Efficiently Constructing Convex Approximation Sets in Multiobjective Optimization Problems

Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the approximation set up to a multiplicative factor in each component, a convex approximation set only requires this multiplicative approximation to be achieved by some convex combination of finitely many images of solutions in the set. This makes convex approximation sets efficiently computable for a wide range of multiobjective problems - even for many problems for which (classic) approximations sets are hard to compute. In this article, we propose a polynomial-time algorithm to compute convex approximation sets that builds upon an exact or approximate algorithm for the weighted sum scalarization and is, therefore, applicable to a large variety of multiobjective optimization problems. The provided convex approximation quality is arbitrarily close to the approximation quality of the underlying algorithm for the weighted sum scalarization. In essence, our algorithm can be interpreted as an approximate variant of the dual variant of Benson's Outer Approximation Algorithm. Thus, in contrast to existing convex approximation algorithms from the literature, information on solutions obtained during the approximation process is utilized to significantly reduce both the practical running time and the cardinality of the returned solution sets while still guaranteeing the same worst-case approximation quality. We underpin these advantages by the first comparison of all existing convex approximation algorithms on several instances of the triobjective knapsack problem and the triobjective symmetric metric traveling salesman problem

Techniques for Multiobjective Optimization with Discrete Variables: Boxed Line Method and Tchebychev Weight Set Decomposition

Many real-world applications involve multiple competing objectives, but due to conflict between the objectives, it is generally impossible to find a feasible solution that optimizes all, simultaneously. In contrast to single objective optimization, the goal in multiobjective optimization is to generate a set of solutions that induces the nondominated (ND) frontier. This thesis presents two techniques for multiobjective optimization problems with discrete decision variables. First, the Boxed Line Method is an exact, criterion space search algorithm for biobjective mixed integer programs (Chapter 2). A basic version of the algorithm is presented with a recursive variant and other enhancements. The basic and recursive variants permit complexity analysis, which yields the first complexity results for this class of algorithms. Additionally, a new instance generation method is presented, and a rigorous computational study is conducted. Second, a novel weight space decomposition method for integer programs with three (or more) objectives is presented with unique geometric properties (Chapter 3). The weighted Tchebychev scalarization used for this weight space decomposition provides the benefit of including unsupported ND images but at the cost of convexity of weight set components. This work proves convexity-related properties of the weight space components, including star-shapedness. Further, a polytopal decomposition is used to properly define dimension for these nonconvex components. The weighted Tchebychev weight set decomposition is then applied as a “dual” perspective on the class of multiobjective “primal” algorithms (Chapter 4). It is shown that existing algorithms do not yield enough information for a complete decomposition, and the necessary modifications required to yield the missing information is proven. Modifications for primal algorithms to compute inner and outer approximations of the weight space components are presented. Lastly, a primal algorithm is restricted to solving for a subset of the ND frontier, where this subset represents the compromise between multiple decision makers’ weight vectors.Ph.D