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Why do nonlinear optimization algorithms use a sequence of guesses called iterates?

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optimization problem — optimization Constraint set bounded by the five lines x₁ = 0, x₂ = 0, x₁ = 8, x₂ = 5, and x₁ + x₂ = 10. These enclose an infinite number of points that represent feasible solutions.

Nonlinear optimization algorithms utilize a sequence of guesses, known as iterates, because these problems involve objective functions or constraints that cannot be expressed as linear functions. Unlike linear programming, where solutions can be reached more directly, nonlinear problems often require an iterative approach to navigate complex, curved solution spaces.12

The primary purpose of using iterates is to systematically approach an optimal value of the variable vector $x$ . Each subsequent iterate is chosen based on information gathered about the objective and constraint functions at previous points in the sequence. Because it is frequently impractical to identify a globally optimal solution in nonlinear contexts, these algorithms often aim to converge on a local optimum—the best value within a specific region of feasible solutions.1

By refining these guesses through successive steps, the algorithms can effectively manage the complexities of nonlinear systems, such as those found in finance, engineering, and chemistry, where variables interact in non-proportional ways.12