Introduction
Determination of the minimum of a real valued function of one variable, and the location of that minimum, plays an important role in nonlinear optimization. A one-dimensional minimization routine may be called several times in a multivariable problem. We show later that at the minimum point of a sufficiently smooth function, the slope is zero. If the slope and curvature information is available, minimum may be obtained by finding the location where the slope is zero and the curvature is positive. The need to determine the zero of a function occurs frequently in nonlinear optimization. Reliable and efficient ways of finding the minimum or a zero of a function are necessary for developing robust techniques for solving multivariable problems. We present the basic concepts involved in single variable minimization and zero finding.
Theory Related to Single Variable (Univariate) Minimization
We present the minimization ideas by considering a simple example. The first step is to determine the objective function that is to be optimized.
Example 2.1
Determine the objective function for building a minimum cost cylindrical refrigeration tank of volume 50 m3, if the circular ends cost $10 per m2, the cylindrical wall costs $6 per mm2, and it costs $80 per m2 to refrigerate over the useful life.
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