A linear bound on the number of scalarizations needed to solve discrete tricriteria optimization problems

Multi-objective optimization problems are often solved by a sequence of parametric single-objective problems, so-called scalarizations. If the set of nondominated points is finite, the entire nondominated set can be generated in this way. In the bicriteria case it is well known that this can be realized by an adaptive approach which requires the solution of at most $$2|Z_{N}|-1$$2|ZN|-1 subproblems, where $$Z_{N}$$ZN denotes the nondominated set of the underlying problem and a subproblem corresponds to a scalarized problem. For problems with more than two criteria, no methods were known up to now for which the number of subproblems depends linearly on the number of nondominated points. We present a new procedure for finding the entire nondominated set of tricriteria optimization problems for which the number of subproblems to be solved is bounded by $$3 |Z_{N}|-2$$3|ZN|-2, hence, depends linearly on the number of nondominated points. The approach includes an iterative update of the search region that, given a (sub-)set of nondominated points, describes the area in which additional nondominated points may be located. If the $$\varepsilon $$ε-constraint method is chosen as scalarization, the upper bound can be improved to $$2 |Z_{N}|-1$$2|ZN|-1.