I know that Dynare (which sits on top of Matlab) can solve many kinds of dynamic stochastic general equilibrium (DSGE) and overlapping generations (OLG) models. I also know that Dynare can handle some sorts of adjustment costs. For example, I have seen convex adjustment cost examples in Dynare. In particular, The Macroeconomic Model Data Base provides on the order of 50 models compatible with Dynare and the user manual indicates several models (e.g. NK_IR04 and US_NFED0) with quadratic (a type of convex) adjustment costs.
Can Dynare solve models with non-convex adjustment costs like An equilibrium model of lumpy housing investment (Iacoviello and Pavan (2008)) or Housing and debt over the life cycle and over the business cycle (Iacoviello and Pavan (2013))? Non-convex has a specific mathematical meaning, but in the context of these papers it indicates that adjustment costs that are not proportional to the amount of adjustment. Instead, the adjustment costs have a fixed-cost proportional to the current asset value. However, there are other forms of non-convex adjustment cost. If Dynare can solve any model with any sort of non-convex adjustment costs that is of interest.
If models with these adjustment costs can be solved with Dynare, please provide an example or a link to an example (if possible). If Dynare currently cannot solve these models is there any published code that can do so? Even sample code for a specific model solution rather than a general product like Dynare would be helpful.
More details on non-convex adjustment costs:
I draw my language here from a A Model of Housing in the Presence of Adjustment Costs: A Structural Interpretation of Habit Persistence (Flavin and Nakagawa (2008))
At the instant the house is sold, the household pays a transactions cost proportional to the value of the house sold, so that wealth also changes discontinuously....The housing model developed in Section I invokes a fourth set of assumptions: utility depends nonseparably on nondurable consumption and on housing, nondurable consumption is costlessly adjustable, but housing is subject to a nonconvex adjustment cost ($\lambda > 0 $).
Perhaps this language is non-standard but that's a quote from a paper in the AER, and when I've discussed it with others people seem to know what I'm talking about. The two mentioned papers don't use that language but do have the same rough form, that transaction costs are not increasing in the degree of the adjustment but rather that any use of adjustment (other than a small bit, perhaps for depreciation or unit improvement perhaps) triggers a cost related to the state variables instead of the control variables. The paper On the Nature of Capital Adjustment Costs (Cooper and Haltiwanger (2005)) seems to use nonconvex adjustment costs in the same way in a firm capital setting.
Building upon the analysis of Abel and Eberly , Cooper, Haltiwanger and Power  and Caballero and Engel , during periods of investment plants incur a fixed adjustment cost. Generally, these non-convex costs of adjustment are intended to capture indivisibilities in capital, increasing returns to the installation of new capital and increasing returns to retraining and restructuring of production activity. These fixed adjustment costs represent the need for plant restructuring, worker retraining and organizational restructuring during periods of intensive investment