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

Multi-objective optimization using statistical models

In this paper we consider multi-objective optimization problems (MOOP) from the point of view of Bayesian analysis. MOOP problems can be considered equivalent to certain statistical models associated with the specific objectives and constraints. MOOP that can explore accurately the Pareto frontier are Generalized Data Envelopment Analysis and Goal Programming. In turn, posterior analysis of their associated statistical models can be implemented using Markov Chain Monte Carlo (MCMC) simulation. In addition, we consider the minimax regret problem which provides robust solutions and we develop similar MCMC posterior simulators without the need to define scenarios. The new techniques are shown to work well in four examples involving non-convex and disconnected Pareto problems and to a real world portfolio optimization problem where the purpose is to optimize simultaneously average return, mean absolute deviation, positive and negative skewness of portfolio returns. Globally minimum regret can also be implemented based on post-processing of MCMC draws. © 2019 Elsevier B.V

Network Interdiction under Uncertainty

We consider variants to one of the most common network interdiction formulations: the shortest path interdiction problem. This problem involves leader and a follower playing a zero-sum game over a directed network. The leader interdicts a set of arcs, and arc costs increase each time they are interdicted. The follower observes the leader\u27s actions and selects a shortest path in response. The leader\u27s optimal interdiction strategy maximizes the follower\u27s minimum-cost path. Our first variant allows the follower to improve the network after the interdiction by lowering the costs of some arcs, and the leader is uncertain regarding the follower\u27s cardinality budget restricting the arc improvements. We propose a multiobjective approach for this problem, with each objective corresponding to a different possible improvement budget value. To this end, we also present the modified augmented weighted Tchebychev norm, which can be used to generate a complete efficient set of solutions to a discrete multi-objective optimization problem, and which tends to scale better than competing methods as the number of objectives grows. In our second variant, the leader selects a policy of randomized interdiction actions, and the follower uses the probability of where interdictions are deployed on the network to select a path having the minimum expected cost. We show that this continuous non-convex problem becomes strongly NP-hard when the cost functions are convex or when they are concave. After formally describing each variant, we present various algorithms for solving them, and we examine the efficacy of all our algorithms on test beds of randomly generated instances

Using Scalarizations for the Approximation of Multiobjective Optimization Problems: Towards a General Theory

We study the approximation of general multiobjective optimization problems with the help of scalarizations. Existing results state that multiobjective minimization problems can be approximated well by norm-based scalarizations. However, for multiobjective maximization problems, only impossibility results are known so far. Countering this, we show that all multiobjective optimization problems can, in principle, be approximated equally well by scalarizations. In this context, we introduce a transformation theory for scalarizations that establishes the following: Suppose there exists a scalarization that yields an approximation of a certain quality for arbitrary instances of multiobjective optimization problems with a given decomposition specifying which objective functions are to be minimized / maximized. Then, for each other decomposition, our transformation yields another scalarization that yields the same approximation quality for arbitrary instances of problems with this other decomposition. In this sense, the existing results about the approximation via scalarizations for minimization problems carry over to any other objective decomposition -- in particular, to maximization problems -- when suitably adapting the employed scalarization. We further provide necessary and sufficient conditions on a scalarization such that its optimal solutions achieve a constant approximation quality. We give an upper bound on the best achievable approximation quality that applies to general scalarizations and is tight for the majority of norm-based scalarizations applied in the context of multiobjective optimization. As a consequence, none of these norm-based scalarizations can induce approximation sets for optimization problems with maximization objectives, which unifies and generalizes the existing impossibility results concerning the approximation of maximization problems

Effective anytime algorithm for multiobjective combinatorial optimization problems

In multiobjective optimization, the result of an optimization algorithm is a set of efficient solutions from which the decision maker selects one. It is common that not all the efficient solutions can be computed in a short time and the search algorithm has to be stopped prematurely to analyze the solutions found so far. A set of efficient solutions that are well-spread in the objective space is preferred to provide the decision maker with a great variety of solutions. However, just a few exact algorithms in the literature exist with the ability to provide such a well-spread set of solutions at any moment: we call them anytime algorithms. We propose a new exact anytime algorithm for multiobjective combinatorial optimization combining three novel ideas to enhance the anytime behavior. We compare the proposed algorithm with those in the state-of-the-art for anytime multiobjective combinatorial optimization using a set of 480 instances from different well-known benchmarks and four different performance measures: the overall non-dominated vector generation ratio, the hypervolume, the general spread and the additive epsilon indicator. A comprehensive experimental study reveals that our proposal outperforms the previous algorithms in most of the instances.This research has been partially funded by the Spanish Ministry of Economy and Competitiveness (MINECO) and the European Regional Development Fund (FEDER) under contract TIN2017-88213-R (6city project), the European Research Council under contract H2020-ICT-2019-3 (TAILOR project), the University of Málaga, Consejería de Economía y Conocimiento de la Junta de Andalucía and FEDER under contract UMA18-FEDERJA-003 (PRECOG project), the Ministry of Science, Innovation and Universities and FEDER under contract RTC-2017-6714-5, and the University of Málaga under contract PPIT.UMA.B1.2017/07 (EXHAURO Project)

Supporting strategy selection in multiobjective decision problems under uncertainty and hidden requirements

Decision-makers are often faced with multi-faceted problems that require making trade-offs between multiple, conflicting objectives under various uncertainties. The task is even more difficult when considering dynamic, non-linear processes and when the decisions themselves are complex, for instance in the case of selecting trajectories for multiple decision variables. These types of problems are often solved using multiobjective optimization (MOO). A typical problem in MOO is that the number of Pareto optimal solutions can be very large, whereby the selection process of a single preferred solution is cumbersome. Moreover, preference between model-based solutions may not be determined only by their objective function values, but also in terms of how robust and implementable these solutions are. In this paper, we develop a methodological framework to support the identification of a small but diverse set of robust Pareto optimal solutions. In particular, we eliminate non-robust solutions from the Pareto front and cluster the remaining solutions based on their similarity in the decision variable space. This enables a manageable visual inspection of the remaining solutions to compare them in terms of practical implementability. We illustrate the framework and its benefits by means of an epidemic control problem that minimizes deaths and economic impacts, and a screening program for colorectal cancer that minimizes cancer prevalence and costs. These examples highlight the general applicability of the framework for disparate types of decision problems and process models

Air Force Institute of Technology Research Report 2019

This Research Report presents the FY19 research statistics and contributions of the Graduate School of Engineering and Management (EN) at AFIT. AFIT research interests and faculty expertise cover a broad spectrum of technical areas related to USAF needs, as reflected by the range of topics addressed in the faculty and student publications listed in this report. In most cases, the research work reported herein is directly sponsored by one or more USAF or DOD agencies. AFIT welcomes the opportunity to conduct research on additional topics of interest to the USAF, DOD, and other federal organizations when adequate manpower and financial resources are available and/or provided by a sponsor. In addition, AFIT provides research collaboration and technology transfer benefits to the public through Cooperative Research and Development Agreements (CRADAs). Interested individuals may discuss ideas for new research collaborations, potential CRADAs, or research proposals with individual faculty using the contact information in this document