Lagrange multipliers and two constraints

In summary, Lagrange multipliers are essential for efficient numerical constrained optimization, and solving the Lagrangian equations numerically is one of the standard methods to solve constrained optimization problems.
  • #1
autre
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So I need to find the min and max values of f(x,y,z) = x^2 + 2y^2 + 3z^2 given the constraints x + y + z = 1 and x - y + 2z =2. I've gotten as far as (2x, 4y, 6z) = (u,u,u) + (m,-m,2m). I'm stuck trying to solve this system of equations. Any hints?
 
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  • #2
You have the constraints as two more equations: x+y+z=1 and x-y+2z=2.

ehild
 
  • #3
autre said:
So I need to find the min and max values of f(x,y,z) = x^2 + 2y^2 + 3z^2 given the constraints x + y + z = 1 and x - y + 2z =2. I've gotten as far as (2x, 4y, 6z) = (u,u,u) + (m,-m,2m). I'm stuck trying to solve this system of equations. Any hints?

Write them out: 2x = u+m, 4y = u-m, 6z = u+2m. Solving for x, y and z as functions of u and m is quite easy. Now you need two more equations to determine u and m. Can you guess what they are?

RGV
 
  • #4
whilst lagrange is a good idea, and the comments above from ehild & Ray are good ideas about where to head - i think a geometric method would be a tiny bit quicker here... though its upto preference i guess, and all the same at the end of the day

the intersection of two planes is a line. If you find the equation of that line it reduces to optimising a single variable function.
 
  • #5
how would you go about solving for x,y,z as functions of u and m?
 
  • #6
lanedance said:
whilst lagrange is a good idea, and the comments above from ehild & Ray are good ideas about where to head - i think a geometric method would be a tiny bit quicker here... though its upto preference i guess, and all the same at the end of the day

the intersection of two planes is a line. If you find the equation of that line it reduces to optimising a single variable function.

The use of Lagrange multipliers is crucial in achieving robust, efficient numerical constrained optimization algorithms. Often, the convergence speed of a recursive search algorithm can be increased by orders of magnitude by updating not only the estimates of x,y,z,... but also by intelligently updating the estimates of Lagrange multipliers. In many problems involving inequality constraints, one proceeds by assuming some of the constraints are equalities---the "active set---and ignoring the others; knowing the signs of the corresponding Lagrange multipliers is crucial in checking whether one has the correct active set. Some of the very best constrained optimization codes combine Lagrange multiplier and "penalty" methods---the so-called method of multipliers---to deal with difficult constrained optimization problems. Finally, solving the Lagrangian equations numerically is one of the standard methods to solve constrained optimization problems numerically.

So, rather than being an unnecessary distraction, Lagrange multipliers play an extremely important part in modern optimization theory and practice. They should be known by every serious student who will ever do optimization in his/her professional life. That said, it is often the case that the material is not well-taught, and the actual geometric content of the method is perhaps not spelled out the way it ought to be, but that may be more the fault of textbook choice rather than of the material itself. Of course, if the method just appears as a small section in a Calculus course, the crucial role of the method in optimization practice may be missed or not sufficiently emphasized.

RGV
 

1. What are Lagrange multipliers and how are they used in optimization?

Lagrange multipliers are a mathematical tool used in optimization problems to find the optimal values of multiple variables subject to one or more constraints. They involve creating a new equation (the Lagrangian) that combines the objective function with the constraints, and then solving for the variables that satisfy the necessary conditions for optimality.

2. Can you explain the concept of two constraints in Lagrange multipliers?

In Lagrange multipliers, the concept of two constraints refers to the situation where there are two or more constraints that need to be satisfied in order to find the optimal values of the variables. This requires the use of multiple Lagrange multipliers, one for each constraint, in the Lagrangian equation.

3. What are the necessary conditions for optimality in Lagrange multipliers with two constraints?

The necessary conditions for optimality with two constraints in Lagrange multipliers are known as the KKT conditions. These include the gradient of the objective function being equal to the weighted sum of the gradients of the constraints, as well as the constraints being satisfied and the Lagrange multipliers being non-negative.

4. How do Lagrange multipliers with two constraints differ from those with only one constraint?

The main difference between Lagrange multipliers with one and two constraints is that with two constraints, multiple Lagrange multipliers are used, while with one constraint, only a single Lagrange multiplier is needed. This makes the solving process more complex, as it requires solving a system of equations instead of just one.

5. What are some real-world applications of Lagrange multipliers with two constraints?

Lagrange multipliers with two constraints have many practical applications, such as in economics for optimizing production levels subject to resource limitations, in engineering for designing structures that need to meet multiple design constraints, and in physics for finding the optimal path for a particle subject to two or more forces.

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