Global optimization subject to constraints

In summary, the maximum value of f(x,y,z)=(xyz)1/3 is k/3, where k is the constant defined by x+y+z=k. Using this result, we can show that for any nonnegative numbers x,y,z, the value of (xyz)1/3 is less than the average of x, y, and z, or (x+y+z)/3. This holds true even if the constraint x+y+z=k is removed.
  • #1
kingwinner
1,270
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1a) Determine the maximum value of f(x,y,z)=(xyz)1/3 given that x,y,z are nonnegative numbers and x+y+z=k, k a constant.

1b) Use the result in (a) to show that if x,y,z are nonnegative numbers, then (xyz)1/3 < (x+y+z)/3


Attempt:
1a) Using the Lagrange Multiplier method, I get that the absolute maximum of f subject to the constraints x+y+z=k and x,y,z>0 is k/3

1b) Here, it seems to me that one of the constraints, namely x+y+z=k, is removed. If so, then how can we still use the result of part (a) here?

I need some help on part (b). Any help is appreciated!
 
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  • #2
kingwinner said:
1a) Determine the maximum value of f(x,y,z)=(xyz)1/3 given that x,y,z are nonnegative numbers and x+y+z=k, k a constant.

1b) Use the result in (a) to show that if x,y,z are nonnegative numbers, then (xyz)1/3 < (x+y+z)/3


Attempt:
1a) Using the Lagrange Multiplier method, I get that the absolute maximum of f subject to the constraints x+y+z=k and x,y,z>0 is k/3

1b) Here, it seems to me that one of the constraints, namely x+y+z=k, is removed. If so, then how can we still use the result of part (a) here?

I need some help on part (b). Any help is appreciated!

For any x, y, z, x+ y+ z is something isn't it? For any x, y, z, define k= x+ y+ z. Then you have shown by a that [itex](xyz)^{1/3}\le k/3= (x+ y+ z)/3[/itex].
 
  • #3
HallsofIvy said:
For any x, y, z, x+ y+ z is something isn't it? For any x, y, z, define k= x+ y+ z. Then you have shown by a that [itex](xyz)^{1/3}\le k/3= (x+ y+ z)/3[/itex].

For simplicity, let's take k=5.

In part b, x,y,z are only required to be nonnegative numbers. There is no restriction that x+y+z=5 as there is in part a.
Take e.g. x=5, y=5, z=5 which are nonnegative
But x+y+z=15, which is not equal to 5.

It seems to me that (xyz)1/3 < (x+y+z)/3 is true only if x+y+z=k, but NOT true for ANY nonnegative numbers, and in part b we have to prove the latter.

Can someone explain more, please?
 
  • #4
You really didn't listen to Halls, did you? You proved the max for ALL k. If k=5 then the max is at x=y=z=5/3. And (xyz)^(1/3)<=5=x+y+z. If you take x=y=z=5 you'd better set k=15. Then (xyz)^(1/3)<=15<=x+y+z. Any other values of k you'd like me to address individually? I think you should think about this a little more before posting another question.
 

1. What is global optimization subject to constraints?

Global optimization subject to constraints is a mathematical problem-solving technique that involves finding the best solution for a given problem while satisfying a set of constraints. This technique is commonly used in various fields such as engineering, economics, and computer science.

2. How is global optimization subject to constraints different from other optimization methods?

Unlike other optimization methods that focus on finding the best solution within a given range, global optimization subject to constraints aims to find the global optimal solution that gives the best possible outcome for the given problem. It also considers constraints that must be satisfied in the solution.

3. What are some common constraints in global optimization?

Some common constraints in global optimization include inequality constraints, equality constraints, and bound constraints. Inequality constraints limit the values that the variables can take, equality constraints impose a specific relationship between variables, and bound constraints restrict the variables within a certain range.

4. What are the applications of global optimization subject to constraints?

Global optimization subject to constraints has various applications in different fields. It is commonly used in engineering for optimal design of structures, in economics for portfolio optimization, and in computer science for optimizing algorithms and data structures. It can also be applied in other areas such as logistics, transportation, and energy systems.

5. How is global optimization subject to constraints solved?

Global optimization subject to constraints can be solved using various algorithms such as branch and bound, simulated annealing, and genetic algorithms. These methods involve systematically exploring the solution space to find the global optimal solution while satisfying the given constraints. The choice of algorithm depends on the specific problem and its constraints.

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