A decomposable self-adaptive projection-based prediction–correction algorithm for convex time space network flow problem
Introduction
In real-life applications, many problems can be modeled as dynamical network flow problems. Time space network (TSN) is one effective technique for modeling a dynamic network; it provides a holistic framework to handle the dynamic and real-time decision problems in practice. Unlike traditional static connection-based network flow model, TSN allows representation of network flows over time (or ‘dynamic flows’) by extending the entire network structure in both time and space dimensions [1]. Essentially, TSN uses a rolling horizon approach that can easily incorporate real-time information and decision updates [2]. For this reason, TSN has been widely used to formulate various industrial and engineering problems, such as flight scheduling [3], vehicle and crew scheduling [4], arc capacity assignment [5], and emergency distribution optimization [6], [7]. However, the size of network structure and the number of node constraints needed to ensure flow conservation in a TSN grow rapidly with the number of physical nodes and the length of time horizon. Moreover, the huge network structure usually results in large scale matrices which are difficult to calculate in practical situations. In this paper, we focus on solving a special class of TSN flow problem that has a convex decomposable objective function and contains coupling linear constraints. The abstract model for this problem can be generalized as a linearly constrained convex programming, which is usually difficult to solve if the problem is large scale. A feasible technique is to decompose the large scale problem into several smaller sub-problems through utilizing the decomposability embedded in the problem structure, so as to reduce the complexity and scale.
Some classical methods like Benders decomposition [8] and Lagrangian relaxation [9] are commonly used to deal with those complex problems with decomposable structure, but a main issue among these methods is that it is difficult to guarantee the convergence, as well as to attain a feasible solution in some real-world applications. Although alternating direction method (ADM) can guarantee the convergence during the iterations [10], [11], [12], the decomposed subproblems are usually not easy to solve in practice. Proximal point algorithm (PPA) [13], [14] as another useful method to decompose primal–dual optimization problem, however, the computational cost for its subproblems are much high in the most. Besides, extra-gradient algorithm (EGA) [15] and projection-based prediction–correction algorithm (PPCA) [16] are implementable in solving large-scale optimization problems. They not only have distinct advantages in guaranteeing the convergence and solving large-scale complex problems, but also can dynamically find a quick and feasible convergent direction during the iterations. However, a main shortcoming of EGA is that it is hard to determine the stepsize in advance. While in the works conducted by He et al. [16], and Xu et al. [17], they investigated how to choose an appropriate stepsize in the developed PPCA, but their studies treated the primal and dual variables as a whole and neglected the scale balance problem between them. Thus, in [18], they further investigated the decomposable feature of the primal and dual variables, and developed a self-adaptive projection-based prediction–correction algorithm (SPPCA) to explore the scale balance problem between them.
In this study, we propose a decomposable self-adaptive projection-based prediction–correction algorithm (DSPPCA) through exploring the decomposable structure of primal decision variables in the convex TSN flow model. Our work differs from the research of [18] in two aspects. First, in addition to consider the decomposition between primal and dual variables as they addressed, we further explore the separability of primal variables and decompose them into two parts. In their work, they investigated a simple situation that exists in a decomposition between primal and dual problems, but did not consider the separable structure among primal variables. Second, the non-zero residual in our iteration procedure only contains two terms. However, there are three non-zero residual terms in the work conducted by Fu and He [18], which may cause more inaccuracy of the current iteration. The main contributions of this paper are twofold: (1) We propose a novel DSPPCA to solve a convex time space network flow problem with a decomposable objective function and linear coupling constraints, which can also be extended to solve other convex optimization problems with a decomposable structure; (2) The proposed DSPPCA only contains two non-zero residual terms, which provides enhanced accuracy and convergence,compared with the results of EGA and SPPCA.
Our paper is organized as follows. In Section 2, we describe the convex TSN flow problem, and discuss how to solve it as a variational inequality problem. In Section 3, we focus on how to solve the variational inequality. We first illustrate the implementation procedures of EGA; then, we explore the decomposable structure in the TSN flow problem and propose a novel DSPPCA. Finally, we provide the convergent theory of DSPPCA. In Section 4, we report several computational experiments derived from emergency logistics to verify the computational performance. Finally, we conclude this paper in Section 5.
Section snippets
Problem description
In this section, we first introduce a convex TSN flow problem, and then discuss how to transform its Karush–Kuhn–Tucker (KKT) conditions into a variational inequality.
Decomposable self-adaptive projection-based prediction–correction algorithm
In this section, we first address the EGA implementation procedure to solve variational inequality in Eq. (8), and briefly analyse its main advantages and disadvantages. Then, we demonstrate the proposed DSPPCA and provide its convergent theory.
Computational experiments
In this section, we present computational experiments to compare the efficiency and effectiveness of the proposed DSPPCA with that of EGA and SPPCA. The test examples arise from emergency resource allocation and distribution problems discussed in [20], in which they investigated a quadratic TSN flow model that jointly optimizes emergency resource allocation and emergency distribution simultaneously. In the following discussions, we use this model to test our proposed algorithm and report the
Conclusions
In this paper, our study concentrates on how to solve a convex TSN flow problem with decomposable structure. We demonstrate that the KKT condition of the constrained convex TSN flow model is equivalent to a variational inequality problem. Then, in terms of the decomposable structure embedded in primal decision variables, we propose a novel DSPPCA to solve the variational inequality, in which we decompose the decision variables into two parts and calculate them separately during the iterations.
Acknowledgements
This work is supported by the Natural Science Foundation of China (70671021, 70901018), the Excellent Doctoral Thesis Foundation of Southeast University (YBJJ1015), and Natural Science and Engineering Council of Canada (Discovery Grant 539050). The authors would like to thank the editor and referees for their valuable comments and suggestions that have helped improve the presentation of the paper significantly.
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