Advanced Topics in Optimization. Piecewise Linear Approximation of a Nonlinear Function

Transcription

1 Advanced Tocs n Otmzaton Pecewse Lnear Aroxmaton of a Nonlnear Functon Otmzaton Methods: M8L

2 Introducton and Objectves Introducton There exsts no general algorthm for nonlnear rogrammng due to ts rregular behavor Nonlnear roblems can be solved by frst reresentng the nonlnear functon (both objectve functon and constrants) by a set of lnear functons and then aly smlex method to solve ths usng some restrctons Objectves To dscuss the varous methods to aroxmate a nonlnear functon usng lnear functons To demonstrate ths usng a numercal examle Otmzaton Methods: M8L

3 Pecewse Lnearzaton A nonlnear sngle varable functon f(x) can be aroxmated by a ecewse lnear functon Geometrcally, f(x) can be shown as a curve beng reresented as a set of connected lne segments 3 Otmzaton Methods: M8L

4 Pecewse Lnearzaton: Method Consder an otmzaton functon havng only one nonlnear term f(x) Let the x-axs of the nonlnear functon f(x) be dvded by breakng onts t, t, t,, t Corresondng functon values be f(t ), f(t ),, f(t ) If x can take values n the nterval 0 x X, then the breakng onts can be shown as 0 t < t <... < t X 4 Otmzaton Methods: M8L

5 Pecewse Lnearzaton: Method contd. Exress x as a weghted average of these breakng onts x = w t + w t Functon f(x) can be exressed as. e., x = w t w t 5 f where ( x) = w f ( t ) + w f ( t ) w f ( t ) = w f ( t ) w = Otmzaton Methods: M8L

6 Pecewse Lnearzaton: Method contd. Fnally the model can be exressed as Max or Mn f ( x) = w f ( ) subject to the addtonal constrants w t w = x = t 6 Otmzaton Methods: M8L

7 Pecewse Lnearzaton: Method contd. Ths lnearly aroxmated model can be solved usng smlex method wth some restrctons Restrcted condton: There should not be more than two w n the bass and Two w can take ostve values only f they are adjacent..e., f x takes the value between t and t +, then only w and w + (contrbutng weghts to the value of x ) wll be ostve, rest all weghts be zero In general, for an objectve functon consstng of n varables ( n terms) reresented as ( x) = f ( x ) + f ( x ) +... ( ) Max or Mn f + f n x n 7 Otmzaton Methods: M8L

8 Pecewse Lnearzaton: Method contd. subjected to m constrants g ( x ) + g ( x ) g ( x ) b for j,,..., m j j nj n j = 8 The lnear aroxmaton of ths roblem s Max or Mn subjected to n n k = k = w k w w k k g f k kj ( t ( t k k ) ) b j for j =,,..., m = for k =,,..., n Otmzaton Methods: M8L

9 Pecewse Lnearzaton: Method 9 x s exressed as a sum, nstead of exressng as the weghted sum of the breakng onts as n the revous method x = t + u + u u where u s the ncrement of the varable x n the nterval.e., the bound of u s 0 u t+ The functon f(x) can be exressed as ( x ) = f ( t ) u + α f = where α reresents the sloe of the lnear aroxmaton between the onts t + and t f ( t+ ) f ( t ) α = t t = t + u + t ( t, t + ) Otmzaton Methods: M8L

10 Pecewse Lnearzaton: Method contd. Fnally the model can be exressed as Max or Mn f α ( x) = f ( t ) + subjected to addtonal constrants u t + u = 0 u t+ x t, =,,..., 0 Otmzaton Methods: M8L

11 Pecewse Lnearzaton: Numercal Examle The examle below llustrates the alcaton of method Consder the objectve functon Maxmze f = + 3 x x subject to x 0 x + x x The roblem s already n searable form (.e., each term conssts of only one varable). Otmzaton Methods: M8L

12 Pecewse Lnearzaton: Numercal Examle contd. Slt u the objectve functon and constrant nto two arts where and are treated as lnear varables as they are n lnear form f g f = g = f g ( x ) + f( x ) ( x ) + g ( x ) 3 ( x ) = x ; f( x ) = x ( x ) = x ; g( x ) = x f ( ) ( ) x g x Otmzaton Methods: M8L

13 Pecewse Lnearzaton: Numercal Examle contd. Consder fve breakng onts for x f ( x ) can be wrtten as, f 5 ( x ) w f ( t ) = = w 0 + w + w3 8 + w4 7 + w Otmzaton Methods: M8L

14 Pecewse Lnearzaton: Numercal Examle contd. ( ) g x can be wrtten as, g 5 ( x ) w g ( t ) = = w 0 + w + w3 8 + w4 8 + w5 3 4 Thus, the lnear aroxmaton of the above roblem becomes Maxmze f = w 8 w + x subject to w w w + 8w + w + w3 + 7w w + 8w w 4 0 for =,,...,5 + 3w + w x = 5 + s Otmzaton Methods: M8L = 5

15 Pecewse Lnearzaton: Numercal Examle contd. Ths can be solved usng smlex method n a restrcted bass condton The smlex tableau s shown below 5 Otmzaton Methods: M8L

16 Pecewse Lnearzaton: Numercal Examle contd. From the table, t s clear that w 5 should be the enterng varable s should be the extng varable But accordng to restrcted bass condton and cannot occur together n bass as they are not adjacent Therefore, consder the next best enterng varable Ths also s not ossble, snce should be exted and and cannot occur together The next best varable, t s clear that should be the extng varable w w5 w 4 s w4 w w w3 6 Otmzaton Methods: M8L

17 Pecewse Lnearzaton: Numercal Examle contd. The smlex tableau s shown below The enterng varable s w 5. Then the varable to be exted s s and ths s not accetable snce s not an adjacent ont to w w 4 w5 3 Next varable can be admtted by drong. s 7 Otmzaton Methods: M8L

18 Pecewse Lnearzaton: Numercal Examle contd. The smlex tableau s shown below w w 5 4 w w w3 Now, cannot be admtted snce cannot be droed Smlarly and cannot be entered as cannot be droed 8 Otmzaton Methods: M8L

19 Pecewse Lnearzaton: Numercal Examle contd. Snce there s no more varable to be entered, the rocess ends Therefore, the best soluton s Now, w3 = 0.3; w4 = 0.7 x = w t = w3 + w4 3 =.7 The otmum value s 5 and x = 0 f =. 3 Ths may be an aroxmate soluton to the orgnal nonlnear roblem However, the soluton can be mroved by takng fner breakng onts 9 Otmzaton Methods: M8L

20 Thank You 0 Otmzaton Methods: M8L