Dual-based Approach Research Articles

Dual-Based Approximation Algorithms for Cut-Based Network Connectivity Problems

We consider a variety of NP-Complete network connectivity problems. We introduce a novel dual-based approach to approximating network design problems with cut-based linear programming relaxations. This approach gives a 3/2-approximation to Minimum 2-Edge-Connected Spanning Subgraph that is equivalent to a previously proposed algorithm. One well-studied branch of network design models ad hoc networks where each node can either operate at high or low power. If we allow unidirectional links, we can formalize this into the problem Dual Power Assignment (DPA). Our dual-based approach gives a 3 / 2-approximation to DPA, improving the previous best approximation known of $$11/7\approx 1.57$$ . Another standard network design problem is Minimum Strongly Connected Spanning Subgraph (MSCS). We propose a new problem generalizing MSCS and DPA called Star Strong Connectivity (SSC). Then we show that our dual-based approach achieves a 1.6-approximation ratio on SSC. As a consequence of our dual-based approximations, we prove new upper bounds on the integrality gaps of these problems.

A Dynamic Graph-Based Scheduling and Interference Coordination Approach in Heterogeneous Cellular Networks

To meet the demand of increasing mobile data traffic and provide better user experience, heterogeneous cellular networks (HCNs) have become a promising solution to improve both the system capacity and coverage. However, due to dense self-deployment of small cells in a limited area, serious interference from nearby base stations may occur, which results in severe performance degradation. To mitigate downlink interference and utilize spectrum resources more efficiently, we present a novel graph-based resource allocation and interference management approach in this paper. First, we divide small cells into cell clusters, considering their neighborhood relationships in the scenario. Then, we develop another graph clustering scheme to group user equipment (UE) in each cell cluster into UE clusters with minimum intracluster interference. Finally, we utilize a proportional fairness scheduling scheme to assign subchannels to each UE cluster and allocate power using water-filling method. To show the efficacy and effectiveness of our proposed approach, we propose a dual-based approach to search for optimal solutions as the baseline for comparisons. Furthermore, we compare the graph-based approach with the state of the art and a distributed approach without interference coordination. The simulation results show that our graph-based approach reaches more than 90% of the optimal performance and achieves a significant improvement in spectral efficiency compared with the state of the art and the distributed approach both under cochannel and orthogonal deployments. Moreover, the proposed graph-based approach has low computation complexity, making it feasible for real-time implementation.

An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem

Higham (2002, IMA J. Numer. Anal., 22, 329–343) considered two types of nearest correlation matrix problems, namely the W-weighted case and the H-weighted case. While the W-weighted case has since been well studied to make several Lagrangian dual-based efficient numerical methods available, the H-weighted case remains numerically challenging. The difficulty of extending those methods from the W-weighted case to the H-weighted case lies in the fact that an analytic formula for the metric projection onto the positive semidefinite cone under the H-weight, unlike the case under the W-weight, is not available. In this paper we introduce an augmented Lagrangian dual-based approach that avoids the explicit computation of the metric projection under the H-weight. This method solves a sequence of unconstrained convex optimization problems, each of which can be efficiently solved by an inexact semismooth Newton method combined with the conjugate gradient method. Numerical experiments demonstrate that the augmented Lagrangian dual approach is not only fast but also robust.

Maximizing network utilization with max–min fairness in wireless sensor networks

The state-of-the-art for optimal data-gathering in wireless sensor networks is to use additive increase algorithms to achieve fair rate allocation while implicity trying to maximize network utilization. For the quantification of the problem we present a receiver capacity model to capture the interference existing in a wireless network. We also provide empirical evidence to motivate the applicability of this model to a real CSMA based wireless network. Using this model, we explicitly formulate the problem of maximizing the network utilization subject to a max---min fair rate allocation constraint in the form of two coupled linear programs. We first show how the max---min rate can be computed efficiently for a given network. We then adopt a dual-based approach to maximize the network utilization. The analysis of the dual shows the sub-optimality of previously proposed additive increase algorithms with respect to bandwidth efficiency. Although in theory a dual-based sub-gradient search algorithm can take a long time to converge, we find empirically that setting all shadow prices to an equal and small constant value, results in near-optimal solutions within one iteration (within 2% of the optimum in 99.65% of the cases). This results in a fast heuristic distributed algorithm that has a nice intuitive explanation--rates are allocated sequentially after rank ordering flows based on the number of downstream receivers whose bandwidth they consume. We also investigate the near optimal performance of this heuristic by comparing the rank ordering of the source rates obtained from the heuristic to the solutions obtained by solving the linear program.

An exact solution procedure for a cluster hub location problem

This paper proposes a new (MIP) model formulation and a new solution procedure for the hub network design problem under a non-restrictive policy introduced by Sung and Jin [Sung, C.S., Jin, H.W., 2001. Dual-based approach for a hub network design problem under non-restrictive policy. European Journal of Operational Research 132 (1), 88–105]. The model formulation contains significantly fewer variables so that optimal solutions for the LP-relaxation of the model can be determined for large instances using standard procedures for LP-models. Furthermore, the LP-relaxation provides very tight lower bounds. Computational results are given, which demonstrate that the new model formulation allows for solving much larger instances. It turned out that the new (exact) solution procedure, which utilises the new model formulation, is faster than the heuristic proposed by Sung and Jin (2001). It is also shown that the problem is np-hard.

Adaptive modeling for free-surface flows

This work represents a first step towards the simulation of the motion of water in a complex hydrodynamic configuration, such as a channel network or a river delta, by means of a suitable “combination” of different mathematical models. In this framework a wide spectrum of space and time scales is involved due to the presence of physical phenomena of different nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solve throughout the whole domain the most complex model (with solution ) to accurately describe all the physical features of the problem at hand. In our approach instead, for a user-defined output functional , we aim to approximate, within a prescribed tolerance τ , the value by means of the quantity , being the so-called adapted solution solving the simpler models on most of the computational domain while confining the complex ones only on a restricted region. Moving from the simplified setting where only two hydrodynamic models, fine and coarse, are considered, we provide an efficient tool able to automatically select the regions of the domain where the coarse model rather than the fine one are to be solved, while guaranteeing below the tolerance τ . This goal is achieved via a suitable a posteriori modeling error analysis developed in the framework of a goal-oriented theory. We extend the dual-based approach provided in [Braack and Ern, Multiscale Model Sim. 1 (2003) 221–238], for steady equations to the case of a generic time-dependent problem. Then this analysis is specialized to the case we are interested in, i.e. the free-surface flows simulation, by emphasizing the crucial issue of the time discretization for both the primal and the dual problems. Finally, in the last part of the paper a widespread numerical validation is carried out.

Blood-based screening and light based imaging for the early detection and monitoring of ovarian cancer xenografts.

We report a novel technology for in vivo early detection, identification, and monitoring of ovarian cancer in live mice leading to better treatment outcome. A genetic dualistic reporter system that uses an adenoviral (Ad) vector to transfer the genetic reporters to the ovarian cancer is described. Infection of the cancer cells leads to expression of one reporter that is detected in blood, namely, secreted human placental alkaline phosphatase (SEAP). A second reporter, namely, enhanced green fluorescent protein (GFP) is also delivered by the Ad, leading to expression at the site of ovarian cancer. The SEAP gene under control of the cytomegalovirus (CMV) promoter element is linked to the GFP gene with an IRES element. A diagnostic adenoviral vector (Ad) encoding the SEAP and GFP (Ad5-SEAP-GFP) is produced. Efficacy of newly developed diagnostic vector is tested in cell culture and animal models. SKOV3ip.1 cells are infected with Ad5-SEAP-GFP. Over time the cells are monitored for fluorescence and SEAP is also measured in the growth media supernatant. For animal experiments, SKOV3ip.1 cells are implanted first in nude mice either subcutaneously (SC) or intraperitoneally (IP) separately. After 4-7 days, the Ad5-SEAP-GFP is administered. Control mice do not receive any Ad vector. All mice are imaged with a fluorescent stereomicroscope after 24 h, and blood is collected for SEAP analyses. Increasing green fluorescence is detected in all SKOV3ip.1 cells infected with Ad5-SEAP-GFP, while SEAP levels in growth media increase over monitoring period. Expression of GFP in both SC and IP tumors is detected by 24 h in the live mice. At this time, the SEAP blood levels are more than 2-3 fold greater than blood levels of control group. GFP fluorescence and SEAP levels continue to increase in all mice that are injected with Ad5-SEAP-GFP until termination. Control mice (both SC and IP) do not express GFP or SEAP throughout the experiment. GFP contrast is necessary to differentiate between micro-sized early stage non-palpable ovarian tumor nodules and surrounding normal tissue. While the studies are conducted in mice, it is envisioned that the dual-based approach will eventually be translated into human applications for routine diagnosis and monitoring of ovarian cancer when an ovarian cancer specific promoter will be available. Due to the thickness of the abdominal wall in human laparoscopy or laparotomy will be necessary. This system will provide gynecologic oncologists with a more effective tool for treating patients. The blood-based screening assay provides a quick test to determine the presence of the ovarian cancer at its earliest stage. The location of the ovarian cancer is afforded by the light-based imaging component, which represents a new and improving technology with tremendous advantages of sensitivity and spatial resolution to localize micro-sized tumor nodules. The novelty of the dualistic system is the linkage of blood-based reporter screening as a selection criteria for subsequent light-based imaging procedures. This combination will lead to an accurate and widely applicable method for the early detection and monitoring of ovarian cancer, especially in high-risk women

The plant location and technology acquisition problem

This paper presents an analytical approach for simultaneous optimization of the plant location, capacity acquisition and technology selection decisions in a multi-product environment. The proposed approach can be useful when there is considerable interaction between these structural decisions e.g., in global manufacturing companies. We present a formal definition of the plant location and technology acquisition problem and provide a mathematical model. We describe the analytical properties of the model, which lead to the development of a solution algorithm. Progressive piecewise linear underestimation constitutes the backbone of our solution algorithm. The arising subproblems are amenable to a dual-based approach. We report on a set of experiments that improved our understanding of the interaction among facility design decisions and showed that the computational performance of the proposed solution procedure is quite satisfactory.

Dual-based approach for a hub network design problem under non-restrictive policy

This paper considers a hub network design problem where the network service area partitioned into the predetermined zones (represented by node clusters) is given, and non-stop (direct link) service is allowed. The objective is to design a hub network under non-restrictive networking policy by determining all the required hub locations in the predetermined zones and also all the terminal-to-terminal routes such that the total network cost (hub construction and flow transportation cost) is minimized. For the design problem, a dual-based solution approach is proposed and tested for its effectiveness with numerical examples.

The plant location and technology acquisition problem

This paper presents an analytical approach for simultaneous optimization of the plant location, capacity acquisition and technology selection decisions in a multi-product environment. The proposed approach can be useful when there is considerable interaction between these structural decisions e.g., in global manufacturing companies. We present a formal definition of the plant location and technology acquisition problem and provide a mathematical model. We describe the analytical properties of the model, which lead to the development of a solution algorithm. Progressive piecewise linear underestimation constitutes the backbone of our solution algorithm. The arising subproblems are amenable to a dual-based approach. We report on a set of experiments that improved our understanding of the interaction among facility design decisions and showed that the computational performance of the proposed solution procedure is quite satisfactory.

Deterministic Capacity Expansion Under Deterioration

This paper considers the deterministic capacity expansion problem with deterioration. First, the paper extends the conventional model by incorporating capacity deterioration effects and operating costs that are a function of: (i) the age of the facility and (ii) the extent of utilization of capacity. The model still has the useful properties of the standard model, enabling the use of efficient algorithms. Second, the paper extends well-known planning horizon results to this model using a novel dual-based approach, which also provides additional intuition. Finally, interesting computational results are provided that compare optimal and near-optimal forecast horizon values. The impact of capacity deterioration and convexity in variable costs on forecast horizons is also considered.

The technology and management of multi-species fisheries

This paper presents a dual-based approach for determining forms of stock management in a multi-species or multiproduct fishery. A dual revenue function is specified and estimated for the New Englan...

A Dual-based Heuristic for the Simple Facility Location Problem With Stochastic Demand

Abstract This paper deals with a stochastic version of the simple facility location problem where the demands of customers are random variables. Under the assumption of step function type distribution of demands, the problem is shown to be approximated by a mixed 0–1 linear programming problem. A heuristic procedure is developed to solve the problem, which successfully extends the well known dual-based approach by Bilde & Krarup, and Erlenkotter. Computational results with 20 test problems are presented to demonstrate the effectiveness of the proposed heuristic.

The Multi-Commodity Facilities Location Problem

The Multi-commodity Location Problem was one of the ‘multi-level’ location problems introduced by Warszawski and Peer in their paper on building sites. Methods of solution are developed for this problem via a dual-based approach and via a Lagrangean dual-based approach with hill-climbing. Numerical results are presented.