A local optimum is a position with no better option among the nearby alternatives currently being compared, although it may not be the best position across the whole landscape.
Imagine climbing in thick fog. At every step you choose the steepest upward direction and eventually reach a summit, yet a higher mountain may stand beyond it. A method that permits only immediate improvement cannot leave the present peak, because descending first looks worse. Different starting points may therefore lead to different summits.
A local optimum does not mean the choice was simply wrong. Many real problems cannot be searched exhaustively, and a sufficiently good local answer can be useful. It also differs from path dependence: path dependence concerns how the past changes present switching costs, while a local optimum arises because the comparison remains too local and further improvement may require a temporary decline. The concept asks us to inspect the search range, starting point and room for exploration, not merely whether the next step looks better.
https://inst.eecs.berkeley.edu/~cs188/textbook/search/local.html
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