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:
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CHAPTER THREE
APPLICATION OF LAGRANGE MULTIPLIER
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