# Molly's question at Yahoo Questions regarding constrained optimization

• MHB
• MarkFL
In summary, the Lagrange multiplier gives us two critical points where the objective function is zero, and so we can determine the minimum value of the function.
MarkFL
Gold Member
MHB
Here is the question:

Constrained optimization and Lagrange Multipliers? Help please!?

Find the minimum value of the function f(x,y)=12x^2+7y^2+6xy+8x+2y+4 subject to the constraint 4x^2+2xy=1

Minimum Value: _____

THANK YOU! :)

I have posted a link there to this thread so the OP can view my work.

Hello Molly,

We are given the objective function:

$$\displaystyle f(x,y)=12x^2+7y^2+6xy+8x+2y+4$$

subject to the constraint:

$$\displaystyle g(x,y)=4x^2+2xy-1=0$$

Now, using Lagrange multipliers, we obtain:

$$\displaystyle 24x+6y+8=\lambda\left(8x+2y \right)$$

$$\displaystyle 14y+6x+2=\lambda\left(2x \right)$$

This implies:

$$\displaystyle \lambda=\frac{12x+3y+4}{4x+y}=\frac{7y+3x+1}{x}$$

Cross-multiplication yields:

$$\displaystyle 12x^2+3xy+4x=28xy+12x^2+4x+7y^2+3xy+y$$

This reduces to:

$$\displaystyle 0=28xy+7y^2+y=y(28x+7y+1)$$

This gives us two cases to consider:

i) $$\displaystyle y=0$$

Substituting into the constraint, we find:

$$\displaystyle 4x^2-1=0\implies x=\pm\frac{1}{2}$$

Hence, we find the two critical points:

$$\displaystyle \left(0,\pm\frac{1}{2} \right)$$

ii) $$\displaystyle 28x+7y+1=0\implies y=-\frac{28x+1}{7}$$

Substituting into the constraint, we find:

$$\displaystyle 4x^2+2x\left(-\frac{28x+1}{7} \right)-1=0$$

Multiply through by 7:

$$\displaystyle 28x^2-56x^2-2x-7=0$$

$$\displaystyle 28x^2+2x+7=0$$

Observing that the discriminant is negative, we know there are no real roots, and so this case yields no critical points.

Now, we check the value of the objective function at the two critical points found in the first case:

$$\displaystyle f\left(-\frac{1}{2},0 \right)=3+0+0-4+0+4=3$$

$$\displaystyle f\left(\frac{1}{2},0 \right)=3+0+0+4+0+4=11$$

And so we may conclude that:

$$\displaystyle f_{\min}=f\left(-\frac{1}{2},0 \right)=3$$

We cannot conclude that the other point is a global maximum because the constraint gives us:

$$\displaystyle y=\frac{1}{2x}-2x$$

and substitution into the objective function gives us:

$$\displaystyle f(x)=28x^2+4x-7+\frac{4x+7}{4x^2}$$

which is unbounded as $$\displaystyle x\to\pm\infty$$

If we differentiate this function, and equate the result to zero, we obtain:

$$\displaystyle f'(x)=(2x+1)(2x-1)\left(28x^2+2x+7 \right)=0$$

As before, the yields the critical values:

$$\displaystyle x=\pm\frac{1}{2}$$

Now, the second derivative of the objective function in $x$ is:

$$\displaystyle f''(x)=448x^3+24x^2-2$$

And we find that:

$$\displaystyle f''\left(-\frac{1}{2} \right)<0$$

$$\displaystyle f''\left(\frac{1}{2} \right)>0$$

And so we know there is a relative minimum and a relative maximum, but given the behavior of the function at the extremes, i.e.:

$$\displaystyle \lim_{x\to\pm\infty}f(x)=\infty$$

We must therefore conclude that the relative minimum is the global minimum while the relative maximum is not the global maximum.

## 1. What is constrained optimization?

Constrained optimization is a mathematical process of finding the maximum or minimum value of a function, subject to certain constraints or limitations. It is used in various fields such as economics, engineering, and computer science to make informed decisions based on limited resources and constraints.

## 2. How is constrained optimization different from unconstrained optimization?

Constrained optimization involves finding the optimal solution within a set of constraints, while unconstrained optimization has no limitations and seeks to find the global maximum or minimum of a function. Constrained optimization is more complex and requires more advanced mathematical techniques compared to unconstrained optimization.

## 3. What are some common constraints in constrained optimization?

Some common constraints in constrained optimization include budget constraints, resource limitations, and physical constraints such as time, space, or capacity. Other constraints could include legal restrictions, safety regulations, or environmental factors.

## 4. How is constrained optimization used in real-world applications?

Constrained optimization is used in a wide range of real-world applications such as supply chain management, portfolio optimization, transportation planning, and production scheduling. It is also commonly used in machine learning and artificial intelligence algorithms to find the optimal solution for a given problem.

## 5. What are some methods for solving constrained optimization problems?

There are several methods for solving constrained optimization problems, including the Lagrangian method, the KKT conditions, and the interior point method. These methods use a combination of mathematical techniques such as calculus, linear algebra, and optimization algorithms to find the optimal solution within the given constraints.

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