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Many engineering problems have multiple objectives, and the overall aim is tooptimize a non-linear function of these objectives. In this paper, we formulatethe problem of maximizing a non-linear concave function of multiple long-termobjectives. A policy-gradient based model-free algorithm is proposed for theproblem. To compute an estimate of the gradient, a biased estimator isproposed. The proposed algorithm is shown to achieve convergence to within an$\epsilon$ of the global optima after sampling$\mathcal{O}(\frac{M^4\sigma^2}{(1-\gamma)^8\epsilon^4})$ trajectories where$\gamma$ is the discount factor and $M$ is the number of the agents, thusachieving the same dependence on $\epsilon$ as the policy gradient algorithmfor the standard reinforcement learning.

Joint Optimization of Multi-Objective Reinforcement Learning with Policy Gradient Based Algorithm