Density-based multi-material topology optimization typically restricts design variables to a standard simplex to ensure physical consistency. However, applications involving anisotropic media—such as fiber-reinforced composites or materials with direction-dependent thermal and electrical properties—often require design spaces defined by general convex polytopes. Incorporating these constraints into standard gradient-based algorithms is challenging, as the necessary projection operations onto high-dimensional polytopes are computationally expensive and can hinder efficiency. To address this, we propose a novel first-order optimization algorithm within the framework of mirror descent. We introduce a constructed Legendre function whose effective domain coincides with a general convex polytope defined by its $V$-representation. This formulation establishes a generalized mirror map that links unconstrained latent variables to the bounded design space, effectively eliminating the need for explicit local projections during gradient updates. Global linear constraints, such as multiple mass limits, are rigorously handled via a Bregman Dykstra projection. The resulting algorithm is easy to implement, mesh-independent, and exhibits fast convergence using adaptive step sizes. We demonstrate the versatility of this approach through numerical examples, including standard compliance minimization and a torque maximization problem for electric motor design. In the latter, we leverage a physics-informed interpolation domain to optimize the distribution of ferromagnetic materials and permanent magnets, accounting for discrete magnetization directions.