finitehorizon

Finite horizon refers to optimization problems in which decisions are planned over a fixed number of time steps, after which the process ends. The horizon is defined by a nonnegative integer T, and the objective aggregates costs or rewards over the T stages, possibly including a terminal value at time T. This contrasts with infinite-horizon problems, where planning extends indefinitely.

In formal terms, a finite-horizon problem involves choosing a sequence of actions to optimize a cumulative objective such as the sum of stage costs plus a terminal cost, subject to system dynamics or stochastic transitions. The horizon imposes a terminal condition that influences optimal decisions as the endpoint approaches. The problem can be deterministic or stochastic and is commonly framed in discrete time, though continuous-time variants exist.

Dynamic programming provides a standard solution approach. Value functions are defined for each time t = 0, 1, ..., T, with V_T(s) equal to the terminal cost. Working backward from t = T-1 to t = 0, the Bellman recursion computes V_t(s) = min_a [ c_t(s,a) + E[ V_{t+1}(s') | s,a ] ] or the corresponding max, and the optimal action is the argmin/argmax. Because the optimal policy can depend on time, finite-horizon problems typically yield time-varying (non-stationary) policies.

Applications span robotics and autonomous planning with deadlines, finance and investment planning under a fixed horizon, inventory and supply-chain management, project scheduling, and energy/industrial systems where a terminal objective or deadline is essential. Finite-horizon analysis often informs strategies that perform well within the limited planning window and complements infinite-horizon methods when deadlines or endpoint goals are central.