(PDF) The use of Lagrange multiplier in solving optimization problems

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.
Lagrange multipliers can be used in linear and non linear problems (which is nice because real life situations are often not linear)
Derivatives are used to solve these Lagrange multipliers

PROOF OF LAGRANGE MULTIPLIERS

Here we will give two arguments, one geometric and one analytic, as to why Lagrange multiplier works.

CRITICAL POINTS

For the function w=f(x, y, z) constrained by g(x, y, z)=c (c is a constraint) the critical points are defined as those points, which satisfy the constraint and where f is parallel to g in equations:

STATEMENT OF LAGRANGE MULTIPLIERS

For the constraint system, local maxima and minima collectively extrema, occur at the critical points

GEOMETRIC PROOF FOR LAGRANGE

Lets only consider the two dimensional case, w=f(x, y) with constraint g(x, y)=c for concreteness, we’ve drawn the constraint, g(x, y)=c, as a circle and some level curves for w=f(x, y)=c with explicate values geometrically, we are looking for the points on the circle where we takes it’s maximum or minimum values lets start at the level curve with w=17, which has no points on the circle. So, clearly, the maximum value of w on the constraint circle is less than 17. Move down the level curves until they first touch P. it is clear that P gives a local maximum for W on g=c, because if you move away from P in either direction on the circle you will be on a level curve with a smaller value.

Download Chapters 1 to 5 PDF

CONTENT STRUCTURE OF THE USE OF LAGRANGE MULTIPLIER IN SOLVING OPTIMIZATION PROBLEMS

The abstract contains the research problem, the objectives, methodology, results, and recommendations
Chapter one of this thesis or project materials contains the background to the study, the research problem, the research questions, research objectives, research hypotheses, significance of the study, the scope of the study, organization of the study, and the operational definition of terms.
Chapter two contains relevant literature on the issue under investigation. The chapter is divided into five parts which are the conceptual review, theoretical review, empirical review, conceptual framework, and gaps in research
Chapter three contains the research design, study area, population, sample size and sampling technique, validity, reliability, source of data, operationalization of variables, research models, and data analysis method
Chapter four contains the data analysis and the discussion of the findings
Chapter five contains the summary of findings, conclusions, recommendations, contributions to knowledge, and recommendations for further studies.
References: The references are in APA
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