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In integer programming and combinatorial optimisation, people use the term matheuristics to refer to methods that are heuristic in nature but draw on concepts from the literature on exact methods. We survey the literature on this topic, with a particular emphasis on matheuristics that yield both primal and dual bounds (i.e., upper and lower bounds in the case of a minimisation problem). We also make some comments about possible future developments. [ABSTRACT FROM AUTHOR]

Given a graph G = (V , E) and a set C of unordered pairs of edges regarded as being in conflict, a stable spanning tree in G is a set of edges T inducing a spanning tree in G, such that for each e i , e j ∈ C , at most one of the edges e i and e j is in T. The existing work on Lagrangean algorithms to the NP-hard problem of finding minimum weight stable spanning trees is limited to relaxations with the integrality property. We exploit a new relaxation of this problem: fixed cardinality stable sets in the underlying conflict graph H = (E , C) . We find interesting properties of the corresponding polytope, and determine stronger dual bounds in a Lagrangean decomposition framework, optimizing over the spanning tree polytope of G and the fixed cardinality stable set polytope of H in the subproblems. This is equivalent to dualizing exponentially many subtour elimination constraints, while limiting the number of multipliers in the dual problem to |E|. It is also a proof of concept for combining Lagrangean relaxation with the power of integer programming solvers over strongly NP-hard subproblems. We present encouraging computational results using a dual method that comprises the Volume Algorithm, initialized with multipliers determined by Lagrangean dual-ascent. In particular, the bound is within 5.5% of the optimum in 146 out of 200 benchmark instances; it actually matches the optimum in 75 cases. All of the implementation is made available in a free, open-source repository. [ABSTRACT FROM AUTHOR]

The paper deals with a truck scheduling and transshipment problem in a cross-docking terminal, with one inbound gate and one outbound gate. Each inbound truck carries one or more load units possibly of different commodities, and each outbound truck must be loaded with different commodities as well. The problem is to define the sequence of trucks at the inbound and outbound gates and the transshipment plan, so as to minimize the makespan of the whole process. We propose a Lagrangian relaxation scheme and a dual ascent method to optimize the Lagrangian dual that encloses three heuristics aimed at recovering feasible solutions for the problem at hand from the solution of the Lagrangian problem. Computational experiments are also provided, in order to evaluate the effectiveness of the Lagrangian approach. [ABSTRACT FROM AUTHOR]

Software-defined networking (SDN) has emerged as a solution to the cumbersome structures of classical computer networks. It separates control and data planes to give independence to devices with respect to either traffic routing or network management. The two isolated planes communicate with each other via the help of software modules, which are located in an SDN controller, such as Floodlight, NOX, or Ryu. In this study, Floodlight is used and an SDN topology with 20 switches is constructed with Python code in Mininet. All algorithms have been coded with Java. The default routing algorithm in Floodlight is Dijkstra’s algorithm. Four different network optimization algorithms, namely Bellman-Ford, Ford-Fulkerson, Auction, and Dual Ascent algorithms, are utilized in ordinary network routing instead of Dijkstra’s algorithm. None of these four algorithms were used in SDN before and network implementations using Ford-Fulkerson, Auction, or Dual Ascent algorithms were scarce in the literature. The results are analyzed with multiple types of normalization on a new user interface communicating with Floodlight part via HTTP requests. There has not been a user interface that performs the same operations in Floodlight. In the future, this study may possibly be improved with considering normalization processes based on various proportions among the metric values and accounting the computational time of the algorithms.

Software-defined networking (SDN) has emerged as a solution to the cumbersome structures of classical computer networks. It separates control and data planes to give independence to devices with respect to either traffic routing or network management. The two isolated planes communicate with each other via the help of software modules, which are located in an SDN controller, such as Floodlight, NOX, or Ryu. In this study, Floodlight is used and an SDN topology with 20 switches is constructed with Python code in Mininet. All algorithms have been coded with Java. The default routing algorithm in Floodlight is Dijkstra’s algorithm. Four different network optimization algorithms, namely Bellman-Ford, Ford-Fulkerson, Auction, and Dual Ascent algorithms, are utilized in ordinary network routing instead of Dijkstra’s algorithm. None of these four algorithms were used in SDN before and network implementations using Ford-Fulkerson, Auction, or Dual Ascent algorithms were scarce in the literature. The results are analyzed with multiple types of normalization on a new user interface communicating with Floodlight part via HTTP requests. There has not been a user interface that performs the same operations in Floodlight. In the future, this study may possibly be improved with considering normalization processes based on various proportions among the metric values and accounting the computational time of the algorithms. [ABSTRACT FROM AUTHOR]

We present a branch-and-bound (B&B) framework for the asymmetric prizecollecting Steiner tree problem (APCSTP). Several well-known network design problems can be transformed to the APCSTP, including the Steiner tree problem (STP), prizecollecting Steiner tree problem (PCSTP), maximum-weight connected subgraph problem (MWCS), and node-weighted Steiner tree problem (NWSTP). The main component of our framework is a new dual ascent algorithm for the rooted APCSTP, which generalizes Wong's dual ascent algorithm for the Steiner arborescence problem. The lower bounds and dual information obtained from the algorithm are exploited within powerful boundbased reduction tests and for guiding primal heuristics. The framework is complemented by additional alternative-based reduction tests. Extensive computational results on benchmark instances for the PCSTP,MWCS, andNWSTP indicate the framework's effectiveness, as most instances from literature are solved to optimality within seconds, including most of the (previously unsolved) largest instances from the recent DIMACS Challenge on Steiner trees. Moreover, results on newasymmetric instances for the APCSTP are reported. Since the addressed network design problems are frequently used for modeling various real-world applications (e.g., in bioinformatics), the implementation of the presented B&B framework has been made publicly available. [ABSTRACT FROM AUTHOR]

We present an effective heuristic for the Steiner Problem in Graphs. Its main elements are a multistart algorithm coupled with aggressive combination of elite solutions, both leveraging recently-proposed fast local searches. We also propose a fast implementation of a well-known dual ascent algorithm that not only makes our heuristics more robust (by dealing with easier cases quickly), but can also be used as a building block of an exact branch-and-bound algorithm that is quite effective for some inputs. On all graph classes we consider, our heuristic is competitive with (and sometimes more effective than) any previous approach with similar running times. It is also scalable: with long runs, we could improve or match the best published results for most open instances in the literature. [ABSTRACT FROM AUTHOR]

We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth $$\ell _1$$ regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function. [ABSTRACT FROM AUTHOR]

This paper presents an exciting finding for the power industry: The parameters of secondary frequency control based on integral or proportional integral control can be tuned to achieve economic operation and frequency regulation simultaneously within a considerable short period. We show that if the power imbalance is represented by frequency deviation, an iterative dual-ascent-based economic dispatch solving is equivalent to integral control. An iterative method of multipliers based economic dispatch is equivalent to proportional integral control. Similarly, if the controller parameters of the secondary frequency controls are chosen based on generator cost functions, these secondary frequency controllers achieve both economic operation and frequency regulation simultaneously. [ABSTRACT FROM AUTHOR]

In the traveling salesman problem with pickup, delivery, and ride-time constraints (TSPPD-RT), a vehicle located at a depot is required to service a number of requests where the requests are known before the route is formed. Each request consists of (i) a pickup location (origin), (ii) a delivery location (destination), and (iii) a maximum allowable travel time from the origin to the destination (maximum ride-time). The problem is to design a tour for the vehicle that (i) starts and ends at the depot, (ii) services all requests, (iii) ensures that each request's ride-time does not exceed its maximum ride-time, and (iv) minimizes the total travel time required by the vehicle to service all requests (objective function). A capacity constraint that may be present is that the weight or volume of the undelivered requests on the vehicle must always be no greater than the vehicle's capacity. In this article, we concurrently analyze the TSPPD-RT with capacity constraints and without capacity constraints. We describe two mathematical formulations of the problem. These formulations are used to derive new lower bounds on the solution to the problem. Then, we provide two exact methods for finding the optimal route that minimizes the total travel cost. Our extensive computational analysis on both versions of the TSPPD-RT shows that the proposed algorithms are capable of solving to optimality instances involving up to 50 requests. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(2), 95-110 2016 [ABSTRACT FROM AUTHOR]

This paper describes an application of a matheuristic algorithm to a real-world city logistics problem for a mid-sized town, whose core could be modeled as a multitrip vehicle routing problem with time windows, pickup and deliveries, and heterogeneous fleet. The proposed matheuristic is based on a dual ascent procedure applied to an extended set covering model (SCC) and on a randomized constructive heuristic. It starts by constructing an initial set of feasible solutions. The corresponding routes are the starting subset of columns of SCC. At each iteration, the dual ascent based on a Lagrangian optimization of the SCC computes a near-optimal dual solution that is returned along with a primal solution obtained by a Lagrangian heuristic. The dual costs are used for generating new feasible solutions by means of the constructive heuristic, and the corresponding routes are the new columns added to the SCC. At the end of the algorithm, the best heuristic solution is pruned to get a feasible solution for the city logistics problem. The matheuristic could actually solve different city logistics scenarios to be used as a basis for a successive stakeholders concertation process. A further interesting byproduct of this research is the dual ascent procedure that also proved effective on the classical set covering problem, in addition to its extension considered in this paper. [ABSTRACT FROM AUTHOR]

The Reformulation Linearization Technique (RLT) applied to the Quadratic Assignment Problem yields mixed 0–1 programming problems whose linear relaxations provide a strong bound on the objective value. Nevertheless, in the high level RLT representations the computation requires much effort. In this paper we propose a new compact reformulation for each level of the RLT representation exploiting the structure of the problem. Computational results on some benchmark instances indicate the potential of the new RLT representations as the level of the RLT increases. [ABSTRACT FROM AUTHOR]

We study a variant of the spanning tree problem where we require that, for a given connected graph, the spanning tree to be found has the minimum number of branch vertices (that is vertices of the tree whose degree is greater than two). We provide four different formulations of the problem and compare different relaxations of them, namely Lagrangian relaxation, continuous relaxation, mixed integer-continuous relaxation. We approach the solution of the Lagrangian dual both by means of a standard subgradient method and an ad-hoc finite ascent algorithm based on updating one multiplier at the time. We provide numerical result comparison of all the considered relaxations on a wide set of benchmark instances. A useful follow-up of tackling the Lagrangian dual is the possibility of getting a feasible solution for the original problem with no extra costs. We evaluate the quality of the resulting upper bound by comparison either with the optimal solution, whenever available, or with the feasible solution provided by some existing heuristic algorithms. [ABSTRACT FROM AUTHOR]

In the two-echelon capacitated vehicle routing problem (2E-CVRP), the delivery to customers from a depot uses intermediate depots, called satellites. The 2E-CVRP involves two levels of routing problems. The first level requires a design of the routes for a vehicle fleet located at the depot to transport the customer demands to a subset of the satellites. The second level concerns the routing of a vehicle fleet located at the satellites to serve all customers from the satellites supplied from the depot. The objective is to minimize the sum of routing and handling costs. This paper describes a new mathematical formulation of the 2E-CVRP used to derive valid lower bounds and an exact method that decomposes the 2E-CVRP into a limited set of multidepot capacitated vehicle routing problems with side constraints. Computational results on benchmark instances show that the new exact algorithm outperforms the state-of-the-art exact methods. [ABSTRACT FROM AUTHOR]

This paper presents new lower bounding procedures and an exact method for the m-peripatetic vehicle routing problem (m-PVRP) based on polyhedral and column generation approaches. The branch-and-cut algorithms use three types of valid cuts on the edge-based formulation. The column-generation-based lower bounding procedure is applied on the dual set partitioning formulation and is composed of dual heuristics that estimate good dual variable values and therefore high lower bounds. Computational results on instances from the literature show that these new lower bounds reach on average 97.5% to 99% of the best known upper bounds and optimality is proven for a third of the instances. [ABSTRACT FROM AUTHOR]

We apply the level-3 reformulation-linearization technique (RLT3) to the quadratic assignment problem W (QAP). We then present our experience in calculating lower bounds using an essentially new algorithm based on this RLT3 formulation. Our method is not guaranteed to calculate the RLT3 lower bound exactly, but it approximates this lower bound very closely and reaches it in some instances. For Nugent problem instances up to size 24, our RLT3-based lower bound calculation solves these problem instances exactly or serves to verify the optimal value. Calculating lower bounds for problem sizes larger than size 27 still presents a challenge because of the large amount of memory needed to implement the RLT3 formulation. Our presentation emphasizes the steps taken to significantly conserve memory by using the numerous problem symmetries in the RLT3 formulation of the QAP. We implemented this RLT3-based bound calculation in a branch-and-bound algorithm. Experimental results project significant runtime improvement over all other published QAP branch-and-bound solvers. [ABSTRACT FROM AUTHOR]

The pickup and delivery problem with time windows (PDPTW) is a generalization of the vehicle routing problem with time windows. In the PDPTW, a set of identical vehicles located at a central depot must be optimally routed to service a set of transportation requests subject to capacity, time window, pairing, and precedence constraints. In this paper, we present a new exact algorithm for the PDPTW based on a set-partitioning-like integer formulation, and we describe a bounding procedure that finds a near-optimal dual solution of the LP-relaxation of the formulation by combining two dual ascent heuristics and a cut-and-column generation procedure. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between a known upper bound and the lower bound achieved. If the resulting problem has moderate size, it is solved by an integer programming solver; otherwise, a branch-and-cut-and-price algorithm is used to close the integrality gap. Extensive computational results over the main instances from the literature show the effectiveness of the proposed exact method. [ABSTRACT FROM AUTHOR]

This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved. [ABSTRACT FROM AUTHOR]

This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered. [ABSTRACT FROM AUTHOR]

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We consider the telecommunications network design problem of simultaneously determining at which locations to place a switch, interconnecting all switches with backbone trunks, and connecting each location to some switch by an access circuit. We assume there are no capacity constraints, and minimize the sum of switch, backbone trunk, and access circuit costs using a dual ascent method that solves a sequence of dual uncapacitated facility location problems. A Steiner tree based heuristic provides a primal feasible design. On 15 random problems with 100 locations, the average duality gap is 2.0%. [ABSTRACT FROM AUTHOR]

We apply a recent extension of the Bregman proximal method for convex programming to LP relaxations of 0–1 problems. We allow inexact subproblem solutions obtained via dual ascent, increasing their accuracy successively to retain global convergence. Our framework is applied to relaxations of large-scale set covering problems that arise in airline crew scheduling. Approximate relaxed solutions are used to construct primal feasible solutions via a randomized heuristic. Encouraging preliminary experience is reported. [ABSTRACT FROM AUTHOR]

We analyze the rate of convergence of certain dual ascent methods for the problem of minimizing a strictly convex essentially smooth function subject to linear constraints. Included in our study are dual coordinate ascent methods and dual gradient methods. We show that, under mild assumptions on the problem, these methods attain a linear rate of convergence. Our proof is based on estimating the distance from a feasible dual solution to the optimal dual solution set by the norm of a certain residual. [ABSTRACT FROM AUTHOR]

We propose a new decomposition method for large-scale linear programming. This method dualizes an (arbitrary) subset of the constraints and then maximizes the resulting dual functional by dual ascent. The ascent directions are chosen from a finite set and are generated by a truncated version of the painted index algorithm of Rockafellar. Central to this method is the novel notion of (ϵ, δ)-complementary slackness (ϵ ≥ 0, δ ∈ [0,1]) which allows each Lagrangian subproblem to be solved only approximately with O(ϵδ) accuracy and provides a lower bound of Ω(∈(1 - δ)) on the amount of improvement per dual ascent. By dynamically adjusting ϵ, the subproblems can be solved with increasing accuracy. We show that (i) the method terminates finitely, provided that ϵ and δ are bounded away from 0 and 1, respectively, (ii) the final solution produced by the method is feasible and is within O(ϵ) in cost of the optimal cost, and (iii) the final solution produced by the method is optimal for all ϵ sufficiently small. [ABSTRACT FROM AUTHOR]