<p>Gradient-based optimization methods are widely employed in large-scale unconstrained problems due to their convergence properties, scalability, and efficient sensitivity computation tools. However, as problem complexity increases, evaluations of the objective function and of its sensitivities are often carried out approximately to reduce computational cost. This is particularly evident in differential equation-constrained optimization (DECO) problems formulated in the discrete reduced space. In this setting, the state and adjoint equations are typically solved only approximately using iterative methods, leading to inexact discrete adjoint gradients. This creates a gap between computational practice and convergence theory, which largely relies on exact gradient information. In this context, we propose the Inexact General Direction Method (IGDM), a first-order method with inexact gradient-related directions, together with an adjoint-based adaptive algorithm for computing inexact adjoint gradients with controllable error. For IGDM, stationarity of limit points is established under three hypotheses on the gradient error-bound sequence: square-summable with bounded step sizes, summable with diminishing step sizes, and proportional to the inexact gradient norm with bounded step sizes. In the third case, an admissible step-size interval guaranteeing monotonic descent is identified. The adjoint-based algorithm, in turn, provably enforces the proportionality condition of the third hypothesis. The coupling of IGDM with this algorithm provides an optimization framework for solving DECO problems formulated in reduced space. This framework is applied to two parameter estimation problems governed by differential equations, yielding substantial runtime reductions relative to classical gradient descent with fixed tolerances.</p>