A new penalty function
P(y) has been designed, attempting to be less restrictive than the one used in , which was inspired by the road boundary constraint function in .
Keywords: Pattern search methods, aircraft attacking problem, penalty function
and unconstrained optimization.
is used to remove constraints and establish the fitness function.
Valve point effect and a penalty function
for generator active power violations are added to quadratic cost function in order to provide the more appropriate simulation of fuel cost.
lambda]]* is a penalty function
which depends on a tuning parameter [lambda] > 0.
In order to overcome the nonempty assumption of the interior of feasible region, Bian and Xue  proposed a recurrent neural network for nonsmooth convex optimization based on penalty function
is approximated objective function; X is penalty function
of the designed variable; R, B, L and G are penalty functions
of state variables, in front, back, left and right planes directions, respectively; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are state variables, acoustic intensity in the front, back, left and right plane directions, respectively (Fig.
5d can be incorporated as a penalty function
 in Eqs.
Furthermore, the penalty function
p is not arbitrary, but measures the degree of violation against the dropped requirements.
He addresses 1-D algorithms, the conjugate gradient method, the Broyden-Fletcher-Goldfarb-Shanno algorithm, the Powell method, the penalty function
, the augmented Lagrange multiplier method, sequential quadratic programming, the method of feasible directions, genetic algorithms, particle swarm optimization, simulated annealing, ant colony optimization, and tabu search methods, as well as multiobjective optimization problems, the simplex method and affine-scaling interior point method for solving linear programming problems, dynamic programming, and Gomory's cutting plane method, branch-and-bound method, and Balas' algorithm for integer programming.
Then a heuristic yet effective method, which combines the penalty function
method and quasi-Newton method , is proposed to obtain a local solution iteratively.
In order to compare the performance of the method given in the paper and the penalty function
algorithm, we use the same computer (CPU: AMD Athlon II M340 2.