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Calculus Optimization Methods

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A key application of calculus is in optimization: finding maximum and minimum values of a function, and which points realize these extrema.

Context

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Formally, the field of mathematical optimization is calledmathematical programming, and calculus methods of optimization are basic forms ofnonlinear programming. We will primarily discuss finite-dimensional optimization, illustrating with functions in 1 or 2 variables, and algebraically discussingn variables.We will also indicate some extensions toinfinite-dimensional optimization, such ascalculus of variations, which is a primary application of these methods in physics.

Techniques

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Basic techniques include the first and second derivative test, and their higher-dimensional generalizations.

A more advanced technique is Lagrange multipliers, and generalizations as Karush–Kuhn–Tucker conditions and Lagrange multipliers on Banach spaces.

Applications

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Optimization, particularly via Lagrange multipliers, is particularly used in the following fields:

Further, several areas of mathematics can be understood as generalizations of these methods, notablyMorse theory andcalculus of variations.

Terminology

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  • Input points, output values
  • Maxima, minima, extrema, optima
  • Stationary point, critical point; stationary value, critical value
  • Objective function
  • Constraints – equality and inequality
    • Especially sublevel sets
    • Feasible region, whose points are candidate solutions

Statement

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This tutorial presents an introduction to optimization problems that involve finding a maximum or a minimum value of anobjective functionf(x1,x2,…,xn){\displaystyle f(x_{1},x_{2},\ldots ,x_{n})} subject to a constraint of the formg(x1,x2,…,xn)=k{\displaystyle g(x_{1},x_{2},\ldots ,x_{n})=k}.

Maximum and minimum

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Finding optimum values of the functionf(x1,x2,…,xn){\displaystyle f(x_{1},x_{2},\ldots ,x_{n})} without a constraint is a well known problem dealt with in calculus courses. One would normally use the gradient to findstationary points. Then check all stationary andboundary points to find optimum values.

Example

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f(x,y){\displaystyle f(x,y)} has one stationary point at (0,0).

The Hessian

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A common method of determining whether or not a function has anextreme value at a stationary point is to evaluate the hessian of the function at that point. where thehessian is defined as

H(f)=[∂2f∂x12∂2f∂x1∂x2…∂2f∂x1∂xn∂2f∂x2∂x1∂2f∂x22…∂2f∂x2∂xn⋮⋮⋱⋮∂2f∂xn∂x1∂2f∂xn∂x2…∂2f∂xn2].{\displaystyle H(f)={\begin{bmatrix}{\frac {{\partial }^{2}f}{\partial x_{1}^{2}}}&{\frac {{\partial }^{2}f}{\partial x_{1}\partial x_{2}}}&\dots &{\frac {{\partial }^{2}f}{\partial x_{1}\partial x_{n}}}\\{\frac {{\partial }^{2}f}{\partial x_{2}\partial x_{1}}}&{\frac {{\partial }^{2}f}{\partial x_{2}^{2}}}&\dots &{\frac {{\partial }^{2}f}{\partial x_{2}\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {{\partial }^{2}f}{\partial x_{n}\partial x_{1}}}&{\frac {{\partial }^{2}f}{\partial x_{n}\partial x_{2}}}&\dots &{\frac {{\partial }^{2}f}{\partial x_{n}^{2}}}\\\end{bmatrix}}.}

Second derivative test

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TheSecond derivative test determines the optimality of stationary pointx{\displaystyle x} according to the following rules [2]:

In the above example.

H(f)=[4002].{\displaystyle H(f)={\begin{bmatrix}4&0\\0&2\end{bmatrix}}.}

Thereforef(x,y){\displaystyle f(x,y)} has a minimum at (0,0).

Sections

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References

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[1] T.K. Moon and W.C. Stirling.Mathematical Methods and Algorithms for Signal Processing. Prentice Hall. 2000.
[2]http://www.ece.tamu.edu/~chmbrlnd/Courses/ECEN601/ECEN601-Chap3.pdf
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