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Glacius
OptimizationConcept reference

Minima and minimizers

A minimum is an objective value; an argmin is the set of decisions attaining it.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. The minimum is the best objective value
  3. A translated square gives another way to find a minimum
  4. An argmin is a decision set, so do not replace it with the smallest score
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01The minimum is the best objective value#

The minimum is the best objective value. The argmin is the set of decisions that attain that value.

Several decisions can tie. Include all of them when the question asks for the argmin set.

In this complete table, two different decisions reach cost 3.

The entire feasible domain is the three inputs 1,4,6, with costs 8,3,3. Only these three points are plotted; no line invents values between them. Minimum cost is 3, attained at inputs 4 and 6.The entire feasible domain is the three inputs 1,4,6, with costs 8,3,3. Only these three points are plotted; no line invents values between them. Minimum cost is 3, attained at inputs 4 and 6.
Figure 1The entire feasible domain is the three inputs 1,4,6, with costs 8,3,3. Only these three points are plotted; no line invents values between them. Minimum cost is 3, attained at inputs 4 and 6.
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Check your reasoning

Entire feasible table: x=(−1,2,8)x=(-1, 2, 8), f(x)=(5,9,5)f(x)=(5, 9, 5). Give (argmin set; minimum value).

  1. A{5}\{5\}; −1-1
  2. B{−1,8}\{-1, 8\}; 55
  3. C{−1,2}\{-1, 2\}; 55
Show answer and explanation
{−1,8}\{-1, 8\}; 55

Tie at −1 and 8.

02A translated square gives another way to find a minimum#

A translated square gives another way to find a minimum. With a positive coefficient, the square is smallest when its input is zero.

Then evaluate the remaining constant to find the minimum value.

Check your reasoning

Over all real x, f(x)=5(x−2)2+9f(x)=5(x-2)^2+9. Give (argmin set; minimum value).

  1. A{2}\{2\}; 99
  2. B{9}\{9\}; 22
  3. C{2}\{2\}; 00
Show answer and explanation
{2}\{2\}; 99

The square is zero at x=2; the remaining cost is 9.

03An argmin is a decision set, so do not replace it with the smallest score#

An argmin is a decision set, so do not replace it with the smallest score. A tied row is still a minimizer.

Some problems never attain their best limiting value. These examples use complete finite tables or squares that do attain their minimum.

Check your reasoning

Draft omits a tie. Full feasible table: x=(−3,2,6)x=(-3, 2, 6), f(x)=(4,4,7)f(x)=(4, 4, 7). Give (argmin set; minimum).

  1. A{4}\{4\}; −3-3
  2. B{−3,6}\{-3, 6\}; 44
  3. C{−3,2}\{-3, 2\}; 44
Show answer and explanation
{−3,2}\{-3, 2\}; 44

Tie at −3 and 2.

Key takeaway

Find the smallest score, then collect every decision that reaches it.

  • Distinguish the best objective value from the decision attaining it.

Sources & further reading

  1. [1]
    Boyd & Vandenberghe §4.1.1 ↗Boyd & Vandenberghe §4.1.1 · Article

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Minima and minimizers.” Math behind ML. /learn/o-argmin