Abstract: We investigate a generalized form of submodular maximization, referred to as $k$-submodular maximization, with applications across the domains of social networks and machine learning. In this work, we propose the multilinear extension of $k$-submodular functions and unified Frank-Wolfe-type frameworks based on that. This continuous framework accommodates 1) monotone or non-monotone functions, and 2) various constraint types including matroid constraints, knapsack constraints, and their combinations. Notably, we attain an asymptotically optimal $1/2$-approximation for monotone $k$-submodular maximization problems with knapsack constraints, surpassing previous $1/3$-approximation results, and a factor-$1/3$ approximation for non-monotone $k$-submodular maximization problems with knapsack constraints and matroid constraints which outperforms previous $0.245$-approximation results. The foundation for our analysis stems from new insights into specific linear and monotone properties pertaining to the multilinear extension.