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

General 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, and if an appropriate scalarization is employed, 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, given an appropriate initial search space, requires the solution of at most 2|N | − 1 subproblems, where N denotes the nondominated set of the underlying problem. For higher dimensional problems, the best known bound is O(|N |m−1) with m being the number of objectives. We present a new procedure for finding the entire nondominated set of tricriteria optimization problems for which the number of scalarized subproblems to be solved is bounded by 3|N | − 2. The approach includes an iterative update of the search space that, given a (sub-)set of nondominated points, describes the area in which additional nondominated points may be located. In particular, we show that the number of boxes, into which the search space is decomposed, depends linearly on the number of nondominated points.