Avoiding Traps in Nonconvex Problems
This work tackles optimization convergence issues for researchers in mathematical optimization, though it appears incremental with heuristic interpretations rather than fundamental breakthroughs.
The paper addresses the problem of iterative projection methods getting trapped at non-solutions in nonconvex optimization, showing through examples that proper tuning of hyperparameters and metric parameters is crucial to avoid this behavior.
Iterative projection methods may become trapped at non-solutions when the constraint sets are nonconvex. Two kinds of parameters are available to help avoid this behavior and this study gives examples of both. The first kind of parameter, called a hyperparameter, includes any kind of parameter that appears in the definition of the iteration rule itself. The second kind comprises metric parameters in the definition of the constraint sets, a feature that arises when the problem to be solved has two or more kinds of variables. Through examples we show the importance of properly tuning both kinds of parameters and offer heuristic interpretations of the observed behavior.