Bellman

Bellman is a surname associated with figures in mathematics and computer science. The most influential bearer is Richard Bellman, a pioneer of dynamic programming, after whom several core concepts in optimization and decision making are named.

Richard Bellman was an American applied mathematician whose work established dynamic programming as a formal framework for solving complex multi-step problems. He introduced the Bellman equation and the principle of optimality, and he made broad contributions across control theory, optimization, and numerical methods. His work helped form the theoretical basis for sequential decision making and has influenced fields ranging from operations research to artificial intelligence.

The Bellman–Ford algorithm is a key graph algorithm for computing shortest paths from a single source in graphs that may include negative edge weights. The algorithm relaxes all edges repeatedly, up to the number of vertices minus one times, updating path estimates. It can detect negative cycles, and its running time is O(VE). It is widely used in routing protocols and network analysis, where negative weights may arise in certain models.

In dynamic programming and reinforcement learning, the Bellman equation provides a recursive decomposition of value functions. It expresses the value of a state as the best expected return from that state, considering immediate rewards and the discounted value of successor states. The Bellman principle of optimality underpins many algorithms for solving sequential decision problems.