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Glacius
OptimizationConcept reference

Local and global optima

A local minimum compares a neighborhood; a global minimum compares the entire feasible domain.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. A local minimum wins within some neighborhood of a decision
  3. For separated quadratic branches, find the minimum within each interval
  4. A finite collection of trials is not an exhaustive continuous domain
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01A local minimum wins within some neighborhood of a decision#

A local minimum wins within some neighborhood of a decision. A global minimum wins over the entire feasible domain.

Every global minimum is local. A local minimum can still have a better competitor farther away.

On [−3,−1][-3,-1], cost (x+2)2+7(x+2)^2+7 bottoms at 7. On [1,3][1,3], cost (x−2)2+3(x-2)^2+3 bottoms at 3.

The feasible domain is [-3,−1] union [1,3]. On the left f=(x+2)²+7, with minimum 7 at −2. On the right f=(x−2)²+3, with minimum 3 at 2. The gap is not feasible and has no drawn function. The left minimum is local; the right is global.The feasible domain is [-3,−1] union [1,3]. On the left f=(x+2)²+7, with minimum 7 at −2. On the right f=(x−2)²+3, with minimum 3 at 2. The gap is not feasible and has no drawn function. The left minimum is local; the right is global.
Figure 1The feasible domain is [-3,−1] union [1,3]. On the left f=(x+2)²+7, with minimum 7 at −2. On the right f=(x−2)²+3, with minimum 3 at 2. The gap is not feasible and has no drawn function. The left minimum is local; the right is global.
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Check your reasoning

At feasible a, cost is 8. All feasible points cost at least 8; another ties. Classify a.

  1. ALocal, not global.
  2. BLocal and global.
  3. CNeither.
Show answer and explanation
Local and global.

Ties allow global minima.

02For separated quadratic branches, find the minimum within each interval#

For separated quadratic branches, find the minimum within each interval. Comparing the branch minima determines the best cost over their union.

Different branches may tie; global optimality does not require a unique decision.

Check your reasoning

Feasible x: [−3,−1]∪[1,3][-3,-1]\cup[1,3]. Left cost: (x+2)2+5(x+2)^2+5; right cost: (x−2)2+2(x-2)^2+2. Classify x=−2x=-2.

  1. ALocal, not global.
  2. BLocal and global.
  3. CNot local.
Show answer and explanation
Local, not global.

5 locally; 2 globally.

03A finite collection of trials is not an exhaustive continuous domain#

A finite collection of trials is not an exhaustive continuous domain. Winning those trials does not certify all untested points.

A whole-domain lower bound can certify global optimality. One lower-cost feasible counterexample can disprove it.

Check your reasoning

For f(x)=x2f(x)=x^2 on all real x, a report proves x2≥0x^2\ge0 and calls x=0 global. Verdict?

  1. AOnly local proved.
  2. BNo optimum proved.
  3. CGlobal claim proved.
Show answer and explanation
Global claim proved.

Every cost ≥ f(0)=0.

Key takeaway

Identify the comparison domain and use actual bounds or counterexamples before claiming global optimality.

  • Distinguish neighborhood optimality from optimality over the feasible set.

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. “Local and global optima.” Math behind ML. /learn/o-local-global