Advancements in statistics are normally geared to addressing topics that will either address an existing gap in the field or to render analysis results more accurate/reliable. This work aims to add to existing research by extending from binary Generalized Linear Model (GLM) and Generalized Linear Mixed Model (GLMM) to a multinomial logit, multivariate GLM (MGLM) and multivariate GLMM (MGLMM), subject to ordered equality and inequality constraints. We extended the maximum likelihood estimate (MLE) and likelihood ratio hypothesis testing (LRT) methods for the binary and multinomial GLM and GLMM subject to linear equality and inequality constraints on the parameters of interest. These methods will build on existing literature to allow for more options in hypothesis testing and the construction of confidence intervals. The innovative procedures take advantage of the gradient projection (GP) technique for the MLE, and chi-bar-square statistics for constrained LRTs. The model presented in this thesis yields accurate results since parameter orderings or constraints often occur naturally; and when this occurs, we optimize the efficiency of a statistical method by incorporating the parameter constraints into the MLE and hypothesis testing. More specifically, we use ordered constrained inference for multinomial data whereby including equality and inequality constraints adds value to our predictions. Using real-world data from the Canadian Community Health Survey (CCHS), the methodology of using constraints showed significant improvement on methodology that does not, which substantiates the added value of the work presented here. This work contributes to the field by dealing with inequality constraints in MGLMM, specifically multinomial data, which is the most challenging problem in constrained inference. This helps improve results for researchers in both scientific and non-scientific fields. Keywords: constrained/restricted statistical inference, optimization algorithms, gradient projection theory, quadratic programming, multinomial logit, projective geometry, convex cone.