An auxiliary function is constructed based on a penalty function.
Firstly, the method based on the penalty function aimed to transform a COP into an unconstrained one by adding a penalty term to the original objective function [11, 12].
Based on the penalty function used in optimization theory, a multi-objective cost function with constraint is adopted to provide the best deadzone approximation.
With this background, Fodor et al. [9] gives a near optimum power control method under fairness constraints in CoMP transmission systems based on Lagrangian penalty function, which aims to improve the transmitting power allocation efficiency in multicell spatial multiplexing wireless systems.
Then, for given time-domain performance envelopes, a numerical optimization method based on exact penalty functions and linear programming is presented that identifies the multivariable controller that decouples the closed loop and optimizes its performance.
Along the line of [7, 10], for a class of semivectorial bilevel programming problem, where the upper level is a general scalar-valued optimization problem and the lower level is a linear multi-objective optimization problem, Zheng and Wan [6] presented a new penalty function approach based on the objective penalty function method.
The minimum value of the objective function is calculated based on the mixed penalty function method and the optimal support shape that satisfies the given constraints can be obtained.
Based on the objective penalty function approach for the single level programs, Zheng and Wan [16] propose the corresponding penalty function algorithm, which contains two penalty parameters.
We develop an algorithm based on the smoothed penalty function.
Based on an exact penalty function, Zheng et al. [18] proposed an algorithm for the pessimistic linear bilevel programming problem.
