In other words, is it feasible any of the various penalized regression techniques (such as ridge regression, lasso, and elasticnet) could completely miss the optimal solution for a regression model due to poorly chosen initial values for the model parameters?
My understanding is that initial values (usually taken as zero for non-intercepts) are not that influential for these penalization techniques and models such as the logistic. And 'optimal solution' is not very well defined. You need to tell us more. For these techniques what is usually the toughest decision is that of how much penalization to use.
If the loss and penalty are convex functions of your parameters as in a linear regression or GLM then no, you won't get stuck in the wrong place (although it could take awhile to get where you want to go). On the other hand, if you are training (say) a multilayer neural net with quadratic loss using backprop and an L2 penalty then you can very easily get stuck because the loss is not convex as a function of the parameters. In these cases, people typically are just hoping to get a good local solution (which might generalize better anyways), not the best solution.