An improving procedure of the numerical solution method for interval quadratic programming Wei Li School of Science, Hangzhou Dianzi University, Hangzhou, 310018, P.R.China Email:
[email protected]
Abstract—Recently, some interesting numerical methods for interval quadratic programming problem have been developed in [11] and [12]. However, the solution procedures in [11] and [12] require some additional variables, and hence the size of the original problem is greatly enlarged. This paper propose a practical modification of the numerical solution methods proposed in [11] and [12]. The new method working space is the space of the original variables. No additional variables are required. This modification leads to a considerable reduction of the computation cost. Illustrative numerical examples are also presented.
Keywords: quadratic programming; interval number; numerical method I. I NTRODUCTION Many studies have developed efficient and effective algorithms for solving mathematics programming when the value assigned to each parameter is a known constant. However, in real world applications, this assumption may not be satisfied. The reason is because mathematics programming models usually are formulated to find some future course of action. The parameter values used would be based on a prediction of future conditions which inevitably involves some degree of uncertainty. There are also situations that the data cannot be collected without error. The primary objective is to find the range of the parameters within which the current solution is still optimal. If some parameters are imprecise or uncertain, then some crisp values are usually assigned to those uncertain parameters to make the conventional quadratic program workable. This simplification might result in a derived result which is misleading. To manipulate imprecise parameters in some cases, use of an interval coefficient may be one of the appropriate choices. Fuzzy and stochastic approaches are frequently used to describe and treat imprecise and uncertain elements present in a real decision problem. In fuzzy programming problems it is assumed that the membership functions of fuzzy sets are known [1], [2]. Also, in stochastic programming problems it is assumed that the probability distributions of coefficients ( viewed as random variables) are known [3], [4], [5]. In real world applications, however, it is not always easy to specify the membership function or the probability distribution in an inexact environment. Thus in some of the cases, use of an interval coefficient may serve the purpose better. The solution methods to interval linear programming were explored by some scholars (see, for example, [6], [7], [8], [9], [10]). However, there are little results for solving interval
quadratic programming. Recently, Liu and Wang [11] proposed an interesting numerical solution method for interval quadratic programming, where the coefficients of the quadratic term in the objective are still real numbers. Li and Tian [12] generalize the method proposed by Liu and Wang to general interval quadratic programming problems, where all parameters in the problem, including cost coefficients, constraint coefficients and right-hand sides, are interval numbers. The idea in both [11] and [12] is to find the upper bound and lower bound of the range of the objective values by employing the two-level mathematical programming technique. Although the methods proposed in [11], [12] are interesting and practical, the solution procedure for upper bound of the objective function values are not so efficient. In Fact, the upper bounds is formulated as a two level programming in [11], [12], where the outer program and inner program have different directions for optimization, viz., one for maximization and another for minimization, which leads to some difficulties in the solution procedure. To overcome this difficulty, the dual theory of the quadratic programming and the variable transformation technique are employed in [11], [12], which conduces to some tiresome computation. In this paper, we present a modification in the process of finding the upper upper bound of the range of the objective values. This modification reduces greatly the cost of computation. For completeness, in the next section we briefly present the numerical solution methods for interval quadratic programming described in [11] and [12]. Then in Section 3, a new effective method for interval quadratic programming is developed by improving the procedure of finding the upper bound. Some numerical examples illustrate the new solution algorithm in Section 4. Finally, Section 5 concludes the paper. II. P RELIMINARIES A. Notations A superscript, I, on a quantity indicates that the quantity is an interval (number, vector, or matrix). Quantities without a superscript are real (numbers, vectors, or matrices). The left endpoint of an interval is indicated by a superscript, L, and the right endpoint by a superscript, R. Thus, a scalar interval, aI , is given by [aL , aR ]. The vector of left endpoints of an interval vector, xI , is denoted by xL ; and the vector of right endpoints by xR . Thus, we write an interval vector, xI , as [xL , xR ]. Similarly, we write an interval matrix, AI , as [AL , AR ].
We say that a real vector, x ∈ Rn , is contained in an interval R vector, xI , and we write x ∈ xI , if xL i ≤ xi ≤ xi ; for all i = n×n , is contained in 1, . . . , n. We say that a real matrix, A ∈ R an interval matrix, AI , and we write A ∈ AI , if aL ij ≤ aij ≤ L R L R L aR ; for all i, j = 1, . . . , n. If a = a (x = x , A = AR ), ij I L R I L R I L R then a = [a , a ] (x = [x , x ], A = [A , A ])is a real number (vector, or matrix). B. Previous results Consider the following model (1)
s.t. AI x ≤ bI , x ≥ 0. Liu and Wang [11] formulate a pair of two-level mathematical programs to calculate the upper bound z R and lower bound z L of the objective values of the programming (1). The method used in [11] for finding the upper bound z R is rather complex. First they formulate the problem for finding z R as a two-level mathematical programming as follows max
(c∈cI ,b∈bI ,A∈AI )
To find the interval bounding the objective values, it suffices to find the lower bound and upper bound of the objective values of problem (3). Denote by z L and z R , the exact lower bound and upper bound of the objective values of problem (3), i.e., the best optimum and the worst optimum , respectively. The following lemma shows how to determine the largest feasible region and least feasible region defined by (3.1) with x ≥ 0. Lemma 1 Suppose we have the interval inequality AI x ≤ b where x ≥ 0. Then AL x ≤ bR and AR x ≤ bL are the largest feasible region and least feasible region defined by (3.1), respectively. Inequalities AL x ≤ bR and AR x ≤ bL are called the largest feasible region inequalities and the least feasible region inequalities, respectively. Also, it is easy to obtain the following lemma, which applies to objective functions. I
1 min z = (cI )T x + xT Qx 2
zR =
III. M ODIFIED NUMERICAL SOLUTION METHOD
1 min z = cT x + xT Qx x 2
(2)
s.t. Ax ≤ b, x ≥ 0, where c, A and b vary in the given interval and Q is a real matrix. Then, based on the duality theorem and by applying the variable transformation technique, the two-level mathematical programming (2) is transformed into a classical quadratic programming. Finally, the upper bound z R is obtained by solve the classical quadratic programming. This procedure for finding z R is rather complex and computing inefficient. Recently, Li and Tian generalize Liu and Wang’s method to general interval quadratic programming of the following form 1 min z = (cI )T x + xT QI x 2
Lemma 2 Let z = cT x + 12 xT Qx, where c ∈ cI , Q ∈ Q and x ≥ 0, be an objective function. Then (cL )T x + 1 T L 1 T 1 T R T R T 2 x Q x ≤ c x + 2 x Qx ≤ (c ) x + 2 x Q x for any given solution x ≥ 0. I
The facts stated in Lemma 1 and 2 are very trivial and hence the proof is omitted. For similar concepts for interval linear programming, we refer to [6] and [7]. Functions (cL )T x + 12 xT QL x and (cR )T x + 12 xT QR x are called the most favorable objective function and the least favorable objective function, respectively. Lemma 1 and 2 allow the computation of the best and worst optimal solutions of the programming (3), and hence obtain the best optimum z L and worst optimum z R . Firstly, we use the most favorable version of the objective function and the largest feasible region inequalities to determine the best optimum solution xL and the worst optimum z L . Then, we use the least favorable version of the objective function and the least feasible region inequalities to determine the worst optimum solution xR and the worst optimum z R . Thus, the programming (3) is transformed into two classical quadratic programming 1 z L = min z = (cL )T x + xT QL x 2 s.t. AL x ≤ bR , x≥0
(3) and
I
I
s.t. A x ≤ b ,
(3.1)
x ≥ 0, I where cI = (cI1 , . . . , cIn )T , QI = (qij )n×n and bI = (bI1 , . . . , bIm )T . The method proposed in [12] to calculate the upper bound z R is similar to the method used in [11] and hence the computation cost is also expensive.
(4)
1 z R = min z = (cR )T x + xT QR x 2
(5)
s.t. AR x ≤ bL , x ≥ 0. Combining two solution results to (4) and (5), we conclude that the objective values of the interval quadratic programming (3) lie in the range of [z L , z R ].
It is worth to note that, the formulation of z L and z R in [11] and [12] is actually to find the best optimum solution and the worst optimum solution. Thus, they are the same as z L and z R described here. Also, the method above for computation of the lower bound z L is the same as that used in [11] and [12]. However, our formulation of the upper bound z R is much easier and much more computing efficient.
Solving the programming (8), the worst optimum for the objective value of (7), z R = −1.125 with the worst optimal solution x1 = 0.75 and x2 = 0, are obtained. Now turn to the method described in this paper. The worst optimum z R , according to programming (5), be simply formulated as: z R = min z = −3x1 + 2x2 + 2x21 − 2x1 x2 + 2x22
IV. N UMERICAL EXAMPLES
s.t.
s.t.
(6)
[1, 2]x1 + x2 ≤ [2, 4]
[2, 3]x1 + [−1, −0.5]x2 ≤ [3, 4] x1 , x2 ≥ 0 This is the same example discussed in [11]. The coefficients of the quadratic term in problem (9) are real numbers, which can be viewed as a degenerate interval number. Based on the programming (4), the best optimum, i.e., the lower bound of the objective value z L , can be formulated as: z L = min z = −5x1 + x2 + 2x21 − 2x1 x2 + 2x22 s.t.
(7)
x1 + x2 ≤ 4 2x1 − x2 ≤ 4
x1 , x2 ≥ 0 By using Matlab, we derive the the worst optimum z R = −1.125 which occurs at x1 = 0.75 and x2 = 0. Combining these two results, we conclude that the objective values of this interval quadratic programming lie in the range of [−3, −1.125]. Thus, we obtain completely the same result as that obtained in [11]. However, the method used in this paper requires much less computing. The problem size is the same as original problem, whereas the method in [11] should introduce 10 additional variables. Thus the programming for z R is 6 times the size of the original problem. Example 2 Consider the following interval quadratic programming: min z = [−10, −6]x1 +[2, 3]x2 +[4, 10]x21 +[−1, 1]x1 x2 +[10, 20]x22 (10) s.t. [1, 2]x1 + 3x2 ≤ [1, 10] [−2, 8]x1 + [4, 6]x2 ≤ [4, 6]
x1 , x2 ≥ 0 (There is a typo in [11], where the second inequality 2x1 − x2 ≤ 4 is written as 2x1 − 0.5x2 ≤ 4 by mistake.) Solving this classical quadratic programming by employing Matlab, we obtain the the best optimum z L which occurs at x1 = 1.5 and x2 = 0.5. Now consider the worst optimum z R , i.e., the upper bound of the objective value. Liu and wang’s method [11] formulated z R , by duality theorem and by applying the variable transformation technique as z R = max z = −2x21 + 2x1 x2 − 2x22 − 2λ1 − 3λ2 s.t.
4x1 − 2x2 + r11 + r21 − δ1 = −c1 , − 2x1 + 4x2 + r12 + r22 − δ2 = −c2 , −5 ≤ c1 ≤ −3, 1 ≤ c2 ≤ 2, λ1 ≤ r11 ≤ 2λ1 , 2λ2 ≤ r21 ≤ 3λ2 , r12 = λ1 , − λ2 ≤ r22 ≤ −0.5λ2 , λ1 , λ2 , δ1 , δ2 ≥ 0.
2x1 + x2 ≤ 2 3x1 − 0.5x2 ≤ 3
Example 1 Consider the following interval quadratic programming: min z = [−5, −3]x1 + [1, 2]x2 + 2x21 − 2x1 x2 + 2x22
(9)
(8)
x1 , x2 ≥ 0 This is the same example used in [12]. We only discuss the solution procedure of the upper bound of the objective value, since the method for finding the lower bound is simple and it is the same in [11], [12] and this paper. The method [12] formulated the upper bound of the objective value z R , by duality theorem and by applying the variable transformation technique as z R = max z = −10x21 − x1 x2 − 20x22 − λ1 − 4λ2 x,λ,δ
s.t.
(11)
20x1 + x2 + r11 + r21 − δ1 = 6 x1 + 40x2 + r12 + r22 − δ2 = −3 λ1 ≤ r11 ≤ 2λ1 − 2λ2 ≤ r21 ≤ 8λ2 r12 = λ1 4λ2 ≤ r22 ≤ 6λ2 λ 1 , λ 2 , δ1 , δ2 ≥ 0
By using the function quadprog in Matalab 6.5, we derive the optimum solution z R = −0.9, x1 = 0.3 and x2 = 0.
Now turn to the method described in this paper. The worst optimum, i.e., the upper bound of the objective value z R , z R , can be simply formulated as: z R = min z = −6x1 + 3x2 + 10x21 + x1 x2 + 20x22 s.t.
(12)
2x1 + 3x2 ≤ 1 8x1 + 6x2 ≤ 4 x1 , x2 ≥ 0
By using Matlab, we obtain the worst optimum z R = −0.9, which occurs at x1 = 0.3 and x2 = 0, the same as the result in [12]. Note that there are no additional variables are exploited in the solution procedure. V. C ONCLUSION Interval quadratic programming problems have raised difficulties in the solution procedure. For such problems, the recently proposed algorithms in [11] and [12] is interesting and practical. However, these algorithms require additional variables in the objective function and the constraints. Unfortunately, adding these extraneous variables creates computational instability and increases the computational complexity. In this paper, we proposed a new general solution algorithm, which is easy to understand and implement. Also, the algorithm works in the space of the original variables. It is free from any extraneous variables. This enhances considerably the storage space, and the computational complexity of the proposed solution algorithm. R EFERENCES [1] M. Delgado, J.L. Verdegay, M.A. Vila, A general model for fuzzy linear programming, Fuzzy Sets and Systems 29 (1989) 21–29. [2] R. Slowinski, A multicriteria fuzzy linear programming method for water supply systems development planning, Fuzzy Sets and Systems 19 (1986) 217–237. [3] J.K. Sengupta, Optimal Decision Under Uncertainty, Springer, New York, 1981. [4] J.R. Birge, F. Louveaux, Introduction to Stochastic Programming, Springer, Berlin, 1997. [5] G.-M. Cho, Log-barrier method for two-stage quadratic stochastic programming, Applied Mathematics and Computation 164 (2005) 45–69. [6] Shaocheng Tong, Interval number and fuzzy number linear programmings , Fuzzy Sets and Systems 66 (1994) 301–306 [7] J.W. Chinneck, K. Ramadan, Linear Programming with Interval Coefficients The Journal of the Operational Research Society, 51(2000) 209–220. [8] A. Sengupta, T.K. Pal, D. Chakraborty, Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming. Fuzzy Sets and Systems,119(2001) 129-138. [9] P. Serafini, Linear programming with variable matrix entries, Operations Research Letters, 33 (2005) 165–170. [10] Wei Li, G.-X. Wang, General Solutions for Linear Programming with Interval Right Hand Side, Proceedings of ICMLC 2006,3: 1836-1839. in Xizhao Wang, Daniel Yeung and Xiaolong Wang eds. (Dalian) [11] S.-T. Liu, R.-T. Wang, A numerical solution method to interval quadratic programming, Applied Mathematics and Computation 189 (2007) 1274– 1281. [12] Wei Li, Xiaoli Tian, Numerical solution method for general interval quadratic programming, Applied Mathematics and Computation, (2008), doi: 10.1016/j.amc.2008.02.039
