Optimization inequality constraint

In summary, the conversation discusses minimizing a function with constraints, specifically 2x²+2y²-2xy-9y with constraints 4x + 3y ≤ 10, y - 4x² ≤ -2, x ≥ 0, and y ≥ 0. The conversation also mentions solving an equation, -16x² + 2x + 17 + λ2 = 0, and completing the square. There is a constraint that λ2 cannot be negative, as it violates the nonnegativity constraint. The conversation also calculates the value of λ based on the equation and constraints.
  • #1
22
0

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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  • #2
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.)
 
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  • #3
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.
 
  • #4
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.
 
  • #5
why
λ + 17 + 1/16 > 0?
 

What is an optimization inequality constraint?

An optimization inequality constraint is a type of mathematical constraint that is used in optimization problems to restrict the possible solutions to a set of values that satisfy a particular inequality. It is often used in conjunction with other constraints to find the optimal solution to a problem.

Why are optimization inequality constraints important?

Optimization inequality constraints are important because they allow for more realistic and practical solutions to optimization problems. In many real-world situations, there are limits or restrictions on the values that certain variables can take, and optimization inequality constraints help to incorporate these limits into the optimization process.

What types of optimization problems use inequality constraints?

Inequality constraints are commonly used in linear programming, quadratic programming, and nonlinear programming problems. They are also used in a variety of other optimization problems, such as constrained optimization and dynamic programming.

How do you handle optimization inequality constraints in mathematical models?

There are several methods for handling optimization inequality constraints in mathematical models. These include the substitution method, the Lagrangian method, and the penalty method. The choice of method depends on the specific problem and the type of inequality constraint.

What are some common mistakes when using optimization inequality constraints?

Some common mistakes when using optimization inequality constraints include not properly defining the constraints, using the wrong type of constraint for the problem, and not considering all possible constraints. It is important to carefully analyze the problem and select the appropriate constraints to ensure an accurate and optimal solution.

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