Abstract When solving large-scale multiobjective optimization problems, solvers can get stuck because of memory and/or time limitations. In such cases, one is left with no information on the distance to the best feasible solution, found before the optimization process has stopped, to the true Pareto optimal solution. In this work, we show how to provide such information. To this aim we make use of the concept of lower shells and upper shells, developed in our earlier works. No specific assumptions about the problems to be solved are made. We illustrate the proposed approach on biobjective multidimensional knapsack problems derived from single-objective multidimensional knapsack problems in the Beasley OR Library. We address cases when a top-class commercial mixed-integer linear solver fails to provide Pareto optimal solutions attempted to be derived by scalarization.

Abstract When solving large-scale cardinality-constrained Markowitz mean–variance portfolio investment problems, exact solvers may be unable to derive some efficient portfolios, even within a reasonable time limit. In such cases, information on the distance from the best feasible solution, found before the optimization process has stopped, to the true efficient solution is unavailable. In this article, I demonstrate how to provide such information to the decision maker. I aim to use the concept of lower bounds and upper bounds on objective function values of an efficient portfolio, developed in my earlier works. I illustrate the proposed approach on a large-scale data set based upon real data. I address cases where a top-class commercial mixed-integer quadratic programming solver fails to provide efficient portfolios attempted to be derived by Chebyshev scalarization of the bi-objective optimization problem within a given time limit. In this case, I propose to transform purely technical information provided by the solver into information which can be used in navigation over the efficient frontier of the cardinality-constrained Markowitz mean–variance portfolio investment problem.

In one of our earlier works, we proposed to approximate Pareto fronts to multiobjective optimization problems by two-sided approximations, one from inside and another from outside of the feasible objective set, called, respectively, lower shell and upper shell. We worked there under the assumption that for a given problem an upper shell exists. As it is not always the case, in this paper we give some sufficient conditions for the existence of upper shells. We also investigate how to constructively search infeasible sets to derive upper shells. We approach this issue by means of problem relaxations. We formally show that under certain conditions some subsets of lower shells to relaxed multiobjective optimization problems are upper shells in the respective unrelaxed problems. Results are illustrated by a numerical example representing a small but real mechanical problem. Practical implications of the results are discussed.