2018 IX International Conference on Optimization and Applications (OPTIMA 2018) ISBN: 978-1-60595-587-2
Approximation of the Improper Linear Programming Problem with
Restriction on the Norm of the Correction Matrix of the
Left-Hand Side of the Constraints
Victor GORELIK
1and Tatiana ZOLOTOVA
2,*CC FRC CSC RAS, Ulica Vavilova 40, 119333 Moscow, Russia
Financial University under the Government of RF, Leningradsky Prospekt 49, 125993 Moscow, Russia Corresponding author
Keywords: Approximation of an improper problem, Correction matrix, Linear-quadratic programming.
Abstract. Methods of correction (approximation) for inconsistent systems of linear algebraic
equations and inequalities and improper linear programming problems have been widely used. In this paper, we consider an improper linear programming problem with an empty admissible set. It is formalized in the form of the problem of maximizing the initial criterion with the upper bound on the Frobenius norm of the correction matrix of the left-hand side of the constraints. A range of threshold values is found for which this problem has a solution, and its solution is obtained in an analytical form.
Introduction and Related Work
At present, correction methods for improper and unstable problems are very diverse. A cor- rection of the vector of the right-hand side of the constraints and a correction of the matrix of the left-hand side of the constraints or the augmented matrix composed of both sides of the constraints are applied to inconsistent systems of linear algebraic equations and inequalities and improper lin- ear programming problems (see, for example, [1, 2, 4, 10]). Wherein, the typical approach (used in the mentioned above and other works) is to find a correction vector or a correction matrix which has a minimum norm and leads to a consistent system. But the optimization problem of minimiz- ing the Euclidean norm of the correction vector or the Frobenius norm of the correction matrix (equal to the Euclidean norm when a matrix is expanded into a vector), as a rule, has a unique solution, and the maximization of the initial criterion does not make sense anymore. Therefore, in [3], formalization of the improper linear programming problem (LP) with an inconsistent system of constraints was proposed as the problem of minimizing the spectral norm or the Frobenius norm of the parameter correction matrix under the restriction from below to the initial criterion. This approach can be applied also to the proper LP problems, in which the optimal value of the criterion on the initial admissible set is unsatisfactory from the point of view of the decision-maker. But if these problems are formalized in the form of maximization of the initial criterion with a restriction from above to the spectral norm or the Frobenius norm of the correction matrix, we obtain the problem of maximizing a linear function under a quadratic restriction. We called it the problem of linear-quadratic programming and suggested the method for its solution [7].
1 2
*
There are three main cases of the limited correction problem: a) variation of the right-hand side of the constraints, b) variation of the left-hand side of the constraints, c) variation of both sides of the constraints. The case a) is the most simple in a mathematical sense, however, it is least interesting not only in the theoretical but also in the applied plan (for example, in the economic interpretation it consists in increasing of the resource amounts and/or reducing the production tasks). Cases b) and c), as our studies show, lead to identical in complexity mathematical problems (this complexity is not an end in itself, in the economic interpretation they mean the improvement of technology). In addition, cases of combining the prohibitions of correction of some rows or columns of the constraint matrix are possible (for a minimal correction, the corresponding results are given in [4]), as well as the use of different norms.
In this paper, we consider the problem of correction (approximation) for an improper LP problem with the restriction to the Frobenius norm of the left-hand side correction matrix, which has certain specificity. A range of threshold values is found for which this problem has a solution, and its solution is obtained in an analytical form.
The Formulation of the Problem and the Existence of the Solution
To begin with, we consider the problem of limited correction without the condition of non-negativity of the variables. Initial an improper LP problem, namely, a problem with an inconsistent system of constraints of the equality type has the form:
hc, xi →max
x∈X, X ={x|Ax=b}=∅. (1)
We introduce the correction matrix H of the left-hand side of the constraints of the problem (1). The problem of bounded correction (by the square of the Frobenius norm of a matrix with a threshold value ε) has the form
max
x,H {hc, xi|(A+H)x=b, ||H||
2 ≤ε}. (2)
For a fixed x, the matrix H must satisfy the condition Hx = b−Ax. The minimum of the Frobenius norm of the matrix H satisfying this equation is ||H|| = ||b−||xAx|||| (here and below the norm of the vectors is Euclidean and the vectors are column-vectors), and the correction matrixH
itself is determined by the formulaH = (b−xAxTx)xT (see [5], superscriptT is the sign of transposition).
In [5] the value of the infimum of the matrix H norm, under which {x|(A+H)x = b} 6=∅, was also obtained, namely,
inf
x,H||H||
2 =µ
min(AT(I−Pb)A), (3)
where Pb = bb
T
||b||2 is the matrix of projection onto the vector b, I is the identity matrix of the
corresponding dimension,µmin(AT(I−Pb)A) is the minimal eigenvalue of the matrixAT(I−Pb)A,
which is obviously symmetric and non-negative definite. It was also shown in [5], that the infimum in the left-hand side of (3) is attained if and only if there exists a unit eigenvectore∗, corresponding to µmin(AT(I − Pb)A), such that bTAe∗ 6= 0 (otherwise the system of constraints (1) becomes
consistent in the limit for ||x|| → ∞). In this case, the corresponding solution of the system (A+H∗)x∗ = b is x∗ = bbTTAebe∗∗, and the correction matrix is H
∗ = (b−Ax∗)x∗T
x∗Tx∗ . Since multiplying
an eigenvector of the matrix by any number, we again obtain an eigenvector, then the condition for the existence of a solution of the minimal correction problem can also be written in the form bTAe∗ >0 (the solution x∗ and H∗ does not change in this case).
We introduce the notation for the admissible set of the problem (2)
Z ={H, x|(A+H)x=b, ||H||2 ≤ε}.
Theorem 1. Suppose that
X ={x|Ax=b}=∅, µmin(AT(I−Pb)A)< µmin(ATA)
and parameter ε satisfies the conditionµmin(AT(I−Pb)A) < ε < µmin(ATA). Then the set Z6=∅, is closed and bounded, and the problem (2) has a solution.
Proof. We have
||H||2 = ||b−Ax||2
||x||2 =
bTb−2bTAx+xTATAx
xTx ≤ε. (4)
We represent the vector x in the formx=αe, where ||e||=1, α ≥0. Then inequality (4) can be represented in the form
bTb−2αbTAe+α2(eTATAe−ε)≤0. (5)
As the parameter ε satisfies the condition ε < µmin(ATA), theneTATAe−ε >0 ∀e and it follows from (5) that the parameter α is bounded above.
We note that the identities hold
eTAT(I−P
b)Ae≡eTAT(I− bb
T
||b||2)Ae≡
≡eTATAe−eTAT bbT
||b||2Ae≡eTATAe− (bTAe)2
||b||2 .
As min
e (e
TAT(I−P
b)Ae) = µmin(AT(I−Pb)A) and
min
e (e
TATAe− (bTAe)2
||b||2 )≤mine (e
TATAe) = µ
min(ATA),
then µmin(AT(I−Pb)A)≤µmin(ATA). On the other hand,
µmin(ATA) = min
e (A T
(I −Pb)A+
(bTAe)2
||b||2 )≤µmin(A
T
(I−Pb)A) +
(bTAe∗)2
||b||2 ,
where e* is any unit eigenvector corresponding to µmin(AT(I−Pb)A) (as further examples show,
this inequality can be either an equality or a strict inequality).
By assumption, we have the inequality µmin(AT(I−Pb)A)< µmin(ATA). This strict inequality is a sufficient condition for the existence of a minimal correction matrix, since in this case, from the inequalities proved, there follows the existence of e* for which bTAe∗ 6= 0 (it is also the sufficient condition for uniqueness of the solution of the minimal correction problem [1]). The vector x∗ = bbTTAebe∗∗ and the matrix H
∗
= (b−xAx∗T∗x)∗x∗T corresponding to this vector e* satisfy the
constraints of the problem (2). Thus, the setZ is not empty closed and bounded, and the problem (2) has a solution, Q.E.D.
Since in this case µmin(ATA) > 0, the matrix ATA is positive definite and the Gramian det(ATA), composed of the column-vectors of the matrix A, is positive. Hence, if the matrix A has a dimension m×n, then rankA=n, and m > n (if m =n the system of equations Ax=b is consistent).
Thus, the problem of bounded correction (2) has a solution if the parameter ε belongs to the intervalµmin(AT(I−Pb)A)< ε < µmin(ATA).
Forε < µmin(AT(I−Pb)A) the setZ=∅, and ifε=µmin(AT(I−Pb)A), we have a unique solution
of the corrected systemx* and (2) turns into a minimal correction problem. Whenε ≥µmin(ATA) the problem (2) has no solution, because when substituting the eigenvector e** corresponding µmin(ATA) in (5), its norm α is not bounded, so hc, xi is not bounded above (if hc, e∗∗i 6= 0 and bTAe∗∗>0).
Thus, we reduced the problem (2) to the problem
max
α,e {αhc, ei|b
Tb−2αbTAe+α2(eTATAe−ε)≤0}. (6)
Since for a fixed e the linear function reaches its extremum on the boundary, the restriction in (6) must be satisfied as an equality, that is α satisfies the quadratic equation
bTb−2αbTAe+α2(eTATAe−ε) = 0.
The roots of this equation are
α1,2 =
bTAe±p
(bTAe)2−(eTATAe−ε)bTb
(eTATAe−ε) =
=
bTAe±
q
bTb(ε−eTAT(I −bbT
bTb)Ae)
(eTATAe−ε) .
The denominator of this expression for a given range of values of ε is positive, and the vector
e must satisfy the condition ε−eTAT(I− bbT
bTb)Ae ≥ 0. Such vectors exist for a given range of ε,
for them both roots are real and positive values (if we choose such e that bTAe > 0). If wherein
hc, ei>0, then α is a larger root, and otherwise – smaller.
Method of Solution
Returning to the variable x in (6), we have the problem of linear-quadratic programming
max
x {hc, xi| b
Tb−2bTAx+xT(ATA−εI)x≤0}. (7)
Under the conditions of Theorem 1, this is a convex programming problem, because by the choice ofεthe matrix (ATA−εI) is positive definite. On an optimal plan, the constraint in problem
(7) is satisfied as an equality, since the linear function reaches an extremum at the boundary. Let’s introduce the Lagrangian function L(x, λ) =hc, xi+12λ(xT(ATA−εI)x−2bTAx).
Necessary and sufficient conditions for an extremum lead to the system of equations
c+λ((ATA−εI)x−ATb) = 0, bTb−2bTAx+xT(ATA−εI)x= 0.
The matrix (ATA−εI) has an inverse matrix (ATA−εI)−1, which is also positive definite. We express from the first equation x byλ:
x= (ATA−εI)−1ATb− (A
TA−εI)−1c
λ ,
and substitute it into the second equation:
bTb−2bTA(ATA−εI)−1ATb−(ATA−εI)−1c
λ
+
+(ATA−εI)−1ATb− (ATA−εI)−1c
λ
T
(ATA−εI)×
×(ATA−εI)−1ATb−(ATA−εI)−1c
λ
=
=bTb−2bTA(ATA−εI)−1ATb− (ATA−λεI)−1c+
+ ATb− c λ
(ATA−εI)−1ATb− (ATA−εI)−1c λ
=
=bTb− hATb,(ATA−εI)−1ATbi+AT(ATAλ2−εI)−1c = 0.
We obtain the quadratic equation for the determination of λ, the solution of which is
λ=±
s
cT(ATA−εI)−1c
hATb,(ATA−εI)−1ATbi −bTb. (8)
We substitute x in the goal function:
hc, xi=cT((ATA−εI)−1ATb)−c
T(ATA−εI)−1c
λ .
In view of the positive definiteness of the matrix (ATA−εI)−1, the negative sign in front of the root in (8) corresponds to the maximum of the goal function in the problem (7), and the positive sign – to the minimum (however, the requirement λ ≤ 0 for the Lagrange multiplier for the problem (7) is included in the optimality conditions). As a result, we obtain the solution of problem (7):
x0 = (ATA−εI)−1ATb−(A
TA−εI)−1c
λ0 , (9)
where
λ0 =−
s
cT(ATA−εI)−1c
hATb,(ATA−εI)−1ATbi −bTb.
Correction of the Left-Hand Side of the Constraints with an Additional Condition of Non-Negativity
Let us consider the development of this approach with the condition of non-negativity x ≥ 0 in the constraints of the problem (1), which is natural for the LP problem. In a general case, it introduces a significant complication in the conditions of existence and in the method of solving the problem of limited correction.
So, consider the problem
hc, xi →max
x∈X, X ={x|Ax=b, x≥0}=∅. (10)
The solution of the problem of minimal correction of the left-hand side of the constraints (10) with respect to the Frobenius norm, namely, the problem
min{kHk |XH 6=∅}, XH ={x|(A+H)x=b, x≥0}, (11)
is given in [6] : the value of the problem (11) is
inf{||H|| |XH 6=∅}=
p
min(µ1, µ2) (12)
where
µ1 = min{hDe, ei |e≥0,||e||= 1, bTAe≥0}, (13)
µ2 = min{hBe, ei |e≥0,||e||= 1, bTAe≤0}, (14)
D=AT(I−P
b)A, B =ATA.
The lower bound in (12) is attained if and only if µ1 ≤ µ2 and there exists a solution of the problem (13)e∗1 ≥0, such that bTAe∗1 >0. Moreover, e∗1 there is an eigenvector of the matrixD or some of its square submatrix obtained fromD by deleting rows and columns with the same numbers (complemented by zero components at the places of deleted rows), corresponding to the minimal of eigenvalues of these matrices for which there exist nonnegative eigenvectors. The solution of the problem (11) is x∗ = (bTb)e∗1
bTAe∗ 1 , H
∗ = (b−Ax∗)x∗T x∗Tx∗ .
Thus, the verification of the existence of a solution of the problem (11) and its finding reduces to solving two problems of quadratic programming (13) and (14). In the case when a solution of the problem (11) does not exist, the improper problem (10) can be approximated by the LP problem with a nonempty admissible set by correction matrix of the left-hand side of constraints with a norm arbitrarily close to the value (12) under ||x|| → ∞. Thus the resulting LP problem again is not a proper one (if hc, xi → ∞).
We now return to the problem of limited correction of the left-hand side of the constraints of the problem (10), namely, we consider the problem
max
x,H {hc, xi|(A+H)x=b, x≥0,||H||
2 ≤ε}. (15)
Let us find the conditions and the range of possible values of ε for which this problem has a solution. We will need the value
µ∗2 = min{hBe, ei |e≥0,||e||= 1} ≤µ2.
Theorem 2. If there exists a solution of the problem (13) e∗1 ≥ 0 such that bTAe∗
1 > 0 and µ1 < ε < µ∗2, then a solution of the problem (15) exists.
Proof. We represent x≥0 in the form x= αe, α≥0,e≥0, ||e||= 1 and introduce the set
Z+ ={α, e|α≥0, e≥0,||e||= 1, bTb−2αbTAe+α2(eTATAe−ε)≤0}.
The set Z+ 6= ∅, since it contains e∗
1 and the corresponding α. We have eTATAe−ε > 0 for any e ≥ 0, so if e ∈ Z+, then bTAe > 0. By what has been proved earlier, for such e inequality
holds
α≤ b
TAe+qbTb(ε−eTAT(I−bbT
bTb)Ae)
(eTATAe−ε) .
Hence the set Z+ is bounded. In addition, since
eTAT(I−Pb)Ae=eTATAe−
(bTAe)2
||b||2 ≤ε,
then (bTAe)2 ≥bTb(µ∗2−ε), i.e. Z+ is closed.
But the set Z+ uniquely corresponds to the admissible set of the problem (15), hence, it has a solution, Q.E.D.
We describe the method of solving the problem (15). It is reduced to the problem
max
x {hc, xi|x ≥0, b
TAx ≥0, bTb−2bTAx+xT(ATA−εI)x≤0}. (16)
The problem (16) is similar to the problem (7); however, in the conditions of Theorem 2 (for a new range of values ofε), in general, it is not a convex programming problem. Due to the linearity of the objective function, the restriction bTb−2bTAx+xT(ATA−εI)x≤0 must be active at the
solution, and the restriction bTAx ≥ 0, by virtue of what was said above, must be satisfied as a strict inequality. Therefore, it can be omitted from the Lagrange function (the corresponding factor is equal to zero), but it is required to verify that it is a strict inequality. If bTA >0, then
this condition is fulfilled automatically, if bTA ≤0, then the problem has no solution. The extremum conditions in this case lead to a system of equations and inequalities:
(c+λ((ATA−εI)x−ATb))x= 0, bTb−2bTAx+xT(ATA−εI)x= 0,
c+λ((ATA−εI)x−ATb)≤0, bTAx >0.
(17)
If the matrix ATA−εI and its square submatrices obtained by deleting rows and columns with
the same numbers have inverse matrices at the selected value of ε, then the solution can be found by formulas (9) by enumeration of the square submatrices of the matrixATA−εI (including itself) with the subsequent verification of the inequalities in (17) for the solutions found (supplemented by zero components at the places of deleted rows).
For relatively moderate dimensions such enumeration for modern computers does not present a problem (for example, for n = 20 it is a proximally 106 steps). For larger dimensions, you can use specialized optimization software.
Numerical Experiments
Example 1. Consider the problem
x1+x2 →max
x∈X, X ={x| x1+x2 = 1, x1−x2 = 1,−x1 +x2 = 1}=∅.
Here c =
1 1
, A =
1 1 1 −1
−1 1
, b =
1 1 1
. We find the set of values of ε for which
problem (7) has a solution. On the one hand, µmin(AT(I − bb
T
||b||2)A) = 4
3. The corresponding eigenvector e∗ = (0.707, 0.707)T, the solution of the problem of minimum correction is
x∗ = (1.5, 1.5)T, H∗ =
−0.667 −0.667 0.333 0.333 0.333 0.333
,hc, x ∗i
= 3.
On the other hand, µmin(ATA) = µmin(AT(I −Pb)A) + (b
TAe∗)2
||b||2 ) = 2 (here these values are
equal). Therefore, the range of possible values of ε: 43 < ε < 2. We take ε=1.5. Let us find the solution of the problem (7) according to the formula (9): x0 = (3, 3)T, correction matrix
H0 =
−0.833 −0.833 0.167 0.167 0.167 0.167
, hc, x0i= 6.
If we put ε = 43, we get the solution x* and the minimal correction matrix H*, and if ε →2, then ||x|| → ∞.
Note that if we directly solve the correction problem (2) with the built-in numerical optimization method in MathCad, it turns out that it is sensitive to the choice of the initial approximation.
Thus, for initial x= (1, 1)T,H =
1 1 1 1 1 1
we get
x0 = (2.999, 3.001)T, H =
−0.833 −0.833 0.167 0.167 0.166 0.166
,
and for initialx= (1, 1)T,H =
1 1 1 0 1 1
we getx0 = (2.599, 2.578)T,H =
−0.81 −0.803 0.067 0.312 0.103 0.292
.
Example 2. Consider the problem
x1+x2 →max
x∈X, X ={x| x1 +x2 = 1, x1−x2 = 1,−x1+x2 = 1, x1 = 1}=∅.
Here c =
1 1
, A =
1 1 1 −1
−1 1 1 0
, b =
1 1 1 1
. We find the set of values of ε for which
problem (7) has a solution.
On the one hand, µmin(AT(I− bb
T
||b||2)A) = 1.370.
The corresponding eigenvector e∗ = (0.677, 0.736)T.
On the other hand, µmin(ATA) = 2.382 and µmin(AT(I−Pb)A) + (b
TAe∗)2
||b||2 ) = 2.462 (here these
values are not equal). Therefore, the range of possible values of ε: 1.370 < ε < 2.382. We take
ε=2. Let us find the solution of the problem (7) according to the formula (9):
x0 = (5.191, 7.286)T, H0 =
−0.744 −1.045 0.201 0.282
−0.071 −0.1
−0.272 −0.382
,hc, x0i= 12.477.
If ε= 1.370, we get the minimal correction matrix H∗ =
−0.603 −0.655 0.394 0.428 0.314 0.341
−0.105 −0.114
, and the solution
x∗ = (1.296, 1.408)T, hc, x∗i= 2.704, and if ε→2.382 then ||x|| → ∞.
In the above examples, the condition x≥0 was fulfilled automatically. Let us now consider an example in which the addition of this condition is essential.
Example 3. Consider the problem (first without the condition of nonnegativity)
x1+x2 →max
x∈X, X ={x|5x1+ 4x2 = 1,2x1+x2 = 1,7x1+ 4x2 = 1,2x1 = 1}=∅.
Herec=
1 1
,A=
5 4 2 1 7 4 2 0
,b=
1 1 1 1
. We find the set of values of εfor which problem
(7) has a solution. The matrices
D=AT(I− bb T
||b||2)A=
18 14 14 12.75
, B =ATA=
82 50 50 33
.
On the one hand, µmin(D) = 1.131, and the corresponding eigenvector ise∗ = (0.639, 0.770)T. On the other hand, µmin(B) = 1.820 and the corresponding eigenvector e∗ = (−0.529, 0.849)T. Therefore, the range of possible values of ε: 1.131 < ε < 1.820. We take ε=1.5. Let us find the solution of the problem (7) according to the formula (9): x0 = (0.548,−0.525)T, the correction
matrix H0 =
0.343 −0.329 0.408 −0.391
−0.7 0.671
−0.092 0.088
, hc, x0i= 0.023.
We now add the condition of nonnegativityx≥0. HerebTA = (16, 9), therefore, the condition
bTAx > 0 is fulfilled automatically ∀x ≥ 0, x 6= 0. We calculate the range of possible values
ε: µ1 = 12.75 and µ∗2 = 33. Let’s take ε = 20. The matrix ATA −εI =
62 50 50 13
. For
x1 > 0, x2 > 0 the conditions (17) are not satisfied. Therefore, we perform a search of the square submatrices of the matrix ATA −εI. We have two submatrices and by formula (9) for them x = (0, 1.107)T and x = (0.304, 0)T. The solution of the problem with the condition
of non-negativity is the vector x∗ = (0, 1.107)T, the correction matrix H∗ =
0 −3.096 0 −0.096 0 −3.096 0 0.904 ,
hc, x∗i= 1.107.
Forε= 30 the matrixATA−εI =
52 50 50 3
. We perform a search of the square submatrices
of the matrixATA−εI, including the matrix itself (here, for the casex
1 >0,x2 >0 conditions (17) are satisfied). For the matrixATA−εI, using formula (9), we have x= (0.191, 0.141)T, the value of the objective function at this point is hc, xi = 0.332. For two submatrices, using formula (9), we obtain x = (0, 5.769)T and x = (0.441, 0)T. The solution of the problem with the condition
of nonnegativity is the vector x∗ = (0, 5.769)T , the correction matrix H∗ =
0 −3.827 0 −0.827 0 −3.827 0 0.173
,
hc, x∗i= 5.769.
Conclusion
The initial problem is not a convex programming problem (it contains bilinear constraints). Therefore, a direct solution of this problem by numerical methods of optimization can, as shown by computational experiments, lead to local extrema and sensitivity from the choice of the initial approximation. The developed method for reducing this problem to the problem of linear-quadratic programming and its solution found in the analytical form made it possible to use standard matrix calculations for solving the problem of limited correction. In the case when the original improper LP problem does not contain the condition of non-negativity of variables, we arrive at the problem of convex programming, the solution of which by the developed method is very simple and reduces to finding an inverse matrix. Such a procedure, as well as finding the eigenvalues of matrices (which, however, is required only for determining the range of threshold values) is effectively implemented in modern application software (for example, MatLab). In the case of the presence of the condition of non-negativity of variables, the method leads to a search of submatrices. For relatively small dimensions this is not essential in the computational aspect. The question of the effectiveness of the method for large dimensions is still open. A direct comparison with existing works on methods of minimal correction (for example, [1]) is hardly justified, since the problem of limited correction is more complicated than the problem of minimal correction, because of the presence of quadratic restriction.
The main thing in this work is a fundamentally new formulation of the correction problem for the improper LP problem. This approach can be applied to other classes of improper and unstable problems. In particular, in the framework of this approach, we obtained the solutions of the correction problems for the improper LP problem with the restriction to the Euclidean norm of the right-hand side correction vector and the Frobenius norm of the correction matrix of both sides of constraints. These results are being prepared for publication.
Note that methods of minimal correction (approximation) for different sets of parameters are widely used in data processing, where overdetermined systems of equations naturally arise. An overview of current results and a bibliography on these questions can be found, for example, in [8, 9]. We propose to apply the above approach to this field. The idea is, on the one hand, to replace the exact approximation with ε-approximation, and on the other hand, to supplement it with the requirement of the proximity of the test and validation samples trends. Since, in this case, the non-negativity conditions are not usually imposed on parameters of the model, the proposed method becomes especially efficient.
Acknowledgements
This work was performed within the State project no. 1.8535.2017 of the Ministry for Science and Education of the Russian Federation.
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