A nonequilibrium simulated annealing (NESA) algorithm is presented for solving combinatorial minimization problems. The original Metropolis algorithm and its variant due to Glauber were modified by enforcing the cooling schedule as soon as an improved solution is obtained, without the need to reach near-equilibrium conditions at each temperature level. The original and modified algorithms were applied to the classical traveling salesman problem and to the optimization of a pressure relief header network. Statistical evaluation of the performance of the algorithms revealed that the proposed modification resulted in a faster approach to the global optimum. A simple stopping criterion, based on an averaged gradient of the objective function with respect to the number of function evaluations, provides a convenient method to control the convergence of the modified algorithms.