An Adaptive Scheme to Generate the Pareto Front Based on the Epsilon-Constraint Method

This paper presents a scheme for generating the Pareto front of multiobjective optimization problems by solving a sequence of constrained single-objective problems. Since the necessity of determining the constraint value a priori can be a serious drawback of the original epsilon-constraint method, our scheme generates appropriate constraint values adaptively during the run. A simple example problem is presented where the running time (measured by the number of constrained singleobjective sub-problems to be solved) of the original epsilon-constraint method is exponential in the problem size (number of decision variables), although the size of the Pareto set grows only linearly. For our method we show that — independent of the problem or the problem size — the time complexity is O(km−1), where k is the number of Pareto-optimal solutions to be found and m the number of objectives. Using the algorithm together with a standard ILP solver for the constrained single-objective problems, the exact Pareto front is generated for the three-objective 0/1 knapsack problem with up to 100 decision variables. Links to problem instances and a reference implementation of the algorithm are provided.