I have to maximize the following function - Max A <msubsup> C 1 <mrow class="M

Laila Andrews 2022-05-07 Answered
I have to maximize the following function -
Max A C 1 m / m + (1-A) C 2 m / m
Subject to,
C 1 ≤ 5(1-x) + x
C 2 ≤ 3(1-x) + 7x
1≤x≤10
I wrote it as: L(x) = f(x) - λ 1 ( C 1 - 5(1-x) + x) - λ 2 ( C 2 - 3(1-x) + 7x) - λ 3 (x-1) - λ 4 (x-10)
Can I write last constraint as partioned into λ 3 and λ 4 . Is there some other way to introduce such box constraints into the same problem?
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Answers (2)

Elliana Shelton
Answered 2022-05-08 Author has 10 answers
Considering
L ( x , λ , ϵ ) = λ 1 ( C 1 5 ( 1 x ) x ϵ 1 2 ) + λ 2 ( C 2 3 ( 1 x ) 7 x ϵ 2 2 ) + λ 3 ( 1 x ϵ 3 2 ) + λ 4 ( x 10 ϵ 4 2 )
with ϵ k convenient slack variables, the stationary points are the solutions for
L = 0 = { 4 λ 1 4 λ 2 λ 3 + λ 4 = 0 C 1 5 ( 1 x ) x ϵ 1 2 = 0 C 2 3 ( 1 x ) 7 x ϵ 2 2 = 0 1 x ϵ 3 2 = 0 x 10 ϵ 4 2 = 0 λ 1 ϵ 1 = 0 λ 2 ϵ 2 = 0 λ 3 ϵ 3 = 0 λ 4 ϵ 4 = 0
and the feasible solutions are those observing ϵ k 2 0 for k = 1 , 2 , 3 , 4
( x λ 1 λ 2 λ 3 λ 4 ϵ 1 2 ϵ 2 2 ϵ 3 2 ϵ 4 2 5 C 1 4 0 0 0 0 0 C 1 + C 2 8 C 1 1 4 1 4 ( C 2 + 35 ) C 2 3 4 0 0 0 0 C 1 + C 2 8 0 7 C 2 4 C 2 43 4 x 0 0 0 0 C 1 + 4 x 5 C 2 4 x 3 1 x x 10 )
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agrejas0hxpx
Answered 2022-05-09 Author has 4 answers
Is your function constant? If I read correctly you have
maximise w.r.t x R A C 1 m m ( 1 A ) C 2 m m = γ subj.to h 1 ( x ) 0 h 2 ( x ) ( x ) 0 α h 3 ( x ) β
Maximizing a constant function under some constraints is called a satisfiability problem; any reachable value will attain the maximum (and minimum) of γ. Your method of putting
α h 3 ( x ) β α h 3 ( x ) 0 and h 3 ( x ) β 0
Before putting the problem into Lagrangian form is the go-to solution in most cases if f is non constant!
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