Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization

We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1+epsilon by a piecewise-linear minimization problem over the same feasible set. Our main result is that when the feasible set is a polyhedron, the number of resulting pieces is polynomial in the input size of the polyhedron and linear in 1/epsilon. For many practical concave cost problems, the resulting piecewise-linear cost problem can be formulated as a well-studied discrete optimization problem. As a result, a variety of polynomial-time exact algorithms, approximation algorithms, and polynomial-time heuristics for discrete optimization problems immediately yield fully polynomial-time approximation schemes, approximation algorithms, and polynomial-time heuristics for the corresponding concave cost problems. We illustrate our approach on two problems. For the concave cost multicommodity flow problem, we devise a new heuristic and study its performance using computational experiments. We are able to approximately solve significantly larger test instances than previously possible, and obtain solutions on average within 4.27% of optimality. For the concave cost facility location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape

Improved Local Search Algorithms with Multi-Cycle Reduction for Minimum Concave Cost Network Flow Problems

The minimum concave cost network flow problem (MCCNFP) has many applications in areas such as telecommunication network design, facility location, production and inventory planning, and traffic scheduling and control. However, it is a well known NP-hard problem, and all existing search based exact algorithms are not practical for networks with even moderate numbers of vertices. Therefore, the research community also focuses on approximation algorithms to tackle the problems in practice. In this paper, we present an improved local search algorithm for the minimum concave cost network flow problem based on multi-cycle reduction. The original cycle reduction local search algorithm as proposed by Gallo and Sodini considers only negative cost single cycles; however, we find that such cycle reduction is not complete. We show that negative cost multi-cycles may exist in a network with concave edge costs that has no negative cost cycles, and an existing flow can be reduced to an adjacent neighboring flow with lower cost by redirecting flows along these negative multi-cycles. In this paper, we present an improved local search algorithm based on multi-cycle reduction. We evaluate our proposed algorithm in networks with a simple concave edge cost in different topologies and sizes. The experimental results show that the original cycle reduction algorithms can improve the quality of solutions obtained from a simple minimum cost augmentation approximation heuristic (LDF), and that a multi-cycle reduction yields more improvements; however, it reaches a point of diminished returns when we attempt to reduce more than bicycles

Computational results for Constrained Minimum Spanning Trees in Flow Networks

In this work, we address the problem of finding a minimum cost spanning tree on a single source flow network. The tree must span all vertices in the given network and satisfy customer demands at a minimum cost. The total cost is given by the summation of the arc setup costs and of the nonlinear flow routing costs over all used arcs. Furthermore, we restrict the trees of interest by imposing a maximum number of arcs on the longest arc emanating from the single source vertex. We propose a dynamic programming model an solution procedure to solve this problem exactly. Intensive computational experiments were performed using randomly generated test problems and the results obtained are reported. From them we can conclude that the method performance is independent of the type of cost functions considered and improves with the tightness of the constrains.Dynamic programming, network flows, constrained trees, general nonlinear costs

Models and Methods for Merge-In-Transit Operations

We develop integer programming formulations and solution methods for addressing operational issues in merge-in-transit distribution systems. The models account for various complex problem features including the integration of inventory and transportation decisions, the dynamic and multimodal components of the application, and the non-convex piecewise linear structure of the cost functions. To accurately model the cost functions, we introduce disaggregation techniques that allow us to derive a hierarchy of linear programming relaxations. To solve these relaxations, we propose a cutting-plane procedure that combines constraint and variable generation with rounding and branch-and-bound heuristics. We demonstrate the effectiveness of this approach on a large set of test problems with instances with up to almost 500,000 integer variables derived from actual data from the computer industry. Key words : Merge-in-transit distribution systems, logistics, transportation, integer programming, disaggregation, cutting-plane method

용량 제약이 없는 부보상 문제의 혼합 이진 이차 문제로의 모형화를 통한 해법

학위논문(석사) -- 서울대학교대학원 : 공과대학 산업공학과, 2022. 8. 홍성필.부보상 문제는 비순환 유향 그래프 상에서 출발, 도착 마디를 잇는 경로와 그 경로 상의 흐름을 결정하는 문제이다. 부보상은 도시 1에서 n까지 이동하면서 각 도시를 경유하 거나 지나치며, 경유하는 도시에서만 상품을 매매할 수 있지만 이동 거리와 상품량에 따른 비용 또한 지불해야 한다. 이 때, 부보상은 자신의 수익, 즉 총 상품의 판매량에서 얻는 수익과 지불 비용의 차를 최대화하고자 한다. 본 연구에서는 용량 제약이 없는 경우만을 다루며 기존 부보상 문제를 혼합이진이차문제으로 재모형화하여 분지절단법 으로 문제를 푼다. 이 때 목적 함수를 볼록화하고 연속 완화시켜 얻을 수 있는 상한을 비교하기 위해 여러 볼록화 방법들을 비교하고 비교실험한 결과 또한 제시한다.Bubosang Problem is a problem set on a directed acyclic graph path concerning both the path and multi-commodity flow decisions. A merchant travels from city 1 through n, either transiting through a city and trading products or passing by the city to the next city on his route. He wants to choose the path and trading product quantity to maximize his net profit which is defined by the difference between the total sales revenue and the traveling cost. The scope of the study considers only the uncapacitated case. In this study, we reformulate BP into a mixed binary quadratic problem to employ the branch-and-cut algorithm to solve the problem. Specifically, we compare the upper bound obtained through the continuous relaxation and convexification of the objective by studying different convexification methods. Computational results of the comparison are also provided.Chapter 1 Introduction 1 1.1 Background 1 1.2 Literature Review 3 1.3 Research Motivations 5 1.4 Organization of the Thesis 6 Chapter 2 Problem Definition and Mathematical Formulations 7 2.1 Problem Definition 7 2.2 Flow Arc Formulation 8 2.3 MBQP Formulation 11 2.3.1 MBQP 13 2.4 Branch-and-Cut Algorithm 14 2.4.1 Overall Setting 14 2.4.2 Cutset Inequality 14 2.4.3 Lower Bound 15 2.4.4 Upper Bound 18 Chapter 3 Convexification Methods 19 3.1 One Coefficient Case : Eigenvalue Method 21 3.2 Criteria for Convexification Evaluation 22 3.2.1 Criterion for Unweighted Methods 22 3.3 Two Coefficient Case : (α, β) - SDP method 23 3.4 Two Coefficient Case : (α, β) - Sum of Squares Method 24 3.5 Four Coefficient Case : (α, β, γ, δ) - method 26 3.5.1 (α, β, γ, δ) - SDP method 26 3.5.2 (α, β, γ, δ) - Sum of Squares method 28 3.6 Five Coefficient Case : (α, β, γ, δ, τ ) - Sum of Squares method 29 3.7 Weighted methods 30 3.7.1 Criterion for Weighted Methods 30 Chapter 4 Computational Experiments 32 Chapter 5 Conclusion 36 Bibliography 37 국문초록 41석

Multilevel Lot-Sizing with Inventory Bounds

We consider a single-item multilevel lot-sizing problem with a serial structure where one of the levels has an inventory capacity (the bottleneck level). We propose a novel dynamic programming algorithm combining Zangwill’s approach for the uncapacitated problem and the basis-path approach for the production capacitated problem. Under reasonable assumptions on the cost parameters the time complexity of the algorithm is O(LT6) with L the number of levels in the supply chain and T the length of the planning horizon. Computational tests show that our algorithm is significantly faster than the commercial solver CPLEX applied to a standard formulation and can solve reasonably sized instances up to 48 periods and 12 levels in a few minutes.</p

A strongly polynomial algorithm for generalized flow maximization

A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. As a consequence of the result, we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix.Comment: minor correction