# Minimize function subject to constraint

• aaaa202
In summary, the conversation discusses a function of the form f(x,y,z) = ax + by + cz with a constraint x+y=k and the method of minimizing it using the equations dF = adx + bdy + cdz = 0 and dy=-dx. However, the equations do not lead to a solution and further information may be needed. The conversation also mentions using Lagrange multipliers as an alternative method for solving such problems.
aaaa202
Suppose given a function of the form:

f(x,y,z) = ax + by + cz
with the constrain x+y=k

My book minimizes this function by a way I am not completely familiar with:

dF = adx + bdy + cdz 0

and since dy=-dx we can write:

dF = (a-b)dx + cdz = 0
=>
a-b = c dz/dx (1)

How I would minimize is simply plug y=k-x into the definition of f:

f(x,z) = (a-b)x + bk + cz

And take partial derivatives

df/dx = a-b + cdz/dx

df/dz = (a-b) dx/dz + c

And seting both equal to zero yields a system of equations which does not reduce to (1).
What is wrong?

(1) should be
a-b+c dz/dx=0
all the equations are equivalent

What about the dx/dz term i get?

aaaa202 said:
Suppose given a function of the form:

f(x,y,z) = ax + by + cz
with the constrain x+y=k

My book minimizes this function by a way I am not completely familiar with:

dF = adx + bdy + cdz 0

and since dy=-dx we can write:

dF = (a-b)dx + cdz = 0
=>
a-b = c dz/dx (1)

How I would minimize is simply plug y=k-x into the definition of f:

f(x,z) = (a-b)x + bk + cz

And take partial derivatives

df/dx = a-b + cdz/dx

df/dz = (a-b) dx/dz + c

And seting both equal to zero yields a system of equations which does not reduce to (1).
What is wrong?

I hope you have left out some important information, because if we take your problem exactly as you have written it, it has no solution; that is, if a, b, c are all non-zero, you can find a sequence of ##x_n, y_n, z_n## giving ##x_n + y_n = k## for all ##n##, but ##ax_n + by_n+ cz_n \to -\infty ## as ##n \to \infty##. In other words, there is no finite minimum.

Are you sure you have stated the problem completely and exactly?

Last edited:
aaaa202 said:
How I would minimize is simply plug y=k-x into the definition of f:

f(x,z) = (a-b)x + bk + cz

Which has level sets at z = Ax + B for some suitable constants A and B. I.e. the function is linear and does not have a minimum.

I would generally tackle these problems using Lagrange multipliers by the way, if you haven't seen them check them out, they are cool ;)

## What is a "Minimize function subject to constraint"?

A "Minimize function subject to constraint" is a mathematical optimization problem where the goal is to find the minimum value of a function while satisfying a set of constraints. The function is typically a cost or objective function, and the constraints limit the possible values of the variables that can be used to minimize the function.

## What are some common examples of "Minimize function subject to constraint" problems?

Some common examples of "Minimize function subject to constraint" problems include finding the shortest distance between two points while staying within a budget or minimizing the amount of material needed to construct a structure while meeting strength requirements.

## How is "Minimize function subject to constraint" different from other optimization problems?

"Minimize function subject to constraint" is a type of constrained optimization problem, which means that there are limitations on the values that the variables can take. This makes the problem more complex and challenging to solve compared to unconstrained optimization problems, where there are no restrictions on the variables.

## What methods are commonly used to solve "Minimize function subject to constraint" problems?

There are various methods that can be used to solve "Minimize function subject to constraint" problems, such as the Lagrange multiplier method, the method of feasible directions, and the interior-point method. These methods use mathematical algorithms to iteratively find the optimal solution that satisfies the constraints.

## What are the real-world applications of "Minimize function subject to constraint" problems?

"Minimize function subject to constraint" problems have various real-world applications, such as in engineering, economics, and operations research. For example, these types of problems can be used to optimize production processes, resource allocation, and transportation routes, among others. They are also commonly used in financial portfolio optimization and machine learning algorithms.

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