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We consider a nonlinear delay-differential system with unknown state-delays. Our goal is to identify these state-delays using experimental data. To this end, we formulate a dynamic optimization problem in which the state-delays are decision variables and the cost function measures the discrepancy between predicted and observed system output. We then show that the gradient of this problem's cost function can be computed by solving an auxiliary delay-differential system. By exploiting this result, the state-delay identification problem can be solved efficiently using a gradient-based optimization method.