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SECTION 02 / 8

Supremum and infimum: the language of optimization

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Figure 31. Maximum versus supremum.
Figure 31. Maximum versus supremum.
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Illustration: Supremum and infimum: the language of optimization

A maximum must be achieved by an element of the set. A supremum is the least upper bound and need not be achieved. Likewise, a minimum is attained; an infimum need not be.

Worked example

Attained versus unattained

  1. For A=[0,1]A=[0,1], max⁡A=1\max A=1 and sup⁡A=1\sup A=1 because 1 belongs to A.

  2. For B=(0,1)B=(0,1), sup⁡B=1\sup B=1, but max B does not exist because 1 is excluded.

Figure 32. Minimum versus infimum.
Figure 32. Minimum versus infimum.
Open image ↗
Illustration: Supremum and infimum: the language of optimization

  1. For C=(2,5)C=(2,5), inf⁡C=2\inf C=2, but min C does not exist because 2 is excluded.

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