -parameters there arerealizations of ∃ -parameters such that (3) has a solution. Formally, (4) is AE solvable if ∀ A ∀ ∈ A ∀ , ∀ B ∀ ∈ B ∀ , ∀ C ∀ ∈ C ∀ , ∀ D ∀ ∈ D ∀ , ∀ a ∀ ∈ a ∀ , ∀ b ∀ ∈ b ∀ , ∃ A ∃ ∈ A ∃ , ∃ B ∃ ∈ B ∃ , ∃ C ∃ ∈ C ∃ , ∃ D ∃ ∈ D ∃ , ∃ a ∃ ∈ a ∃ , ∃ b ∃ ∈ b ∃ : ( A ∀ + A ∃ ) x = b ∀ + b ∃ has a solution. Notice that as long as (4) has an AE solution, then it is AE solvable, but the converse implicationdoes not hold in general. For example, the interval system of equations x + x = [1 , A x ≤ b , or A x ≤ b , x ≥ Theorem 2 ([7]) . The interval system A x + B y ≤ b , x ≥ is strongly solvable if and only if Ax + By − By ≤ b, x, y , y ≥ is solvable. For AE solvability of interval inequalities, we have the following generalization.
Proposition 6.
For the interval system A x + B ∀ y ≤ b , x ≥ the following are equivalent(i) x, y is an AE solution of (9) ,(ii) x, y solves ( A ∃ + A ∀ ) x + B ∀ c y + B ∀ ∆ | y | ≤ b ∃ + b ∀ , x ≥ , (10) (iii) x, y solves y = y − y , ( A ∃ + A ∀ ) x + B ∀ y − B ∀ y ≤ b ∃ + b ∀ , x, y , y ≥ . (11) Proof.
The equivalence “( i ) ⇔ ( ii )” follows from Proposition 2.“( ii ) ⇒ ( iii )” Let x, y be a solution of (10). Put y := max( y,
0) the positive part and y := max( − y,
0) the negative part of y . Then y = y − y , | y | = y + y , and (10) takes the formof ( A ∃ + A ∀ ) x + B ∀ c ( y − y ) + B ∀ ∆ ( y + y ) ≤ b ∃ + b ∀ , x, y , y ≥ , which is equivalent to (11).“( ii ) ⇐ ( iii )” Let x, y , y be a solution of (11), and put y := y − y . Then( A ∃ + A ∀ ) x + B ∀ c y + B ∀ ∆ | y | = ( A ∃ + A ∀ ) x + B ∀ c ( y − y ) + B ∀ ∆ | y − y |≤ ( A ∃ + A ∀ ) x + B ∀ c ( y − y ) + B ∀ ∆ ( y + y ) ≤ b ∃ + b ∀ , x ≥ , meaning that x, y solves (10). 6 roposition 7. The interval system (9) is AE solvable if and only if it has an AE solution.Proof.
First we show that (9) is AE solvable if and only if( A ∃ + A ∀ ) x + B ∀ y ≤ b ∃ + b ∀ , x ≥ A ∀ ∈ A ∀ , B ∀ ∈ B ∀ and b ∀ ∈ b ∀ ,and for the choice A ∃ := A ∃ and b ∃ := b ∃ , the system( A ∃ + A ∀ ) x + B ∀ y ≤ b ∃ + b ∀ , x ≥ A ∀ ∈ A ∀ , B ∀ ∈ B ∀ and b ∀ ∈ b ∀ ,there are A ∃ ∈ A ∃ and b ∃ ∈ b ∃ such that the system (13) is solvable. This implies that( A ∃ + A ∀ ) x + B ∀ y ≤ b ∃ + b ∀ , x ≥ Example 1.
Consider the interval system of inequalities A ∃ x ≤ − , A ∀ x ≤ , where A ∃ , A ∀ ∈ [ − , ≤ | x | , | x | ≤ , has no solution.In contrast, the interval system is AE solvable. If A ∀ ≥
0, then we can take A ∃ := 1 and x := −
2. If A ∀ ≤
0, then we can take A ∃ := − x := 2. In summary, the interval system ofinequalities is AE solvable, but has no AE solution. Let us recall the characterization of weak solutions for a general system of interval equations andinequalities from Hlad´ık [7].
Theorem 3.
Let A ∈ IR m × n , B ∈ IR m × n ′ , C ∈ IR m ′ × n , D ∈ IR m ′ × n ′ , a ∈ IR m , and b ∈ IR m ′ .A pair ( x, y ) , x ∈ R m , y ∈ R n , is a weak solution to the interval system A x + B y = a , C x + D y ≤ b , x ≥ . if and only if there is s ∈ {± } n such that Ax + ( B c − B ∆ diag( s )) y ≤ b, − Ax − ( B c + B ∆ diag( s )) y ≤ − b,Cx + ( D c − D ∆ diag( s )) y ≤ d, x ≥ . We will also employ the well known Farkas lemma. In particular, we utilize the following formfrom Hlad´ık [7]. 7 emma 1.
Exactly one of the linear systems Ax + By = b, Cx + Dy ≤ d, x ≥ and A T p + C T q ≥ , B T p + D T q = 0 , b T p + d T q ≤ − , q ≥ , is solvable. As long as B ∃ ∆ = 0 and D ∃ ∆ = 0, we have a sufficient and necessary characterization of AEsolvability for the general model (4). Proposition 8.
Suppose that B ∃ ∆ = 0 and D ∃ ∆ = 0 . Then (4) is AE solvable if and only if foreach s ∈ {± } m the system ( A ∃ + A ∀ c + diag( s ) A ∀ ∆ ) x + ( B ∀ c + diag( s ) B ∀ ∆ ) y (14a) − ( B ∀ c − diag( s ) B ∀ ∆ ) y ≤ a ∃ + a ∀ c − diag( s ) a ∀ ∆ , (14b) − ( A ∃ + A ∀ c + diag( s ) A ∀ ∆ ) T x − ( B ∀ c + diag( s ) B ∀ ∆ ) y (14c)+( B ∀ c − diag( s ) B ∀ ∆ ) y ≤ − a ∃ − a ∀ c + diag( s ) a ∀ ∆ , (14d)( C ∀ + C ∃ ) x + D ∀ y − D ∀ y ≤ b ∀ + b ∃ , (14e) x, y , y ≥ is solvable.Proof. The interval system (4) is not AE solvable if and only if there are A ∀ ∈ A ∀ , B ∀ ∈ B ∀ , C ∀ ∈ C ∀ , D ∀ ∈ D ∀ , a ∀ ∈ a ∀ and b ∀ ∈ b ∀ such that the interval system( A ∀ + A ∃ ) x + B ∀ y = a ∀ + a ∃ , ( C ∀ + C ∃ ) x + D ∀ y ≤ b ∀ + b ∃ , x ≥ A ∀ + A ∃ ) x + B ∀ y ≤ a ∀ + a ∃ , − ( A ∀ + A ∃ ) x − B ∀ y ≤ − a ∀ − a ∃ , ( C ∀ + C ∃ ) x + D ∀ y ≤ b ∀ + b ∃ ,x ≥ A ∀ + A ∃ ) T u − ( A ∀ + A ∃ ) T v + ( C ∀ + C ∃ ) T w ≥ , ( B ∀ ) T u − ( B ∀ ) T v + ( D ∀ ) T w = 0 , ( a ∀ + a ∃ ) T u − ( a ∀ + a ∃ ) T v + ( b ∀ + b ∃ ) T w ≤ − ,u, v, w ≥ A ∃ ) T u − ( A ∃ ) T v + ( A ∀ ) T ( u − v ) + ( C ∀ + C ∃ ) T w ≥ , ( B ∀ ) T ( u − v ) + ( D ∀ ) T w = 0 , ( a ∃ ) T u − ( a ∃ ) T v + ( a ∀ ) T ( u − v ) + ( b ∀ + b ∃ ) T w ≤ − ,u, v, w ≥ . ∀ -parameters, we have equivalently that theinterval system ( A ∃ ) T u − ( A ∃ ) T v + ( A ∀ ) T ( u − v ) + ( C ∀ + C ∃ ) T w ≥ , ( B ∀ ) T ( u − v ) + ( D ∀ ) T w = 0 , ( a ∃ ) T u − ( a ∃ ) T v + ( a ∀ ) T ( u − v ) + ( b ∀ + b ∃ ) T w ≤ − ,u, v, w ≥ s ∈ {± } m such that( A ∃ ) T u − ( A ∃ ) T v + ( A ∀ c + diag( s ) A ∀ ∆ ) T ( u − v ) + ( C ∀ + C ∃ ) T w ≥ , ( B ∀ c + B ∀ ∆ diag( s )) T ( u − v ) + D ∀ w ≥ , − ( B ∀ c − diag( s ) B ∀ ∆ ) T ( u − v ) − ( D ∀ ) T w ≥ , ( a ∃ ) T u − ( a ∃ ) T v + ( a ∀ c − diag( s ) a ∀ ∆ ) T ( u − v ) + ( b ∀ + b ∃ ) T w ≤ − ,u, v, w ≥ . is solvable. By the Farkas lemma again, this system is solvable if and only if the system( A ∃ + A ∀ c + diag( s ) A ∀ ∆ ) x + ( B ∀ c + diag( s ) B ∀ ∆ ) y − ( B ∀ c − diag( s ) B ∀ ∆ ) y ≤ a ∃ + a ∀ c − diag( s ) a ∀ ∆ , − ( A ∃ + A ∀ c + diag( s ) A ∀ ∆ ) T x − ( B ∀ c + diag( s ) B ∀ ∆ ) y +( B ∀ c − diag( s ) B ∀ ∆ ) y ≤ − a ∃ − a ∀ c + diag( s ) a ∀ ∆ , ( C ∀ + C ∃ ) x + D ∀ y − D ∀ y ≤ b ∀ + b ∃ x, y , y ≥ Proposition 9.
The interval system (4) is AE solvable if there is z ∈ {± } n ′ such that for each s ∈ {± } m the system ( A ∃ + A ∀ c + diag( s ) A ∀ ∆ ) x (15a)+( B ∃ c − B ∃ ∆ diag( z ) + B ∀ c + diag( s ) B ∀ ∆ ) y (15b) − ( B ∃ c − B ∃ ∆ diag( z ) + B ∀ c − diag( s ) B ∀ ∆ ) y ≤ a ∃ + a ∀ c − diag( s ) a ∀ ∆ , (15c) − ( A ∃ + A ∀ c + diag( s ) A ∀ ∆ ) T x (15d) − ( B ∃ c + B ∃ ∆ diag( z ) + B ∀ c + diag( s ) B ∀ ∆ ) y (15e)+( B ∃ c + B ∃ ∆ diag( z ) + B ∀ c − diag( s ) B ∀ ∆ ) y ≤ − a ∃ − a ∀ c + diag( s ) a ∀ ∆ , (15f)( C ∀ + C ∃ ) x + ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) y (15g) − ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) y ≤ b ∀ + b ∃ (15h) x, y , y ≥ is solvable.Proof. If the interval system (4) is not AE solvable, then there are A ∀ ∈ A ∀ , B ∀ ∈ B ∀ , C ∀ ∈ C ∀ , D ∀ ∈ D ∀ , a ∀ ∈ a ∀ and b ∀ ∈ b ∀ such that the interval system( A ∀ + A ∃ ) x + ( B ∀ + B ∃ ) y = a ∀ + a ∃ , ( C ∀ + C ∃ ) x + ( D ∀ + D ∃ ) y ≤ b ∀ + b ∃ , x ≥
09s not weakly solvable. By Theorem 3, for each z ∈ {± } n ′ the real system( A ∀ + A ∃ ) x + ( B ∀ + B ∃ c − B ∃ ∆ diag( z )) y ≤ a ∀ + a ∃ , − ( A ∀ + A ∃ ) x − ( B ∀ + B ∃ c + B ∃ ∆ diag( z )) y ≤ − a ∀ − a ∃ , ( C ∀ + C ∃ ) x + ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) y ≤ b ∀ + b ∃ ,x ≥ A ∀ + A ∃ ) T u − ( A ∀ + A ∃ ) T v + ( C ∀ + C ∃ ) T w ≥ , ( B ∀ + B ∃ c − B ∃ ∆ diag( z )) T u − ( B ∀ + B ∃ c + B ∃ ∆ diag( z )) T v +( D ∀ + D ∃ c − D ∃ ∆ diag( z )) T w = 0 , ( a ∀ + a ∃ ) T u − ( a ∀ + a ∃ ) T v + ( b ∀ + b ∃ ) T w ≤ − ,u, v, w ≥ A ∃ ) T u − ( A ∃ ) T v + ( A ∀ ) T ( u − v ) + ( C ∀ + C ∃ ) T w ≥ , ( B ∃ c − B ∃ ∆ diag( z )) T u − ( B ∃ c + B ∃ ∆ diag( z )) T v +( B ∀ ) T ( u − v ) + ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) T w = 0 , ( a ∃ ) T u − ( a ∃ ) T v + ( a ∀ ) T ( u − v ) + ( b ∀ + b ∃ ) T w ≤ − ,u, v, w ≥ . Now, there is the point where the equivalence cannot be easily establish. We can conclude thatfor each z ∈ {± } n ′ the interval system( A ∃ ) T u − ( A ∃ ) T v + ( A ∀ ) T ( u − v ) + ( C ∀ + C ∃ ) T w ≥ , ( B ∃ c − B ∃ ∆ diag( z )) T u − ( B ∃ c + B ∃ ∆ diag( z )) T v +( B ∀ ) T ( u − v ) + ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) T w = 0 , ( a ∃ ) T u − ( a ∃ ) T v + ( a ∀ ) T ( u − v ) + ( b ∀ + b ∃ ) T w ≤ − ,u, v, w ≥ . is weakly solvable. By Theorem 3, for each such system there is s ∈ {± } m such that( A ∃ ) T u − ( A ∃ ) T v + ( A ∀ c + diag( s ) A ∀ ∆ ) T ( u − v ) + ( C ∀ + C ∃ ) T w ≥ , ( B ∃ c − B ∃ ∆ diag( z )) T u − ( B ∃ c + B ∃ ∆ diag( z )) T v +( B ∀ c + diag( s ) B ∀ ∆ ) T ( u − v ) + ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) T w ≥ , − ( B ∃ c − B ∃ ∆ diag( z )) T u + ( B ∃ c + B ∃ ∆ diag( z )) T v − ( B ∀ c − diag( s ) B ∀ ∆ ) T ( u − v ) − ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) T w ≥ , ( a ∃ ) T u − ( a ∃ ) T v + ( a ∀ c − diag( s ) a ∀ ∆ ) T ( u − v ) + ( b ∀ + b ∃ ) T w ≤ − ,u, v, w ≥ . is solvable. By the Farkas lemma again, the system( A ∃ + A ∀ c + diag( s ) A ∀ ∆ ) x + ( B ∃ c − B ∃ ∆ diag( z ) + B ∀ c + diag( s ) B ∀ ∆ ) y − ( B ∃ c − B ∃ ∆ diag( z ) + B ∀ c − diag( s ) B ∀ ∆ ) y ≤ a ∃ + a ∀ c − diag( s ) a ∀ ∆ , − ( A ∃ + A ∀ c + diag( s ) A ∀ ∆ ) T x − ( B ∃ c + B ∃ ∆ diag( z ) + B ∀ c + diag( s ) B ∀ ∆ ) y +( B ∃ c + B ∃ ∆ diag( z ) + B ∀ c − diag( s ) B ∀ ∆ ) y ≤ − a ∃ − a ∀ c + diag( s ) a ∀ ∆ , ( C ∀ + C ∃ ) x + ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) y − ( D ∀ + D ∃ c − D ∃ ∆ diag( z )) y ≤ b ∀ + b ∃ x, y , y ≥ − zy + zy ≤ − , y + y ≤ , y , y ≥ z ∈ {± } . Proposition 8 generalizes several classical results on solvability of interval systems. In particular,for A ∈ IR m × n and b ∈ IR m we have: • For strong solvability of interval equations we obtain the same result as the characterizationby Rohn [25, 26]. The interval system A x = b is strongly solvable if and only if the system( A c + diag( s ) A ∆ ) x − ( A c − diag( s ) A ∆ ) x = b c − diag( s ) b ∆ , x , x ≥ s ∈ {± } m . • For strong solvability of interval equations with nonnegative variables we obtain the sameresult as the characterization by Rohn [23, 26]. The interval system A x = b , x ≥ A c + diag( s ) A ∆ ) x = b c − diag( s ) b ∆ , x ≥ s ∈ {± } m . • For strong solvability of interval inequalities we obtain the same result as the characteriza-tion by Rohn & Kreslov´a [27, 26]. The interval system A x ≤ b is strongly solvable if andonly if the system Ax − Ax ≤ b, x , x ≥ • For strong solvability of interval inequalities with nonnegative variables we obtain the sameresult as the classical characterization by Machost [17, 26]. The interval system A x ≤ b , x ≥ Ax ≤ b, x ≥ • For weak solvability of interval equations with nonnegative variables we obtain the sameresult as the consequence of the classical characterization by Oettli & Prager [20, 26]. Theinterval system A x = b , x ≥ Ax ≤ b, Ax ≥ b, x ≥ • For weak solvability of interval inequalities with nonnegative variables we obtain also thesame result as the well known characterization; see, e.g., [26]. The interval system A x ≤ b , x ≥ Ax ≤ b, x ≥ • For each A ∈ A there is b ∈ b such that Ax = b is solvable if and only if b ≤ ( A c + diag( s ) A ∆ ) x − ( A c − diag( s ) A ∆ ) x ≤ b, x , x ≥ s ∈ {± } m . • For each A ∈ A there is b ∈ b such that Ax = b , x ≥ b ≤ ( A c + diag( s ) A ∆ ) x ≤ b, x ≥ s ∈ {± } m . • For each b ∈ b there is A ∈ A such that Ax = b , x ≥ Ax ≤ b c − diag( s ) b ∆ ≤ Ax, x ≥ s ∈ {± } m . • For each A ∈ A there is b ∈ b such that Ax ≤ b is solvable if and only if Ax − Ax ≤ b, x , x ≥ • For each A ∈ A there is b ∈ b such that Ax ≤ b , x ≥ Ax ≤ b, x ≥ • For each b ∈ b there is A ∈ A such that Ax ≤ b , x ≥ Ax ≤ b, x ≥ We characterized AE solutions and for a certain sub-class of problems we also characterized AEsolvability. For general problems, we presented only a sufficient condition for AE solvability. Acomplete characterization of AE solvability remains an open problem.
References [1] G. Alefeld and J. Herzberger.
Introduction to Interval Computations . Computer Science andApplied Mathematics. Academic Press, New York, 1983.[2] M. Allahdadi and H. Mishmast Nehi. The optimal solution set of the interval linear pro-gramming problems.
Optim. Lett. , 7(8):1893–1911, 2013.[3] M. Fiedler, J. Nedoma, J. Ram´ık, J. Rohn, and K. Zimmermann.
Linear optimizationproblems with inexact data . Springer, New York, 2006.[4] A. Goldsztejn. A right-preconditioning process for the formal-algebraic approach to innerand outer estimation of AE-solution sets.
Reliab. Comput. , 11(6):443–478, 2005.125] A. Goldsztejn and G. Chabert. On the approximation of linear AE-solution sets. In
Post-proceedings of 12th GAMM–IMACS International Symposion on Scientific Computing, Com-puter Arithmetic and Validated Numerics, Duisburg, Germany . IEEE, 2006.[6] M. Hlad´ık. Interval linear programming: A survey. In Z. A. Mann, editor,
Linear Program-ming - New Frontiers in Theory and Applications , chapter 2, pages 85–120. Nova SciencePublishers, New York, 2012.[7] M. Hlad´ık. Weak and strong solvability of interval linear systems of equations and inequal-ities.
Linear Algebra Appl. , 438(11):4156–4165, 2013.[8] M. Hlad´ık. How to determine basis stability in interval linear programming.
Optim. Lett. ,8(1):375–389, 2014.[9] M. Hlad´ık. On approximation of the best case optimal value in interval linear programming.
Optim. Lett. , pages 1–13, 2014. DOI: 10.1007/s11590-013-0715-5.[10] M. Hlad´ık and J. Hor´aˇcek. Interval linear programming techniques in constraint program-ming and global optimization. In M. Ceberio and V. Kreinovich, editors,
Constraint Pro-gramming and Decision Making , volume 539 of
Studies in Computational Intelligence , pages47–59. Springer, 2014.[11] V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl.
Computational Complexity and Feasibilityof Data Processing and Interval Computations . Kluwer, 1998.[12] A. V. Lakeev and S. I. Noskov. On the solution set of a linear equation with the right-handside and operator given by intervals.
Sib. Math. J. , 35(5):957–966, 1994.[13] H. Li, J. Luo, and Q. Wang. Solvability and feasibility of interval linear equations andinequalities. submitted to Linear Algebra Appl., 2014.[14] W. Li, J. Luo, and C. Deng. Necessary and sufficient conditions of some strong optimalsolutions to the interval linear programming.
Linear Algebra Appl. , 439(10):3241–3255,2013.[15] J. Luo and W. Li. Strong optimal solutions of interval linear programming.
Linear AlgebraAppl. , 439(8):2479–2493, 2013.[16] J. Luo, W. Li, and Q. Wang. Checking strong optimality of interval linear programming withinequality constraints and nonnegative constraints.
J. Comput. Appl. Math. , 260:180–190,2014.[17] B. Machost. Numerische Behandlung des Simplexverfahrens mit intervallanalytischen Meth-oden. Technical Report 30, Berichte der Gesellschaft f¨ur Mathematik und Datenverar-beitung, 54 pages, Bonn, 1970.[18] R. E. Moore, R. B. Kearfott, and M. J. Cloud.
Introduction to interval analysis . SIAM,Philadelphia, PA, 2009.[19] A. Neumaier.
Interval methods for systems of equations . Cambridge University Press, Cam-bridge, 1990.[20] W. Oettli and W. Prager. Compatibility of approximate solution of linear equations withgiven error bounds for coefficients and right-hand sides.
Numer. Math. , 6:405–409, 1964.[21] E. D. Popova. Explicit description of ae solution sets for parametric linear systems. SIAMJ. Matrix Anal. Appl. , 33(4):1172–1189, 2012.[22] E. D. Popova and M. Hlad´ık. Outer enclosures to the parametric AE solution set.
SoftComput. , 17(8):1403–1414, 2013. 1323] J. Rohn. Strong solvability of interval linear programming problems.
Comput. , 26:79–82,1981.[24] J. Rohn. Inner solutions of linear interval systems. In K. Nickel, editor,
Proceedings of theInternational Symposium on interval mathematics on Interval mathematics 1985 , volume212 of
LNCS , pages 157–158. Springer, Berlin, 1986.[25] J. Rohn. Solvability of systems of linear interval equations.
SIAM J. Matrix Anal. Appl. ,25(1):237–245, 2003.[26] J. Rohn. Solvability of systems of interval linear equations and inequalities. In M. Fiedler,J. Nedoma, J. Ram´ık, J. Rohn, and K. Zimmermann, editors,
Linear optimization problemswith inexact data , chapter 2, pages 35–77. Springer, New York, 2006.[27] J. Rohn and J. Kreslov´a. Linear interval inequalities.
Linear Multilinear Algebra , 38(1-2):79–82, 1994.[28] S. M. Rump. Verification methods: Rigorous results using floating-point arithmetic.
ActaNumer. , 19:287–449, 2010.[29] A. Schrijver.
Theory of linear and integer programming. Repr.
Wiley, Chichester, 1998.[30] S. P. Shary. On controlled solution set of interval algebraic systems.
Interval Comput. ,4(6):66–75, 1992.[31] S. P. Shary. A new technique in systems analysis under interval uncertainty and ambiguity.
Reliab. Comput. , 8(5):321–418, 2002.[32] S. P. Shary. An interval linear tolerance problem.