Whether it is in the field of production, logistics, in medicine or biology; everywhere the global optimal solution or the set of global optimal solutions is sought. However, most real-world problems are of nonlinear nature and naturally multimodal which poses severe problems to global optimization. Multimodality, the existence of multiple (local) optima, is regarded as one of the biggest challenges for continuous single-objective problems . A lot of algorithms get stuck searching for the global optimum or are requiring many function evaluations to escape local optima. One of the most popular strategies for dealing with multimodal problems are population-based methods like evolutionary algorithms due to their global search abilities . In this paper we will examine another approach of coping with local traps, namely multiobjectivization. By transforming a single-objective into a multi-objective problem, we aim at exploiting the properties of multi-objective landscapes. So far, the characteristics of single-objective optimization problems have often been directly transferred to the multiobjective domain.
As potential-based reward shaping functions (heuristic signals conflicts may exist between objectives, there is in general guiding exploration) (Brys et al. 2014a). We prove that this a need to identify (a set of) tradeoff solutions. The set modification preserves the total order, and thus also optimality, of optimal, i.e. non-dominated, incomparable solutions is of policies, mainly relying on the results by Ng, Harada, called the Pareto-front. We identify multi-objective problems and Russell (1999). This insight - that any MDP can be with correlated objectives (CMOP) as a specific subclass framed as a CMOMDP - significantly increases the importance of multi-objective problems, defined to contain those of this problem class, as well as techniques developed MOPs whose Pareto-front is so limited that one can barely for it, as these could be used to solve regular single-objective speak of tradeoffs (Brys et al. 2014b). By consequence, MDPs faster and better, provided several meaningful shapings the system designer does not care about which of the very can be devised.
Sequential decision-making problems with multiple objectives arise naturally in practice and pose unique challenges for research in decision-theoretic planning and learning, which has largely focused on single-objective settings. This article surveys algorithms designed for sequential decision-making problems with multiple objectives. Though there is a growing body of literature on this subject, little of it makes explicit under what circumstances special methods are needed to solve multi-objective problems. Therefore, we identify three distinct scenarios in which converting such a problem to a single-objective one is impossible, infeasible, or undesirable. Furthermore, we propose a taxonomy that classifies multi-objective methods according to the applicable scenario, the nature of the scalarization function (which projects multi-objective values to scalar ones), and the type of policies considered. We show how these factors determine the nature of an optimal solution, which can be a single policy, a convex hull, or a Pareto front. Using this taxonomy, we survey the literature on multi-objective methods for planning and learning. Finally, we discuss key applications of such methods and outline opportunities for future work.
Using some real-world examples I illustrate the important role of multiobjective optimization in decision making and its interface with preference handling. I explain what optimization in the presence of multiple objectives means and discuss some of the most common methods of solving multiobjective optimization problems using transformations to single-objective optimization problems. Finally, I address linear and combinatorial optimization problems with multiple objectives and summarize techniques for solving them. Throughout the article I refer to the real-world examples introduced at the beginning. There are infinitely many ways to invest money and infinitely many possible radiotherapy treatments, but the number of feasible crew schedules is finite, albeit astronomical in practice.
Moreover, the investor, the oncologist, and the airline manager are all in a situation where the number of available options or alternatives is very large or even infinite. There are infinitely many ways to invest money and infinitely many possible radiotherapy treatments, but the number of feasible crew schedules is finite, albeit astronomical in practice. The alternatives are therefore described by constraints, rather than explicitly known: the sums invested in every stock must equal the total invested; the radiotherapy treatment must meet physical and clinical constraints; crew schedules must ensure that each flight has exactly one crew assigned to operate it. Mathematically, the alternatives are described by vectors in variable or decision space; the set of all vectors satisfying the constraints is called the feasible set in decision space. The consequences or attributes of the alternatives are described as vectors in objective or outcome space, where outcome (objective) vectors are a function of the decision (variable) vectors.