Constant penalty functions to simplify optimization of the Choquet integral under constraints

The Choquet integral have been used in many decision making problems as the optimization function that needs to be maximized or minimized. The complexity of these optimization problems is often increased by numerous constraints that are imposed to the solution. Several methods have been developed for optimization under constraints, one of them being penalty functions method. This method transforms a constrained optimization problem in an equivalent unconstrained optimization problem by adding penalties to the solutions that do not satisfy one of more constraints. Penalty functions method is an iterative process, which adjusts penalty functions in each iteration, often yielding a large number of iterations until a satisfying solution is found. To speed the process of the Choquet integral optimization under constraints, we propose to use constant penalty functions. This approach requires only one iteration, and is therefore much faster than the generic penalty function method. In this paper, we derive the constant penalty function that guarantees that the optimal solution found in the first iteration satisfies all constraints.