Mathematical Methods of Operations Research最新文献

Abstract

In multi-criteria optimization problems that originate from real-world decision making tasks, we often find the following structure: There is an underlying continuous, possibly even convex model for the multiple outcome measures depending on the design variables, but these outcomes are additionally assigned to discrete categories according to their desirability for the decision maker. Multi-criteria deliberations may then take place at the level of these discrete labels, while the calculation of a specific design remains a continuous problem. In this work, we analyze this type of problem and provide theoretical results about its solution set. We prove that the discrete decision problem can be tackled by solving scalarizations of the underlying continuous model. Based on our analysis we propose multiple algorithmic approaches that are specifically suited to handle these problems. We compare the algorithms based on a set of test problems. Furthermore, we apply our methods to a real-world radiotherapy planning example.

Abstract

A variety of approaches has been developed to deal with uncertain optimization problems. Often, they start with a given set of uncertainties and then try to minimize the influence of these uncertainties. The reverse view is to first set a budget for the price one is willing to pay and then find the most robust solution. In this article, we aim to unify these inverse approaches to robustness. We provide a general problem definition and a proof of the existence of its solution. We study properties of this solution such as closedness, convexity, and boundedness. We also provide a comparison with existing robustness concepts such as the stability radius, the resilience radius, and the robust feasibility radius. We show that the general definition unifies these approaches. We conclude with an example that demonstrates the flexibility of the introduced concept.

In this paper, we take an in-depth look at the complexity of a hitherto unexplored multiobjective minimum weight minimum stretch spanner problem; or in short multiobjective spanner (MSp) problem. The MSp is a multiobjective generalization of the well-studied minimum t-spanner problem. This multiobjective approach allows to find solutions that offer a viable compromise between cost and utility—a property that is usually neglected in singleobjective optimization. Thus, the MSp can be a powerful modeling tool when it comes to, e.g., the planning of transportation or communication networks. This holds especially in disaster management, where both responsiveness and practicality are crucial. We show that for degree-3 bounded outerplanar instances the MSp is intractable and computing the non-dominated set is BUCO-hard. Additionally, we prove that if ({textbf{P}} ne textbf{NP}), the set of extreme points cannot be computed in output-polynomial time, for instances with unit costs and arbitrary graphs. Furthermore, we consider the directed versions of the cases above.

Abstract

In many optimization problems arising from machine learning, image processing, and statistics communities, the objective functions possess a special form involving huge amounts of data, which encourages the application of stochastic algorithms. In this paper, we study such a broad class of nonconvex nonsmooth minimization problems, whose objective function is the sum of a smooth function of the entire variables and two nonsmooth functions of each variable. We propose to solve this problem with a stochastic Gauss–Seidel type inertial proximal alternating linearized minimization (denoted by SGiPALM) algorithm. We prove that under Kurdyka–Łojasiewicz (KŁ) property and some mild conditions, each bounded sequence generated by SGiPALM with the variance-reduced stochastic gradient estimator globally converges to a critical point after a finite number of iterations, or almost surely satisfies the finite length property. We also apply the SGiPALM algorithm to the proximal neural networks (PNN) with 4 layers for classification tasks on the MNIST dataset and compare it with other deterministic and stochastic optimization algorithms, the results illustrate the effectiveness of the proposed algorithm.

We present a simple iterative method for solving quasimonotone as well as classical variational inequalities without monotonicity. Strong convergence analysis is given under mild conditions and thus generalize the few existing results that only present weak convergence methods under restrictive assumptions. We give finite and infinite dimensional numerical examples to compare and illustrate the simplicity and computational advantages of the proposed scheme.

In this paper, we present an outer approximation algorithm for computing the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems (MOMILPs). It produces the extreme points (i.e., the vertices) as well as the facets of the Edgeworth–Pareto hull. We note that these extreme points are the extreme supported non-dominated points of a MOMILP. We also show how to extend the concept of geometric duality for multi-objective linear programming problems to the Edgeworth–Pareto hull of MOMILPs and use this extension to develop the algorithm. The algorithm relies on a novel oracle that solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the extreme supported non-dominated points in the case of MOMILPs. Thus, for MOMILPs for which the weighted-sum problem is solvable in polynomial time, the facets can be computed with incremental-polynomial delay—a result that was formerly only known for the computation of extreme supported non-dominated points. Our algorithm can be an attractive option to compute lower bound sets within multi-objective branch-and-bound algorithms for solving MOMILPs. This is for several reasons as (i) the algorithm starts from a trivial valid lower bound set then iteratively improves it, thus at any iteration of the algorithm a lower bound set is available; (ii) the algorithm also produces efficient solutions (i.e., solutions in the decision space); (iii) in any iteration of the algorithm, a relaxation of the MOMILP can be solved, and the obtained points and facets still provide a valid lower bound set. Moreover, for the special case of multi-objective linear programming problems, the algorithm solves the problem to global optimality. A computational study on a set of benchmark instances from the literature is provided.

In this paper, we construct a new spline smoothing homotopy method for solving general nonlinear programming problems with a large number of complicated constraints. We transform the equality constraints into the inequality constraints by introducing two parameters. Subsequently, we use smooth spline functions to approximate the inequality constraints. The smooth spline functions involve only few inequality constraints. In other words, the method introduces an active set technique. Under some weaker conditions, we obtain the global convergence of the new spline smoothing homotopy method. We perform numerical tests to compare the new method to other methods, and the numerical results show that the new spline smoothing homotopy method is highly efficient.

Most queueing models and their analysis have a rich history and follow a process of increased generality and complexity. In this paper, we introduce a new model, namely a multiclass queueing model where service times depend on the presence of one of the classes. Our model is motivated by road traffic, where the presence of heavy vehicles in a queue slows down the entire system, or, in contrast, where the presence of emergency vehicles may speed up the service. The specific assumption we impose is that the service time of each customer depends on whether at least one customer of that particular class is present in the system at the time of service. Although we study a fairly simple discrete-time model, we show that analysis is not straightforward. Furthermore, numerical examples expose that the impact of particular customers in the system can lead to a substantial slow down (or, in contrast, speed up) of the entire system.

We investigate, in the framework of set optimization, some issues that are well studied in vectorial setting, that is, penalization procedures, properness of solutions and optimality conditions on primal spaces. Therefore, with this study we aim at completing the literature dedicated to set optimization with some results that have well established correspondence in the classical vector optimization.