multi variable optimization

8/4/2019 Multi Variable Optimization

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Multivariable OptimizationOptimization problems usually can be cast in the fol-lowing mathematical form. There is an objectivefunction f(x1, . . . , xn), a real valued function ofnvariables whose value should be maximized (or mini-mized). For example, a firms profits are a function of its

input and output quantities. There is also a constraintset or opportunity set S that is some subset ofn.For example, a consumer cannot buy larger quantities of

different goods than is allowed by the budget constraint,given the prices that have to be paid and the wealththere is available to spend. Then the problem is to find

maximum or minimum points off in S, provided suchpoints exist.


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By specifying the set Sappropriately, several differenttypes of optimization problem can be covered. Iff has anoptimum at an interior point ofS, we talk about classicalcase. IfS is the set of all points (x1, . . . , xn) that satisfya number of equations, we have the Lagrangean prob-

lem of maximizing (or minimizing) a function subject toequality constraints. The general programming prob-lem is obtained ifS consists of all points (x1, . . . , xn) in

n that satisfy mconstraints in the form of inequalities(including, possibly, nonnegativity conditions on x1,..., xn).If the objective function and all the constraints are linear

in (x1

, . . . , xn

), then we have a l inear programmingproblem .


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Simple Two Variable Optimization


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Constrained OptimizationA typical economic example concerns a consumer whochooses how much of the available income mto spend ona good xwhose price isp, and how much income to leaveover for expenditure yon other goods. Note that theconsumer then faces the budget constraint px+ y= m.

Suppose that preferences are represented by the utilityfunction u(x,y). In mathematical terms, therefore, theconsumer faces the problrm of choosing (x,y) in order to

maximize u(x,y) subject to px+ y= m. This is a typi-cal constrained maximization problem. In this case,because y= mpx, the same problem can be expres-sed as the unconstrained maximization of the functionf(x) = u(x,mpx) with respect to the single variable x.


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When the constraint is a complicated function, this

substitution method might be difficult or impossible tocarry out in practise. In such cases, other techniquesshould be used. In particular, economists make muchuse of the method of Lagrange multipliers. The

reason is that Lagrange multipliers have importanteconomic Interpretations.

The method is named after its discoverer, the French mathematicianJoseph Louis Lagrange (1736 1813). The Danish economist HaraldWestergaard seems to have been the first who used it in economics,in 1876.


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The Lagrangean Method


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Joseph-Louis Lagrange

Born: 25 Jan 1736 in Turin, Sardinia-Piedmont

Died: 10 April 1813 in Paris, France
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Lagrange.html

multi variable optimization

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