# 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[edit | edit source]

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

## Convergence[edit | edit source]

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[edit | edit source]

- ↑
^{1.0}^{1.1}Jerome H. Friedman (1985). "Classification and Multiple Regression through Projection Pursuit" (PDF). Journal of the American Statistical Association. Retrieved 2018-01-31. - ↑ James C. Bezdek (2003). "Convergence of alternating optimization". Neural, Parallel & Scientific Computations. Retrieved 2018-01-31.

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