Can there be several global maximizers?

Yes. Different feasible points can share the same highest objective value.

Does the best result from many random starts prove global optimality?

No. It may provide useful empirical evidence but does not generally rule out better unvisited points.

Does every bounded objective attain a maximum?

No. The open-interval example approaches its upper bound without reaching it.

AI Glossary / Graphs and symbolic AI / Intermediate

Global maximum

Definition

A global maximum is the greatest value that an objective function actually attains anywhere in its specified feasible set. A point attaining that value is a global maximizer. In AI, the objective might measure reward, fit, or another explicitly defined score. “Global” compares the result with every feasible alternative, not merely with nearby candidates or those already tested.

What it means

The feasible set determines the scope of the claim. A setting can be globally best among five permitted options but not among all continuous parameter values. Changing constraints or the objective changes the optimization problem and can change its maximum.
The maximum value and the maximizing point are different objects. Several points can attain the same greatest value. Conversely, some problems have no attained maximum: values may grow without bound or approach an upper limit that no feasible point reaches. A solver's best observed result is not automatically a global maximum.

How it works

To establish a global maximum, find a feasible candidate and show that no feasible alternative has a larger objective value. For a small finite set, exhaustive evaluation can provide that comparison. Other problems permit mathematical proofs, bounds, or specialized optimization methods.
Maximizing f is equivalent to minimizing -f over the same feasible set. If f is concave on a convex feasible set, any local maximizer is global under the standard optimization definition. In more general landscapes, local search can stop at a peak with a higher peak elsewhere. The applicable structure and proof conditions matter more than the algorithm's name.

Three practical examples

A tiny game model enumerates all allowed actions and identifies the action with the highest exactly computed value.
A parameter-tuning exercise distinguishes the best tested grid point from the maximum over a continuous range.
An optimization report uses a feasible score and a valid upper bound to quantify how far the current solution might be from global optimality.

Think of it this way

Think of the highest summit within a clearly marked region on a map. The highest point on your current hill may not be the region's highest point. The analogy is limited because an AI objective may have thousands of dimensions, discrete choices, or no attained highest value, and the score may only approximate the real outcome of interest.

Worked example

Let f(x) = 9 - (x - 2) squared, with feasible interval 0 ≤ x ≤ 5. Because a square is never negative, f(x) ≤ 9 for every feasible x. At x = 2, the square is zero, so f(2) = 9. This supplies both an upper bound and a feasible point attaining it: the global maximum is 9, and x = 2 is a global maximizer.
Now restrict the feasible interval to 3 ≤ x ≤ 5. The nearest feasible x to 2 is 3, giving f(3) = 8. All larger x increase the squared distance and lower f, so the new global maximum is 8 at x = 3.
The objective formula stayed the same, but the constraint changed the answer. Also note that the second maximizer lies at a boundary where the unconstrained derivative is not zero. Checking only stationary interior points would miss it.

Illustrative case study

In a fictional calibration exercise, the Oak team samples a score at settings 0, 1, 3, and 5. It labels the best sampled setting “the global optimum.” An editor notices that setting 2 was never evaluated and that the team has supplied no bound excluding a better score there.
The team revises the label to “best among tested settings” and, for its toy analytic function, derives the bound shown in the worked example. This invented case shows how evidence changes the strength of an optimization claim. It does not describe measured performance in a real calibration system.

Common misconceptions and mistakes

A local maximum need not be global. A zero derivative is not sufficient to establish a maximum and may also occur at a minimum or saddle point. Boundary solutions require attention, and discrete problems may not have derivatives at all.
Do not confuse a supremum with an attained maximum. For f(x) = x on the open interval 0 < x < 1, the supremum is 1, but no feasible x achieves it. Finally, a global maximum of an inaccurate model is still only optimal for that model; it does not prove the best real-world decision.

Business use

The term helps teams interpret optimization reports honestly. A result should identify the objective, feasible region, and basis for any global claim. If the solver stops early, a best feasible score and a valid bound can be more informative than a vague declaration of optimality.
For example, a feasible maximization score of 92 and a valid upper bound of 100 imply an absolute optimality gap no larger than 8 in those score units. This statement is useful only if the bound and candidate apply to the same problem and the reported numerical tolerances are understood.

Check your understanding

Why is 9 a proved global maximum for the original quadratic example rather than just the best of several sampled values?

Answer: The nonnegative-square argument bounds every feasible value by 9, and x = 2 attains that bound. Sampling alone would not supply the universal comparison.

Related terms

Objective function
Supremum (or least upper bound)
Global minimum
Global maximizer
local maximum
feasible set
concave function
upper bound
optimality gap
supremum

Learn next

Learn next about upper bounds and optimality certificates. For a maximization problem with best feasible score 17 and valid upper bound 20, explain what is known and unknown. Then investigate what changes if the bound is reduced to 17 without changing the candidate.

Sources

Boyd and Vandenberghe, original Convex Optimization slides, local and global optimization: https://web.stanford.edu/~boyd/cvxbook/bv_cvxslides.pdf Russell and Norvig, original local-search slides: https://aima.cs.berkeley.edu/4th-ed/slides-pdf/chapter04.pdf

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