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Thermal hot spots and temperature gradients on the die need to be minimized to manufacture reliable systems while meeting energy and performance constraints. In this work, we solve the task scheduling problem for multiprocessor system-on-chips (MPSoCs) using Integer Linear Programming (ILP). The goal of our optimization is minimizing the hot spots and balancing the temperature distribution on the die for a known set of tasks. Under the given assumptions about task characteristics, the solution is optimal. We compare our technique against optimal scheduling methods for energy minimization, energy balancing, and hot spot minimization, and show that our technique achieves significantly better thermal profiles. We also extend our technique to handle workload variations at runtime.