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# Alternating Optimization

Alternating optimization is a technique for solving optimization functions with many parameters, where no explicit solution exists to solve for all simultaneously. For many such functions there are methods to solve for one or a subset of parameters if the others are considered fixed[1], so it is possible to alternate among these subproblems to iteratively determine a solution to the larger optimization.

## Algorithm

1. Assign values to all parameters. These may be chosen randomly or set carefully.
2. Solve for one or a set of parameters via a known method, holding others constant.
3. Replace the parameter values with those found in (2)
4. Focus attention on the next parameter or set of parameters and repeat from (2)
5. Break when the process has converged.

## Convergence

It is possible an alternating optimization procedure gets stuck in a local minimum[1], but "Under reasonable assumptions, the general AO approach is shown to be locally, q-linearly convergent, and to also exhibit a type of global convergence."[2]

## References

1. Jerome H. Friedman (1985). "Classification and Multiple Regression through Projection Pursuit" (PDF). Journal of the American Statistical Association. Retrieved 2018-01-31.
2. James C. Bezdek (2003). "Convergence of alternating optimization". Neural, Parallel & Scientific Computations. Retrieved 2018-01-31.

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