First-Order Optimization as Minimum-Time Control
We formulate first-order optimization as a minimum-time control problem. The iterate is the state, the update, a combination of the gradients observed so far, is the control, and the points with gradient norm at most a prescribed tolerance form the target set. For a fixed objective and start, the minimum number of oracle queries needed to reach the target is a value function: it measures the complexity of the instance, not the worst case over a class. On a strongly convex quadratic, the conjugate gradient iterates emerge from the discrete Pontryagin conditions, and the value is a controllability index. Beyond the quadratic, a Hessian-generated reachable span replaces the controllability matrix, and feasibility becomes a reachability question: whether a critical point can be reached at all, and in how many steps, is read from the span. Curvature is a resource: a checkable condition certifies that an instance needs fewer steps than the controllability index of its quadratic model at the minimizer, and the gap can grow without bound with the dimension. The value is thus a benchmark for the intrinsic difficulty of each instance, against which any first-order method can be measured.