This paper deals with a class of optimization problems by designing and analyzing a finite-time particle swarm optimization (FPSO) algorithm. Two versions of the FPSO algorithm, which consist of a continuous-time FPSO algorithm and a discrete-time FPSO algorithm, are proposed. Firstly, the continuous-time FPSO algorithm is derived from the continuous model of the particle swarm optimization (PSO) algorithm by introducing a nonlinear damping item that can enable the continuous-time FPSO algorithm to converge within a finite-time interval and a parameter that can enhance the exploration capability of the continuous-time FPSO algorithm. Secondly, the corresponding discrete-time version of the FPSO algorithm is proposed by employing the same discretization scheme as the generalized particle swarm optimization (GPSO) such that the exploiting capability of the discrete-time FPSO algorithm is improved. Thirdly, a Lyapunov approach is used to analyze the finite-time convergence of the continuous-time FPSO algorithm and the stability region of the discrete-time FPSO algorithm is also given. Finally, the performance capabilities of the proposed discrete-time FPSO algorithm are illustrated by using three wellknown benchmark functions (global minimum surrounded by multiple minima): Griewank, Rastrigin, and Ackley. In terms of numerical simulation results, the proposed continuous-time FPSO algorithm is used to deal with the problem of odor source localization by coordinating a group of robots.