Solutions to 8 th Order Polynomial Minimization Problem

Transcription

1 Solutions to 8 th Order Polynomial Minimization Problem Timothy K. Gao timkgao@gmail.com Blacksburg High School VA 26 USA Abstract This paper presents a special canonical dual transformation method for finding global minimizer of a class of 8 th order polynomials. The method can be used to solve the 7 th order nonlinear algebraic equations that are their derivatives. Applications are illustrated by examples. Primal Problem and its Canonical Dual This paper presents a method to find global minimizer of the following 8 th order polynomial minimization problem where P 8 (x) = 2 a 2(y 2 (x)) 2 + b 2 y 2 (x) + c 2 dx, () y 2 (x) = 2 (y (x)) 2 + b y (x) + c, y (x) = 2 a x 2 + b x + c, (2) where a i, b i, c i (i =,, 2), and d are given constants. In this paper, we assume that a i (i =,, 2). The criticality condition of P 8 (x) leads to a 7 th order nonlinear algebraic equation (a 2 y 2 (x) + b 2 )( y (x) + b )(a x + b ) d =. (3) Clearly, direct methods for solving this 7 th order nonlinear algebraic equation are very difficult, or even impossible. Also, identifying the global minimizer of P 8 (x) is a fundamentally difficult task in global optimization. The canonical duality theory developed in [] is a potentially useful methodology, and can be used to solve a relatively large class of nonlinear algebraic and differential equations. The associated triality theory can be used to identify both global minimizer and local extrema. In the recent paper [2], a special polynomial P n (x) with b i = (i =,, 2,, n) has been studied. The purpose of this paper is to generalize this result to solve the 7 th order nonlinear equation and to identify global minimizer of P 8 (x). The main result of this paper was announced at the 7 Virginia State Science and Engineering Fair and received an award. 27

2 2 Canonical Dual Polynomial From the pattern of the 8 th order polynomial defined in (), we can let y 2 (x) = U (y (x)) = P 4 (x) = 2 (y (x)) 2 + b y (x) + c, (4) which is a quadratic function of y. The dual variable s of y is defined as By solving this duality relation, we have Thus, by using the Legendre transformation the Legendre conjugate function U (s ) is well-defined as In the same way, U 2 is set equal to s = U (y ) = y + b. (5) y (s ) = s b. (6) U (s ) = y (s )s U (y (s )), (7) U (s ) = c b (s b ) + s (s b ) (s b ) 2 = (s b ) 2 c. (8) The duality relation between y 2 and s 2 is Then, the Legendre conjugate of U 2 (y 2 ) is By the fact that the polynomial () can be written as U 2 (y 2 (x)) = 2 a 2(y 2 (x)) 2 + b 2 y 2 (x) + c 2. (9) s 2 = U 2(y 2 ) = a 2 y 2 + b 2. () U 2 (s 2 ) = c 2 b 2(s 2 b 2 ) a 2 + s 2(s 2 b 2 ) a 2 (s 2 b 2 ) 2 2a 2 = (s 2 b 2 ) 2 2a 2 c 2. () U 2 (y 2 ) + U 2 (s 2 ) = y 2 s 2, (2) P 8 (x) = U 2 (x) + c 2 dx = y 2 (x)s 2 U 2 (s 2 ) + c 2 dx. (3) 272

3 Manipulation of equations (4) and (7) gives Therefore, y 2 = U = y s U (s ). (4) P 8 (x) = s 2 (y (x)s U ) U 2 + c 2 dx. (5) Let Ξ(x, s, s 2 ) equal to equation (5). Further simplification yields By solving Ξ/ x we have Ξ(x, s, s 2 ) = s 2 [( 2 a x 2 + b x + c )s U (s )] U 2 (s 2 ) + c 2 dx. (6) x(s, s 2 ) = d a s s 2 b a. (7) Substituting this x into (6) and setting the final equation equal to P d (s, s 2 ), the canonical dual function can be obtained as P d (s, s 2 ) = Ξ(x(s, s 2 ), s, s 2 ) [ = s s 2 c ( ] d s s 2 b ) 2 s 2 U (s ) U2 (s 2 ) + c 2. (8) 2a By combining (4), (6) and () and simplifying, the relation between s and s 2 is given by ( ( ) ) 2 s 2 = a 2 2 a s b s b + b + c + b 2 ( ) s 2 = b c + b 2. (9) By substituting (9) into (8) and letting s = s, the canonical dual function is finally obtained as P d (s) = P d (s, s 2 (s)). By using Mathematica, P d (s) can be calculated as P d (s) = c 2 a ( 2(s 2 + c b 2 ) 2 + b 8a a 2 c + a ) 2(s 2 b 2 ) 2a [ c (s b ( ) )] ) s (c (2a ) d b () s( b 2 + a 2 (s 2 + c b 2 )) From the canonical duality theory [], we have the following result: Theorem If s is a critical point of P d (s), then with x(s) = d a s 2 s b a, (2) s 2 = a 2 ( s2 b 2 + c ) + b 2 (22) is a critical point of P (x) and P (x) = P d (s). Moreover, if s is the largest critical point, then x(s) is a global minimizer of P (x). 273

4 3 Applications 3. Example Let a =.; =.5; a 2 =.5; b = ; b = ; b 2 = ; c = 6; c = ; c 2 =.5; d =.5, we have P 8 (x) =.5.5x ( 6 +.5x 2 ) 4. (23) This is a nonconvex polynomial of degree eight with one global minimizer, one local minimizer and one local maximizer as shown in Fig Figure. Nonconvex P 8 (x) with one global minimizer, one local minimizer and one local maximizer. The canonical dual of this P 8 (x) has a very simple form Its graph is shown in Fig. 2. P d (s) = ( s 6 )s s 4. (24) Figure 2. Canonical dual polynomial P d (s). The criticality condition P d (s) = has three real roots s = > s 2 =.597 > s 3 = (25) From equation (2), we have x = 3.54, x 2 = , and x 3 =.99. It is easy to check that P (x ) = = P d (s ) < P (x 2 ) = = P d (s 2 ) < P (x 3 ) = = P d (s 3 ). (26) 274

5 This shows that x is a global minimizer, x 2 is a local minimizer, and x 3 is a local maximizer (see Fig. 3) Figure 3. Canonical dual polynomialp d (s). 3.2 Example 2 In Example, if we let a =.; =.5; a 2 =.5; b =.; b =.2; b 2 = ; c = 7.5; c = 5; c 2 =.5; d =.5, the graph of the polynomial P 8 (x) is shown in Fig Figure 4. Nonconvex P 8 (x) with c = 7.5. In this case, the canonical dual function P d (s) has a total of seven critical points (see Fig. 5) Figure 5. Canonical dual polynomialp d (s). s 7 = < s 6 = < s 5 = < s 4 = < s 3 =.4673 < s 2 = < s =

6 By the Theorem we know that x = is a global minimizer of P (x) and P (x ) = P d (s ) = (see Fig. 6) Figure 6. Graphs of P (x) and P d (s). 4 Conclusions The results of this paper can be used to find the all critical points of the eighth polynomial derivative of equation (). One advantage of using this way to find the critical values of P 8 is that, when the derivative of P d (s) is set to and the critical values of P d (s) are found, the critical value with the largest value of s is the absolute minimum. Supplementary File A Mathematica file is available for those who wish to experiment with these techniques. Acknowledgements Special thanks go to David Gao, who showed how to use the method on 4 th degree polynomials of the form P 4 = 2 (y ) 2 + b y + c. The author also would like to offer his sincere thanks to Dr. Ning Ruan for suggestive discuss and detailed comments, which improved this paper dramatically. This research project was under supervision of Mr. Michael Collver at Blacksburg High School. References [] Gao, D.Y.(). Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic Publishers, Dordrecht/Boston/London, xviii+454pp. [2] Gao, D.Y. (6). Complete solutions to a class of polynomial minimization problems, J. Global Optimization, 35,