In exponential families the log-likelihood forms a concave function in the canonical parameters. Therefore, any model given by convex constraints in these canonical parameters admits a unique maximum likelihood estimator (MLE). Such models are called convex exponential families. For models that are convex in the mean parameters (e.g. Gaussian covariance graph models) the maximum likelihood estimation is much more complicated, and the likelihood function typically has many local optima. One solution is to replace the MLE with so called dual likelihood estimator, which is uniquely defined and asymptotically has the same distribution as the MLE. In this talk I will consider a much more general setting, where the model is given by convex constraints on some canonical parameters and convex constraints on the remaining mean parameters. We call such models mixed convex exponential families. We propose for these models a 2-step optimization procedure which relies on solving two convex problems. We show that this new estimator has asymptotically the same distribution as the MLE. Our work was motivated by locally associated Gaussian graphical models that form a suitable relaxation of Gaussian totally positive distributions.