The One Max Problem (OMP) is a fundamental optimization challenge that is frequently utilized as a benchmark within the field of evolutionary algorithms and other heuristic search strategies. 
Evolutionary algorithms, inspired by natural selection, such as genetic algorithms, and heuristic methods, which seek practical solutions at the expense of completeness, often leverage OMP to test their efficacy in navigating complex solution spaces.

If you are familiar with genetic algorithms, you may wish to skip directly to @omp-impl.

== The Core Challenge

Consider the task described below:

#quote(block: true)[
  Given a binary string $x = (x_1, x_2, ..., x_n)$ of length $n$, where each $x_i$ is either 0 or 1, identify a binary string that maximizes the sum of its bits (essentially, the count of ones).
]

For instance, in a binary string `1101`, the sum of its bits is 3, as there are three `1`s.
The goal is to maximize this sum.

== Defining Fitness

The fitness function, $phi(x)$, crucial for evaluating potential solutions, is defined as:


$
  phi(x) = sum_(i = 1)^n x_i
$ <omp_fitness>

This function tallies the `1`s in the binary string.
Achieving a fitness score equal to $n$ signifies an optimal solution, where every bit is `1`.
The choice of this fitness function is intuitive, as it directly quantifies the objective of the OMP, making it an ideal measure for optimization.

== The Significance of Unimodality

OMP is characterized as a unimodal problem, which means it contains a singular peak or optimal solution in its landscape - all ones in the binary string.
This feature simplifies the search process since any improvement in fitness unequivocally moves a solution closer to the global optimum.
However, it's this simplicity that also makes OMP an intriguing test case, contrasting with multimodal problems that contain numerous local optima, complicating the path to the global maximum.

== Navigating the Solution Space

Given a string length $n$, the solution space, comprising $2^n$ potential strings, expands exponentially. 
This vastness renders exhaustive search impractical for large $n$.
Efficient optimization algorithms, such as genetic algorithms, employ mechanisms like crossover and mutation - inspired by biological evolution - to explore this space creatively and efficiently, avoiding the computational cost of evaluating every possible solution.

== Broader Implications

While OMP serves primarily as a theoretical benchmark, the strategies and insights derived from solving it are applicable to more complex, real-world problems.
For instance, the principles of incremental improvement and exploration versus exploitation, critical in solving OMP, are equally relevant in optimizing network configurations, financial portfolios, and many other domains where optimal solutions are sought within immense search spaces.