We determine the capacity-optimal transmission strategy for a multiple-input-multiple-output (MIMO) Gaussian channel under multiple power constraints, namely joint sum power constraint (SPC), per group power constraints (PGPC), and per antenna power constraints (PAPC). First, we focus on cases where we can analytically determine the optimal transmit strategy under joint SPC-PGPC-PAPC. We obtain results for the following cases: 1) nt × 1 multiple-input-single-output (MISO); 2) MIMO channel with full column rank and full rank optimal covariance matrix; and 3) 2 × nr MIMO channel. These results generalize some recent results for the special cases of PAPC only and joint SPC-PAPC. Then, we propose a projected factored gradient descent (PFGD) algorithm for the general MIMO Gaussian channel under joint SPC-PGPC-PAPC including the possibility of additional rank constraints. This algorithm matches the solution of standard convex optimization tools with lower complexity. The algorithm also overcomes the limitations of existing algorithms, in terms of accuracy and applicability to low rank channels. © 2019 IEEE.