On the approximation of real rational functions via mixed-integer ...

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120 N. Papamarkos / Appl. Math. Comput. 112 (2000) 113±124p j ˆ V ‡ a 0 jfor j ˆ 1; 2; ...; N;q j ˆ V ‡ b 0 jfor j ˆ 1; 2; ...; M;…20†then, the linear problem (19) is reformulated asmaximize dsubject toG kn ‡ XMjˆ1q j~B j …x k †jˆ1! XN! G k n ‡ XMq j~B j …x k †jˆ1jˆ1‡ XNp j~ Aj …x k †‡d k y 6 1;jˆ1p j~A j …x k †d k y 6 1;! n ‡ XMq j~B j …x k † ‡ d ‡ h k y 6 0 y ˆ 1; q n1n ‡ p j 6 V ; q n2n ‡ p j P V ; j 2‰1 ...NŠ; r m1 n ‡ q j 6 V ; r m2 n ‡ q j P V ; j 2‰1 ...MŠ;…21†where q n1; q n2and r m1 ; r m2 known integers, n 0 P 0, d > 0 and k ˆ 1; 2; ...; K.All the variables are positive, andh k ˆ V XMjˆ1~B j …x k †; …22†d k ˆG k V XM~B j …x k †‡V XNjˆ1jˆ1~A j …x k †ˆG k h k ‡ v XNjˆ1~A j …x k †: …23†Problem (21) is the linear programming problem that must be solved in eachnode. It corresponds to a total number of M ‡ N ‡ 3 variables and 3K ‡ D n ‡ 1linear constraints, where D n is the number of the additional linear constraints inthe node n. Usually, M ‡ N ‡ 3 3K ‡ D n ‡ 1, which means that the linearprogramming problem must be solved by its dual, using a method such as theRSA.According to the above analysis, the mixed integer linear rational approximationalgorithm consists of the following steps:Step 1. The integer and continuous coecients are de®ned and the LRAapproximation problem (13) is formulated. If it is known from the beginning,we can give an estimation of the global-integer solution.Step 2. The LRA problem is solved as a continuous variable problem. Let Xthe vector of variables that must take integer values in the ®nal mixed-integersolution.
N. Papamarkos / Appl. Math. Comput. 112 (2000) 113±124 121Step 3. If all the elements of X are integers then the optimal integer solutionis found.Step 4. A new linear programming sub-problem of the form (21) is formulatedand solved by the RSA.Step 5. If the sub-problem does not have any solution or its solution is worsethan a previous solution, then go to Step 6. If the sub-problem has an integersolution better than a previous integer solution then put X in ˆ X and go to Step6. In any other case, go to Step 7.Step 6. Move to the next node of the tree and go to Step 7. If all the nodes ofthe tree are examined then go to Step 7.Step 7. De®ne the additional linear constraints and go to Step 4.Step 8. The optimal solution is found.3. ExampleThe proposed method was successfully tested on a variety of approximationproblems. In this paper we use the MIRFA algorithm for the approximation ofthe rational function with n ˆ m ˆ 3 coecients. The sampling data are producedfrom the rational function4 ‡ 0:2x ‡ 3x 2f …x† ˆ…24†1 ‡ 2x ‡ 0:8x 2 ‡ 4x 3in the rage [0,2]. Speci®cally, x k ; k ˆ 1; ...; K, with K ˆ 30, equal spaced valuesare used for the independent variable x, which from (24) produce 30 samplingdata for the function f(x). The values of f …x k †; k ˆ 1; ...; K are truncated tothe ®fth decimal point.The A ~ i …x† and ~B i …x† functions have the following form~A i …x† ˆx i1 ; ~B i …x† ˆx i : …25†The application of the RFA algorithm, after 22 loops, gives the followingcontinuous optimal solutiond ˆ 336215:824n ˆ 2:97428 10 6a 1 ˆ 4:0 b 1 ˆ 1:99778a 2 ˆ 0:200093 b 2 ˆ 0:799946a 3 ˆ 3:00022 b 3 ˆ 4:00034Obviously, the values of the optimal coecients are not exactly the initial of(24) due to the truncating procedure.The application of the MIRFA algorithm, with V ˆ 100, gives the followingoptimal integer solution:
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