Optimization inequality constraint

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Homework Help Overview

The problem involves minimizing the function 2x² + 2y² - 2xy - 9y under a set of constraints, including inequalities and nonnegativity conditions for x and y. Participants are exploring the implications of certain equations and their roots in the context of these constraints.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to understand the implications of an equation having no nonnegative roots and how this relates to the nonnegativity constraints. There is also discussion on completing the square as a method to solve a specific quadratic equation, with some participants expressing uncertainty about the process and its relevance to the problem.

Discussion Status

The discussion is ongoing, with participants questioning the reasoning behind certain constraints and exploring mathematical techniques. Some guidance has been offered regarding completing the square, but there is still uncertainty about the implications of the results and how they relate to the constraints.

Contextual Notes

There are constraints related to nonnegativity for both x and y, and participants are grappling with how these constraints affect the solutions to the equations being discussed.

oswald
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Homework Statement



Minimize 2x²+2y²-2xy-9y subject
4x + 3y =,< 10 ,
y - 4x² =,< -2
x >,= 0
and y >,= 0.


I don't undersant this:

"This equation has no nonnegative root, which contradicts a nonnegativity
constraint."
and how we solve
-16x² + 2x + 17 + h2 = 0
 

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oswald said:

Homework Statement



Minimize 2x²+2y²-2xy-9y subject
4x + 3y =,< 10 ,
y - 4x² =,< -2
x >,= 0
and y >,= 0.


I don't undersant this:

"This equation has no nonnegative root, which contradicts a nonnegativity
constraint."

and how we solve
-16x² + 2x + 17 + h2 = 0
Do you know how to complete the square? That would seem to me to be the best way to solve that equation. (The "h2" here is supposed to be [itex]\lambda_2[/itex], one of the Lagrange multipliers.)
 
Last edited by a moderator:
complete the square on -16x² + 2x + 17 + h2 = 0 or -16x² + 2x + 17 = 0, because i found positive and negative root [ x=~+-1] on -16x² + 2x + 17=0 and i don't know how solve with λ2.
 
Okay, since you are basically saying you do NOT know how to complete the square,
[tex]-16x^2+ 2x+ 17= -16(x^2- (1/8)x)= -16(x^2- (1/8)x+ (1/256)- 256)+ 17= -16(x- 1/16)^2+ 17+ 1/16[/tex].
So [itex]]-16x^2+ 2x+ 17+ \lambda_2[/itex] can only be equal to 0 if [itex]\lambda+ 17+ 1/16>0[/itex] or [itex]\lambda< -(17+ 1/16)[/itex]. That's what violates the "nonegativity constraint", that [itex]\lambda[/itex] can't be negative.
 
why
λ + 17 + 1/16 > 0?
 

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