The Use of Lagrange Multiplier’s Method in Solving the Constrained Optimization Problem
Chapter One
AIMS AND OBJECTIVES
The aim and objectives of this research are:
- To find the extrema of a function subject to a fixed constraint through an analytical investigation of Joseph Louis Lagrange’s (1736-1813) work; referred to as Lagrange multiplier’s method.
- To see its (Lagrange multiplier) applications in a variety of fields, including economics, etc.
- To know the method of Lagrange multiplier
- To minimize/maximize a multivariable function subject to one or multiple constraints
- To understand how the method of Lagrange multipliers can be used to find absolute maximums and absolute minimums of a function over a close region
- To understand the application of Lagrange multipliers on economic
CHAPTER TWO
LITERATURE REVIEW
INTRODUCTION:
This chapter basically is concerned with the review of literature on the method of solving the extrema of a function subject to a fixed outside constraint using the method of Lagrange multipliers.
The Lagrange method also known as Lagrange multipliers is named after Joseph Louis Lagrange (1736-1813), an Italian born mathematician. His Lagrange multipliers have applications in a variety of fields, including physics, astronomy and economics.
THE METHOD OF LAGRANGE MULTIPLIERS
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints, for instance, consider the optimization problem.
Maximize f(x, y)
Subject to g(x,) =c
We need both f and g to have continuous first partial derivatives. We introduce a new variable () called a Lagrange multiplier and study the Lagrange function (or Lagrangian) defined by:
(x, y, = f(x, y) + . [g(x, y)-c]
Where the term may be either added or subtracted if f(x0, y0,) is a maximum of f(x, y) for the original constraint problem, then there exists such that (x0, y0,0 ) is a stationary point for the Lagrange function. (Stationary point are those points for the partial derivatives of are zero). However not all stationary points yield a solution of the original problem. Thus, the method of Lagrange multipliers yield a necessary condition for optimality in constrained problems.
One may reformulate the Lagrangian as a Hamiltonian, which case the solutions are local minima for the Hamiltonian, this is done in optimal control theory, in the form of pontryagin’s minimum principle.
The fact that solutions of the Lagrangian are not necessarily extrema also poses difficulties for numerical optimization example.
HANDLING MULTIPLE CONSTRAINTS
The method of Lagrange multipliers is also used for problems with multiple constraints. To see how this is done, we need to re-examine the problem in a slightly different manner. The basic idea remains essentially the same. If we consider only the points that satisfy the constraints (i.e are in the constraints) then a point [P, f(p)] is a stationary point (i.e. a point in a “flat” region) of F if and only if the constrains at that point do not allow movement in a direction where f changes value.
Once we have located the stationary points, we need to further test to see if we have found a minimum, a maximum or just a stationary point that is neither a maximum nor a minimum.
Typically, if given a constraint of the form g = g (x, y) = k, we instead let
g, (x, y) = g(x, y) =k and we the constraint g (x, y) = 0
Thus, Lagrangian are usually of the form
L(x, y, z) = f(x, y, z) – g1 (x, y, z)
Corresponding, to find the extrema of a function f(x, y, z) subject to two constraints,
G(x, y, z) = k, h(x, y, x) = i
LAGRANGE MULTIPLIERS METHOD
Lagrange multipliers are method used for multivariable calculus, it combines the use of derivatives and the techniques used to solve linear programming like linear programming, Lagrange multipliers are used to solve optimization problems that have multiple variables the same principles apply; an objective function is used with constraint to determine an optional solution. But Lagrange multipliers expand on these principles.
What is so special about Lagrange multipliers:
- Lagrange multipliers can solve more complex problems
- Linear programming deals with exclusively linear objectives functions, linear equality and linear inequality.
CHAPTER THREE
APPLICATION OF LAGRANGE MULTIPLIER
On the applications of Lagrange Multiplier compatible modes for controlling accuracy and stability of mixed shell finite elements. Classical hybrid-type formulations for shell finite elements can be developing from the Helinger- Reissner Virational principle combined with use of stress basis functions. The numerical stability and accuracy of resulting formulations having a delicate dependency on the balance between the stress, displacement and Lagrange multiplier fields.
The application of Lagrange multiplier provides a means of gaining control over this balance. It is shown in the context of Mindlin kinematics the use of element based compatible Lagrange multipliers with local bubble basis function can lead to accurate stress.
Constrained optimization, plays a central role in economics for example, the choice problem for a consumer is represented as one of maximizing a utility function subject to a budget constraint. The Lagrange multiplier has an economic interpretation as the shadow price associated with the constraint.
CHAPTER FOUR
SOLVED PROBLEMS AND DISCUSION OF RESULT
ECONOMIC APPLICATION
Problem 1.
- Show that for production function by the ratio of labour to capital. L/K, which minimizes production subjects to a budget constraint, does not depend on the amount of the budget.
CHAPTER FIVE
SUMMARY AND CONCLUSION
This research is an analytic study of the method of Lagrange multiplier which can be used to find the extrema (minimum and maximum value) of a function subject to fixed outside condition or constraint.
- Lagrange multipliers allows you to maximize a function f(x, y) subject to a constraint g(x, y) = k
- It is a method that can be used to find the extreme points of a function on the boundary of a closed region.
- The method requires you to solve for x, y and given the following expressions.
- We can also restrain to more than one constraint g(x, y, x) = k and h (x, y, z) = c when we have three or more variables. In this case we solve for x, y, z, and given the following expressions
- Cobb Douglas production function is a production function that is particularly useful in business and economics because it has a function of two variables labour and capital.
- Lagrange multipliers are used to analyze the Cobb-Douglass production function.
CONCLUSION
From the data analyzed, this research has been able to show the unique position of Lagrange multiplier as a method of solving the optimization problem of a function of variable subject to outside constraint. In instructing of engineering mathematics. The Lagrange multiplier method is most applied in cases when the constraint condition g(x, y) = 0 can not be uniquely expressed explicitly as a function y = f(x) or x=h(y).
Based on this study it is discovered that in optimal control theory the Lagrange multiplier is interpreted as costate variable and it is also reformulated as the minimization of the Hamiltonian, in pontryagin’s minimum principle. Also in economic the Lagrange multiplier is interpreted as the shadow price associated with the constraint.
Also in this study, Lagrange multiplier is used to analyze the Cobb-Douglass production function.
Finally, the research discovered that Lagrange multiplier is a very useful technique in multivariable calculus where one of the most common problems is that of finding the extrema of a function. Lagrange multiplier makes it possible to find a closed form for the function being extremize especially if the function is subject to a constraint(s).
REFERENCES
- Bazaraa M.S, Gould J.J. and Nashed M.Z. (1973) on the Cones of targents with application to mathematical programming, J. Optimization theory application.
- Beltrami E.J. (1973). A Lagrange multiplier rule, American math. Monthly
- Buys J.D, 1972. Dual Algorithms for constrained optimization problems, doctorial dissertation, Univ. Leiden.
- Bertsekas DP. 1982. Constrained optimization and Lagrange multiplier method, academic press.
- Bliss G. A. (1946) Lectures on the calculus of variations. University of Chicago press, web.
- BAZARAA, M.S and C. M. Shetty (1976):foundations of optimization, Springer Verlag. Berlin.
