Checking weak optimality of the solution to linear ... - Semantic Scholar

Checking weak optimality of the solution to linear programming with interval right-hand side Wei Lia , Jiajia Luoa , Qin Wangb , Ya Lia a. Institute of Operational Research & Cybernetics Hangzhou Dianzi University, Hangzhou, 310018, PR China b. College of Science, China Jiliang University, 310018, PR China

Abstract The interval linear programming (IvLP) 1 is a method for decision making under uncertainty. A weak feasible solution to IvLP is called weakly optimal if it is optimal for some scenario of the IvLP. One of the basic and difficult tasks in IvLP is to check whether a given point is weak optimal. In this paper, we investigate linear programming problems with interval right-hand side. Some necessary and sufficient conditions for checking weak optimality of given feasible solutions are established, based on the KKT conditions of linear programming. The proposed methods are simple, easy to implement yet very effective, since they run in polynomial time. Keywords: interval linear programming; weak feasible solution; weak optimal solution; KKT conditions.

1

Introduction

Many practical problems are solved by linear programming. Since real-life problems are subject to uncertainties due to errors, measurements and estimations, we have to reflect it in linear programming methodology and decision making. Interval computations is a popular way for treating uncertainties in data measurements. Instead of exact values we compute with real intervals, which are supposed to involve all gauging errors. An interval matrix is defined as A = [A, A] = {A ≤ A ≤ A}, where A ≤ A ∈ Rm×n are given matrices and matrix inequalities are understood componentwise. n-dimensional interval vectors can be regarded as interval matrices n-by-1. The set of all m-by-n interval matrices is denoted by IRm×n . Consider the IvLP problems min cT x s.t. Ax ≥ (≤, =)b, x ≥ 0, 1

(1)

We use abbreviation Ivlp, instead of ILP, for interval linear programming, to avoid confusing with the abbreviation for integer linear programming.

1

where A ∈ IRm×n , b ∈ IRm and c ∈ IRn . A scenario of IvLP (1) is a concrete realization of interval values, that is, any linear program with A ∈ A, b ∈ b and c ∈ c. Definition 1 A vector x ∈ Rn is called a weak feasible solution of the linear interval program (1) if it satisfies Ax ≥ (≤, =)b, x ≥ 0, for some A ∈ A, b ∈ b. Definition 2 Let x be a weak feasible solution to (1). It is called weakly optimal if it is optimal for some scenario of (1). (or x be a weak optimal solution to (1)). A frequent problem in interval mathematics programming problem is to compute the range of optimal values when the problem quantities vary within intervals. Some interesting results for IvLP and interval quadratic programming (IvQP) problem are obtained by different authors, see [1-9], among others. However, there are few results on the optimal solutions set of IvLP. In 2011, Hlad´ık [7] proposes the following open problem: Given x∗ ∈ Rn , is it optimal for some scenario? Determining the set of all optimal solutions over all scenarios is one of the most difficult problems in IvLP, only some special cases have been discussed. Steuer and Hlad´ık discussed the IvLP problems with interval objective function coefficients. For this special case, Steuer [10] presented three algorithms to compute all weakly optimal solutions; Hlad´ık [7] proposed some polynomial time algorithm to check weak optimality of a given feasible solution x. In what follows, we discuss another special case: Interval right-hand side linear program. The feasible region of IvLP is normally characterized by interval linear systems. Properties of interval linear equations and interval linear inequalities are quite different, see e.g. [11, 12, 13, 2, 15, 16, 17]. Therefore, we will discuss two types of interval right-hand side linear program, with equality constraint and with inequality constraint, separately. Let A ∈ Rm×n , b ∈ IRm and c ∈ Rn . Consider the interval right-hand side IvLP of the following types min cT x s.t. Ax ≥ b, x ≥ 0,

(2)

min cT x s.t. Ax = b, x ≥ 0,

(3)

and min cT x s.t. Ax ≥ b.

(4)

In next section, we propose some polynomial time methods to check weak optimality of a given feasible solution of (2), (3) and (4) respectively.

2

2

Main results

Theorem 1 Let x¯ ∈ Rn be a weak feasible solution to (2) and assume that x¯k1 > 0, . . . , x¯kp > 0, x¯kp+1 = · · · = x¯kn = 0. Denote by F = {ri |i = 1, . . . , q, Ari ,· x¯ > ¯bri },

(5)

where Ai,· is the i-th row of the coefficient matrix A. Then x¯ is a weak optimal solution to (2) if and only if the linear system yA·,kj = ckj , j = 1, . . . , p, yA·,kj ≤ ckj , j = p + 1, . . . , n, yri = 0, i = 1, . . . , q, yri ≥ 0, i = q + 1, . . . , m,

(6.1) (6.2) (6.3) (6.4)

is solvable, where A·,j denotes the jth column of the matrix A and y = (y1 , . . . , ym ) ∈ Rm is a row vector. Proof “If”: Assume that the linear system (6) is solvable and let y¯ = (¯ y1 , . . . , y¯m ) be a solution to (6), we show that x¯ is a weak optimal solution to (2). From (6.1) we have y¯(A·,k1 , A·,k2 , . . . , A·,kp ) = (ck1 , ck2 , . . . , ckp ). and consequently    x¯k1 x¯k1 x¯k  x¯k   2  2 y¯(A·,k1 , A·,k2 , . . . , A·,kp )  ..  = (ck1 , ck2 , . . . , ckp )  ..  ,  .   .  

x¯kp

x¯kp or y¯

p X

A·,kj x¯kj =

j=1

p X

c kj x kj .

(7)

j=1

Note that x¯kp+1 = · · · = x¯kn = 0 we get A¯ x=

n X

A·,j x¯j =

j=1

p X

A·,kj x¯kj ,

(8)

j=1

hence from (7) and (8) we obtain T

c x¯ =

n X j=1

cj x¯j =

p X

ckj x¯kj = y¯

j=1

p X

A·,kj x¯kj = y¯A¯ x.

(9)

j=1

(a) If the set F defined by (5) is empty, i.e., F = {ri |i = 1, . . . , q, Ari ,· x¯ > ¯bri } = ∅, set b0 = A¯ x. (10) Clearly b0 = A¯ x ≥ b, since x¯ is a weak feasible solution to (2). Furthermore we have ¯ A¯ x ≤ b, since F = ∅. Thus b0 ∈ b. 3

(a.1) x¯ solves the system Ax ≥ b0 , x ≥ 0

(11)

yA ≤ c, y ≥ 0,

(12)

due to (10). (a.2) y¯ solves the system since y¯ is a solution to system (6). (a.3) From (9) and (10) we have cT x¯ = y¯b0 .

(13)

Conditions (11), (12) and (13) implies the KKT conditions for linear program min cT x s.t. Ax ≥ b0 , x ≥ 0.

(14)

Thus, x¯ is an optimal solution to (14) and hence x¯ is a weak optimal solution to (2). (b) If F = {ri |i = 1, . . . , q, Ari ,· x¯ > ¯bri } 6= ∅, hence yri = 0, i = 1, . . . , q, by condition (6.3). Set  k ∈ F, bk ∈ bk ,  bk , n 0 P (b )k = akj x¯j , k ∈ / F.  j=1

Clearly, b0 ∈ b. (b.1) Note that (A¯ x)k =

n P

akj x¯j , k = 1, . . . , m. Thus, for k ∈ F , we have

j=1

(A¯ x)k =

n X

akj x¯j > ¯bk > (b0 )k .

j=1

for k ∈ / F , we have (A¯ x)k =

n X

akj x¯j = (b0 )k .

j=1

So, x¯ solves the system Ax ≥ b0 , x ≥ 0.

(15)

y¯A ≤ c, y ≥ 0,

(16)

(b.2) y¯ solves the system since it solves the linear system (6). (b.3) Now we prove that cT x¯ = y¯b0 . In fact,

4

(17)

 p

cT x¯ =

X j=1

ckj x¯kj

n P



 j=1 a1j x¯j      . . = y¯A¯ x = (0, . . . , 0, y¯q+1 , . . . , y¯m )   .  n  P  amj x¯j j=1





b1 ..   .     b   q P  n  a ¯j  = (0, . . . , 0, y¯q+1 , . . . , y¯m )  q+1,j x  = y¯b0 . j=1    ..   .   n  P  amj x¯j j=1

Conditions (15), (16) and (17) implies that KKT conditions for linear program min cT x s.t. Ax ≥ b0 , x ≥ 0.

(18)

Thus, x¯ is an optimal solution to (18) so that x¯ is a weak optimal solution to (2). “Only if”: We prove this part by contradiction. Assume that x¯ is a weak optimal solution to (2), but the system (6) is not solvable. Thus, there exists b0 ∈ b such that x¯ solves the system min cT x s.t. Ax ≥ b0 , x ≥ 0.

(19)

Due to the dual theory of linear programming there exists a y¯ which solves the system max yb0 s.t. yA ≤ cT , y ≥ 0,

(20)

and it holds that (¯ y A − cT )¯ x=0

(21)

y¯(A¯ x − b0 ) = 0.

(22)

and

Thus, (21) leads to (6.1); (20) leads to (6.2) and (6.4); (22) leads to (6.3). So y¯ solves system (6), this is a contradiction. 5

This completes the proof of the theorem. ¤ The method is obviously polynomial, since the system (6) is linear. In a similar way, we can prove the next result for the problem (3). Theorem 2 Let x¯ be a weak feasible solution to (3) and assume that x¯k1 > 0, . . . , x¯kp > 0, x¯kp+1 = · · · = x¯kn = 0. Then x¯ is a weak optimal solution to (3) if and only if the linear system yA·,kj = ckj , j = 1, . . . , p, yA·,kj ≤ ckj , j = p + 1, . . . , n,

(23.1) (23.2)

is solvable, where A·,j denotes the jth column of the matrix A and y = (y1 , . . . , ym ) ∈ Rm is a row vector. Proof “If”: Assume that the linear system (23) is solvable and let y¯ = (¯ y1 , . . . , y¯m ) be a solution to (23), we show that x¯ is a weak optimal solution to (3). From (23.1) we have y¯(A·,k1 , A·,k2 , . . . , A·,kp ) = (ck1 , ck2 , . . . , ckp ), and consequently 

   x¯k1 x¯k1 x¯k  x¯k   2  2 y¯(A·,k1 , A·,k2 , . . . , A·,kp )  ..  = (ck1 , ck2 , . . . , ckp )  ..  ,  .   .  x¯kp or equivalently y¯

p X

x¯kp

A·,kj x¯kj =

j=1

p X

c k j x kj

j=1

holds. According to the conditions x¯kp+1 = · · · = x¯kn = 0 we get A¯ x=

n X

A·,j x¯j =

j=1

and hence T

c x¯ =

n X j=1

cj x¯j =

p X

p X

A·,kj x¯kj

j=1

ckj x¯kj = y¯

p X

A·,kj x¯kj = y¯A¯ x.

(24)

j=1

j=1

Put b0 = A¯ x.

(25)

Clearly b0 ∈ b, since x¯ is a weak feasible solution to (3). 1) x¯ solves the system Ax = b0 , x ≥ 0

(26)

due to (25), hence x¯ is a weak feasible solution to the system (2). 2) y¯ solves the system yA ≤ cT ,

(27)

since y¯ is a solution to system (23). 6

3) From (24) and (25) we have cT x¯ = y¯b0 .

(28)

Conditions (26), (27) and (28) implies that KKT conditions for linear program min cT x s.t. Ax = b0 , x ≥ 0.

(29)

Thus, x¯ is an optimal solution to (29) and hence x¯ is a weak optimal solution to (3). “Only if”: We prove this part by contradiction. Assume that x¯ is a weak optimal solution to (3), but the system (23) is not solvable. Thus, there exists b0 ∈ b such that x¯ solves the system min cT x s.t. Ax = b0 , x ≥ 0.

(30)

Due to the dual theory of linear programming there exists a y¯ which solves the system max yb0 s.t. yA ≤ cT ,

(31)

(¯ y A − cT )¯ x = 0.

(32)

and it holds that

Thus, (32) leads to (23.1); (31) leads to (23.2) So y¯ solves system (23), this is a contradiction. This completes the proof of the theorem. ¤ Remark One basic problem in IvLP is to find its optimal value bounds. It is interesting to note that determining the optimal value bounds of the IvLP with equality and nonnegative constraints is much more difficult than doing so for IvLP with inequality and nonnegative constraints [2, 6, 7]. However, this is not the case in checking the weak optimality for a weak feasible solution in our context. In the proof of Theorem 2, y¯ solves the system (23) and hence it is optimal for problem (31). Thus, the given feasible solution x¯ itself is optimal for problem (30), if we simply set b0 = A¯ x. Thus we may also simply to check whether the given feasible solution is optimal for problem (30). From this point of view, the proposed method is easier for IvLP with equality and nonnegative constraints. Theorem 3 Let x¯ ∈ Rn be a weak feasible solution to (4) and assume that x¯k1 > 0, . . . , x¯kp > 0, x¯kp+1 = · · · = x¯kn = 0. Denote by F = {ri |i = 1, . . . , q, Ari ,· x¯ > ¯bri }.

7

(33)

Then x¯ is a weak optimal solution to (4) if and only if the linear system yA·,j = cj , j = 1, . . . , n, yri = 0, i = 1, . . . , q, yri ≥ 0, i = q + 1, . . . , m,

(34.1) (34.2) (34.3)

is solvable, where A·,j denotes the jth column of the matrix A and y = (y1 , . . . , ym ) ∈ Rm is a row vector. Proof The proof is similar to that of the Theorems 1 and 2 and is thus omitted here. ¤

3

Illustrative examples

Example 1 Consider the interval right-hand side linear program min z = 2x1 + 2x2 + x3 s.t. 2x1 − x2 + x3 ≥ [−1, 3], x1 + x2 − x3 ≥ [−2, 4], xi ≥ 0, i = 1, 2, 3.

(35)

1) Let x¯1 = (1, 0, 1)T . Clearly x¯1 is a weak feasible solution to (35). 1 1 Now construct the corresponding µ system ¶ (6). Note that x¯1 = x¯3 = 1 > 0, 2 1 thus the equation (6.1) is (y1 , y2 ) = (2, 1). Note that x¯12 = 0, thus the 1 −1 µ ¶ −1 inequality (6.2) is (y1 , y2 ) ≤ 2. The set F defined by (5) is empty, since   1 µ ¶ 1 µ ¶ µ ¶ 2 −1 1   3 3 ¯ 0 = A¯ x = ≤ b = . So the equation (6.3) disappears 1 1 −1 0 4 1 and (6.4) is y1 ≥ 0, y2 ≥ 0. Thus, the system (6) in connection with x¯ = (1, 0, 1)T is 2y1 + y2 y1 − y2 −y1 + y2 y1 ≥ 0, y2

= 2, = 1, ≤ 2, ≥ 0.

(36)

The above linear system has a solution (1, 0)T . Thus, x¯1 = (1, 0, 1)T is a µ weak ¶ 3 optimal solution to linear interval program (35). Since F = ∅, let b0 = A¯ x= . 0 Then it is easy to see that x¯1 = (1, 0, 1)T is optimal for the scenario min z = 2x1 + 2x2 + x3 s.t. 2x1 − x2 + x3 ≥ 3, x1 + x2 − x3 ≥ 0, xi ≥ 0, i = 1, 2, 3. 8

(37)

2) Let x¯2 = (3, 1, 0)T . Clearly x¯2 is a weak feasible solution to (35). It is easy to see that the linear system (6) associated with x¯2 is 2y1 + y2 −y1 + y2 y1 − y2 y1 y2

= 2, = 2, ≤ 1, = 0, ≥ 0.

(38)

The above linear system has a solution (0, 2)T . Thus, x¯2 = (3, 1, 0)T is  a weak  µ ¶ 3 2 −1 1   1 = optimal solution to linear interval program (35). Since A¯ x= 1 1 −1 0 µ ¶ 5 ¯ 1 , so we have F = {1}. Thus, according to Theorem 1, and thus 5 > 3 = b 4 number in the interval [−1, 3]. Here the first component of b0 , b01 can be chosen any   µ ¶ 3 2 0 0 . Then we let b1 = 2 ∈ [−1, 3]. Let b2 = (1, 1, −1) 1 = 4. So choose b0 = 4 0 2 T x¯ = (3, 1, 0) is optimal for the scenario min z = 2x1 + 2x2 + x3 s.t. 2x1 − x2 + x3 ≥ 2 x1 + x2 − x3 ≥ 4 xi ≥ 0, i = 1, 2, 3.

(39)

3) Consider x¯4 = (0, 1, 1)T , a weak feasible solution to (35). The linear system (6) associated with x¯4 is 2y1 + y2 ≤ 2, −y1 + y2 = 2, y1 − y2 = 1, y1 , y 2 ≥ 0

(40)

The above linear system has no solution, thus x¯4 = (0, 1, 1)T is not a weak optimal solution to (35). Example 2 Consider the interval right-hand side linear program min z = 2x1 + 4x2 + 2x3 s.t. x1 + 2x2 + x3 ≥ [0, 2], 2x1 + 4x2 + 3x3 ≥ [1, 3],

(41)   µ ¶ 1 1 2 1   T 0 = Let x¯ = (1, 0, 1) be a weak feasible solution to (41). Since A¯ x= 2 4 3 1 µ ¶ 2 ¯ 2 , so we have F = {2}. Note that x¯11 = x¯13 = 1 > 0, thus and thus 5 > 3 = b 5 the corresponding linear system (34) associated with x¯ is 9

y1 + 2y2 2y1 + 4y2 y1 + 3y2 y1 y2

= 2, = 4, = 2, ≥ 0, = 0.

(42)

The above linear system has a solution y¯ = (2, 0)T . Thus, x¯1 = (1, 0, 1)T is a weak optimal solution to linear interval program (41). It is easy to see that x¯1 = (1, 0, 1)T is an optimal solution to the scenario min z = 2x1 + 4x2 + 2x3 s.t. x1 + 2x2 + x3 ≥ 2 2x1 + 4x2 + 3x3 ≥ 3

4

Conclusion

The proposed method checks the optimality of a given weak feasible solution of righthand side IvLP, by solving a linear system. If the problem is: Given x∗ ∈ Rn , is it optimal for some scenario? We can first check the weak feasibility and then check the optimality. Effective methods for checking weak feasibility of equations and inequalities are developed in, see, e.g., [2, 13, 14, 15, 18, 19, 20, 21]. Nevertheless, checking the optimality of a given weak feasible solution for general IvLP is still a difficult open problem. We hope that our results and some recent developments in determining the optimal solution set of the IvLP, based on the theory of B-stability [22, 23], provide researchers with a good basis for a more complex and still open case with interval uncertainties also in the constraint matrix.

Acknowledgements The authors are grateful to the editor and both the anonymous referees for their constructive comments and suggestions, which have improved the presentation of this paper. This project was supported by the National Natural Science Foundation of China (Grant No. 61003194, 11171316).

References [1] J.W.Chinneck , K.Ramadan, Linear programming with interval coefficients. J Oper Res Soc 51(2) (2000) 209-220. [2] M. Fiedler, J. Nedoma, J. Ramik, J. Rohn, K. Zimmermann, Linear optimization problems within exact data, Springer, New York, 2006. [3] S.-T. Liu, R.-T. Wang, A numerical solution method to interval quadratic programming, Applied Mathematics and Computation , 189 (2007) 1274-1281

10

[4] Wei Li, Xiaoli Tian, Numerical solution method for general interval quadratic programming, Applied Mathematics and Computation, 202 (2008) 589-595. [5] Wei Li, Xiaoli Tian, Fault detection in discrete dynamic systems with uncertainty based on interval optimization, Mathematical modelling and analysis, 16(4) (2011) 549-557. [6] M. Hlad´ık, Optimal value range in interval linear programming, Fuzzy Optim. Decis. Mak. 8(3) (2009) 283-294. [7] M. Hlad´ık, Interval Linear Programming: A Survey, In: Zoltan Adam Mann Edits, Linear Programming New Frontiers. Nova Science Publishers Inc., 2011. [8] M. Hlad´ık, Optimal value bounds in nonlinear programming with interval data, TOP 19 (1) (2011) 93-106. [9] Virginie Gabrel, Cecile Murat, and Nabila Remli, Linear programming with interval right hand sides, Int. Trans. Oper. Res. 17 (2010), no. 3, 397-408. [10] Ralph E. Steuer, Algorithms for linear programming problems with interval objective function coefficients, Math. Oper. Res. 6 (1981), 333-348. [11] R.E.Moore, R.B.Kearfott, M.J.Cloud, Introduction to interval analysis, SIAM, Philadelphia, 2009. [12] G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, NewYork, 1983. [13] A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990. [14] J. Rohn, J.Kreslov´a, Linear interval inequalities. Linear Multilinear Algebra 38 (1994) 79-82 [15] J. Rohn, An algorithm for computing the hull of the solution set of interval linear equations, Linear Algebra and Its Applications. 435 (2011) 193-201. [16] J. Rohn, A general method for enclosing solutions of interval linear equations, Optim. Lett., in press. doi:10.1007/s11590-011-0296-0. [17] S. P. Shary, A New Technique in Systems Analysis Under Interval Uncertainty and Ambiguity, Reliable Computing. 8(5) (2002) 321-418. [18] S. P. Shary, Solving the interval linear tolerance problem, Math. Comput Simulation 39 (1995) 53-85. [19] O.A.Prokopyev, S.Butenko, A.Trapp, Checking solvability of systems of interval linear equations and inequalities via mixed integer programming. Eur J Oper Res 199 (2009) 117-121. [20] G. Alefeld, V. Kreinovich, G. Mayer, On the solution sets of particular classes of linear interval systems, J. Comput. Appl. Math. 152 (2003) 1-15. [21] Wei Li, Huping Wang and Qin Wang, Localized solutions to interval linear equations, J. Comput. Appl. Math. 238 (2013) 29-38

11

[22] M. Allahdadi and H. Mishmast Nehi, The optimal solution set of the interval linear programming problems, Optimization Letters, http://dx.doi.org/10.1007/s11590012- 0530-4 [23] M. Hlad´ık, How to determine basis stability in interval linear programming, Optimization Letters, http://dx.doi.org/10.1007/s11590-012-0589-y

12