This paper aims to characterize the memory-rate tradeoff for decentralized caching under nonuniform file popularity and size. We consider a recently proposed decentralized modified coded caching scheme (D-MCCS) and formulate the cache placement optimization problem to minimize the average rate for the D-MCCS. To solve this challenging non-convex optimization problem, we first propose a successive Geometric Programming (GP) approximation algorithm, which guarantees convergence to a stationary point but has high computational complexity. Next, we develop a low-complexity file-group-based approach, where we propose a popularity-first and size-aware (PF-SA) cache placement strategy to partition files into two groups, taking into account the nonuniformity in file popularity and size. Both algorithms do not require the knowledge of active users beforehand for cache placement. Numerical results show that they perform very closely to each other. We further develop a lower bound for decentralized caching under nonuniform file popularity and size as a non-convex optimization problem and solved it using a similar successive GP approximation algorithm. We show that the D-MCCS with the optimized cache placement attains this lower bound when no more than two active users request files at a time. The same is true for files with uniform size but nonuniform popularity and the optimal cache placement being symmetric among files. In these cases, the optimized DMCCS characterizes the exact memory-rate tradeoff for decentralized caching. For general cases, our numerical results show that the average rate achieved by the optimized D-MCCS is very close to the lower bound.