Primal-dual Approach Research Articles

A primal-dual approach of weak vector equilibrium problems

AbstractIn this paper we provide some new sufficient conditions that ensure the existence of the solution of a weak vector equilibrium problem in Hausdorff topological vector spaces ordered by a cone. Further, we introduce a dual problem and we provide conditions that assure the solution set of the original problem and its dual coincide. We show that many known problems from the literature can be treated in our primal-dual model. We provide several coercivity conditions in order to obtain the existence of the solution of the primal-dual problems without compactness assumption. We apply the obtained results to perturbed vector equilibrium problems.

Welfare maximization with production costs: A primal dual approach

We study online auctions with production costs using an online primal dual framework. The seller allocates items to buyers and can produce multiple copies of each item subject to a non-decreasing marginal cost per copy. The buyers have arbitrary valuation functions and arrive one by one online in some arbitrary order. The goal is to design an online mechanism that maximizes the social welfare, that is, the sum of the buyers' values less the total production cost. For any strictly convex and differentiable production cost function, we characterize the optimal competitive ratio achievable by online mechanisms and, more generally, algorithms without incentive guarantees. We show that online posted pricing mechanisms, which are incentive compatible, can achieve competitive ratios arbitrarily close to the optimal, and construct lower bound instances on which no online algorithms, not necessarily incentive compatible, can do better.

An Online Algorithm for Data Collection by Multiple Sinks in Wireless-Sensor Networks

Data collection by multiple sinks is a fundamental problem in wireless-sensor networks. Existing works focused on designing the optimal offline methods provided that the number and positions of sensors and sinks (or trajectories of mobile sinks) are predetermined. This may not be practical, because although sensors are cheap, sinks are quite expensive in reality. A more practical scenario is that sinks are deployed step by step during the network operation due budget constraints, and we do not know the number, positions, and capacities of sinks a priori. In this paper, we investigate the problem of data collection with multiple sinks, and design a suboptimal online algorithm via a primal-dual approach, requiring very little priori knowledge. We theoretically derive the competitive ratio of the online algorithm, and further improve it by finding the optimal sink location with an approximation ratio. We also analyze the computational complexity of the improved approach. Extensive simulations are conducted to demonstrate the performance of the proposed online algorithm and performance-complexity tradeoff.

Joint Dynamic Rate Control and Transmission Scheduling for Scalable Video Multirate Multicast Over Wireless Networks

In this paper we consider the time-varying characteristics of practical wireless networks and propose a joint dynamic rate allocation and transmission scheduling optimization scheme for scalable video multirate multicast based on opportunistic routing (OR) and network coding. With OR the decision of optimal routes for scalable video coding layered streaming is integrated into the joint optimization formulation. The network throughput is also increased by taking advantage of the broadcast nature of the wireless shared medium and by network coding operations in intermediate nodes. To maximize the overall video reception quality among all destinations the proposed scheme can jointly optimize the video reception rate the associated routes to different destinations and the time fraction scheduling of transmitter sets that are concurrently transmitting in the shared wireless medium. By using dual decomposition and primal-dual update approach we develop a cross-layer algorithm in a fully distributed manner. Simulation results demonstrate significant network multicast throughput improvement and adaptation to dynamic network changes relative to existing optimization schemes.

The Value of Being Socially Responsible: A Primal-Dual Approach

This study attempts to formalize and explain the process through which Corporate Social Responsibility (CSR) is created incorporating it into a production framework for characterizing the technology and representing the transformation process of multiple inputs into multiple outputs. CSR is one of these outputs and plays a mitigation role. Our framework allows for analyzing technical efficiency and deriving a system of internal shadow values for CSR to evaluate the overall value as well as the marginal impact of engaging in socially responsible activities for the firm. The empirical application focuses on the food and beverage manufacturing sector which is particularly interesting as it faces a broad array of both very specific and more common CSR challenges. Our results document that in the sample of 175 firms included in the analysis efficiency levels are very high with approximately 75 percent of the firms being technically efficient. We also find that the average shadow value of socially responsible activities is positive, implying that the cost of implementing these activities is compensated by their beneficial mitigating effect. When focusing on the value of CSR at the margin, our analysis shows that the average marginal value of increasing the socially responsible commitment is positive, suggesting that more CSR is considered advantageous for adding value to the firm. Conversely, the average marginal value of decreasing the CSR effort is negative, indicating that lower levels of CSR are perceived so costly and damaging that firms want to be compensated for reducing their engagement in socially responsible activities.

Constraint programming and operations research

We present an overview of the integration of constraint programming (CP) and operations research (OR) to solve combinatorial optimization problems. We interpret CP and OR as relying on a common primal-dual solution approach that provides the basis for integration using four main strategies. The first strategy tightly interweaves propagation from CP and relaxation from OR in a single solver. The second applies OR techniques to domain filtering in CP. The third decomposes the problem into a portion solved by CP and a portion solved by OR, using CP-based column generation or logic-based Benders decomposition. The fourth uses relaxed decision diagrams developed for CP propagation to help solve dynamic programming models in OR. The paper cites a significant fraction of the literature on CP/OR integration and concludes with future perspectives.

Real-Time Procurement Strategies of a Proactive Distribution Company With Aggregator-Based Demand Response

In this paper, we present a real-time trading framework for distribution networks where a rational aggregator is identified as a broker by contracting with individual demands and dealing with the distribution company. Demand response capability is characterized by the coexistence of elastic and inelastic demand components. A one-leader multi-follower bi-level model is proposed to derive the procurement strategies, i.e., the upper-level problem intends to maximize the profit of the proactive distribution company, while the lower-level expresses the profit maximization per rational aggregator. The proposed model is then transformed into a solvable mathematical program with equilibrium constraints through a primal-dual approach. A modified 33-bus distribution network is utilized to demonstrate the effectiveness of the proposed model.

Asynchronous Multiagent Primal-Dual Optimization

We present a framework for asynchronously solving convex optimization problems over networks of agents which are augmented by the presence of a centralized cloud computer. This framework uses a Tikhonov-regularized primal-dual approach in which the agents update the system's primal variables and the cloud updates its dual variables. To minimize coordination requirements placed upon the system, the times of communications and computations among the agents are allowed to be arbitrary, provided they satisfy mild conditions. Communications from the agents to the cloud are likewise carried out without any coordination in their timing. However, we require that the cloud keeps the dual variable's value synchronized across the agents, and a counterexample is provided that demonstrates that this level of synchrony is indeed necessary for convergence. Convergence rate estimates are provided in both the primal and dual spaces, and simulation results are presented that demonstrate the operation and convergence of the proposed algorithm.

Routing Optimization with Efficient Second Order Distributed Approach Using Congestion Control Rules

Distributed joint congestion control and routing optimization has received a significant amount of attention recently. To date, however, most of the existing schemes follow a key idea called the back-pressure algorithm. Despite having many salient features, the first-order sub gradient nature of the back-pressure based schemes results in slow convergence and poor delay performance. To overcome these limitations, the present study was made as first attempt at developing a second-order joint congestion control and routing optimization framework that offers utility-optimality, queue-stability, fast convergence, and low delay. Contributions in this project are three-fold. The present study propose a new second-order joint congestion control and routing framework based on a primal-dual interior-point approach and established utility-optimality and queue-stability of the proposed second-order method. The results of present study showed that how to implement the proposed second-order method in a distributed fashion.

Primal-dual approach to optical tomography with discretized path integral with efficient formulations.

.We propose an efficient optical tomography with discretized path integral. We first introduce the primal-dual approach to solve the inverse problem formulated as a constraint optimization problem. Next, we develop efficient formulations for computing Jacobian and Hessian of the cost function of the constraint nonlinear optimization problem. Numerical experiments show that the proposed formulation is faster () than the previous work with the log-barrier interior point method () for the Shepp–Logan phantom with a grid size of , while keeping the quality of the estimation results (root-mean-square error increasing by up to 12%).

An Efficient Regression Approach to Solving the Dual Problems of Dynamic Programs

In recent years, information relaxation and duality in dynamic programs have been studied extensively, and the resulted primal-dual approach has become a powerful procedure in solving dynamic programs by providing lower-upper bounds on the optimal value function. Theoretically, with the so called value-based optimal dual penalty, the optimal value function could be recovered exactly via strong duality; however, in practice, generating tight dual bounds usually requires good approximations of the optimal dual penalty, which could be time-consuming due to the conditional expectation terms that need to be estimated via nested simulation. In this paper, we will develop an efficient regression approach to approximating the optimal dual penalty in a non-nested manner, by exploring the structure of the feasible dual penalty space. The resulted approximation maintains to be a dual feasible penalty, leading to a valid dual bound on the optimal value function. We show that the proposed approach is computationally efficient, and the resulted dual penalty leads to a numerically tractable dual problem.

Joint optimization of downlink and D2D transmissions for SVC streaming in cooperative cellular networks

We study the problem of broadcasting scalable video coded (SVC) streams in cellular networks, where all user equipments (UEs) require the same video content and cooperate with each other. To take the advantage of channel diversity gains, the base station (BS) uses network coding to generate linear combinations of video packets for broadcasting in the downlink. Once receiving sufficient number of the combinations, a UE can relay packets to others via device-to-device (D2D) connections. The optimal downlink and D2D transmission arrangement problem is formulated as a mixed integer non-linear programming (MINLP) problem, which becomes difficult to solve when the number of UEs increases. Then we convert the MINLP problem to a quasi-convex optimization problem using a continuous step function, and a primal-dual decomposition approach is used to solve it in a distributed way. In addition, we consider the influence of the transmission energy consumption to the optimal solution. Simulation results show that the proposed approach achieves near-optimal performance.

Beating Level-Set Methods for 5-D Seismic Data Interpolation: A Primal-Dual Alternating Approach

Acquisition cost is a crucial bottleneck for seismic workflows, and low-rank formulations for data interpolation allow practitioners to “fill in” data volumes from critically subsampled data acquired in the field. Tremendous size of seismic data volumes required for seismic processing remains a major challenge for these techniques. Residual-constrained formulations require less parameter tuning when the target noise floor is known. We propose a new approach to solve residual constrained formulations for interpolation. We represent the data volume in a compressed manner using low-rank matrix factors, and build a block-coordinate algorithm with constrained convex subproblems that are solved with a primal-dual splitting scheme. The develop optimization framework works on the whole seismic temporal frequency slices and does not require windowing or nontrivial sorting of seismic data. The new approach is competitive with state of the art level-set algorithms that interchange the role of objectives with constraints. We use the new algorithm to successfully interpolate a large scale 5-D seismic data volume (upto $10^{10}$ data points), generated from the geologically complex synthetic 3-D Compass velocity model, where 80% of the data have been removed. We also develop a robust extension of the primal-dual approach to deal with the outliers (or noise) in the data.

Optimal transmission for scalable video coded streaming in cellular wireless networks with the cooperation of local peer-to-peer network

We study the problem of transmitting scalable video coded streams in a hybrid scenario including a global cellular network and a local peer-to-peer network. Given desirable channel condition in cellular downlink, some cellular user equipments can fetch the video chunks via the cellular base station directly, and also provide relay service via the local peer-to-peer network for other user equipments unable to meet their basic video quality. To take the advantage of channel diversity gains, the method of random linear coding is adopted to generate linear combinations of the video chunks for transmission in both the cellular and peer-to-peer networks. To optimize the cellular and peer-to-peer transmission arrangement, we first formulate it as a mixed integer nonlinear programming problem, which becomes more complicated when the number of user equipments increases. Then, we convert it to a quasi-convex optimization problem by approximating the indicator function with a continuous step function, and it can be solved by a centralized approach. Furthermore, a primal-dual decomposition approach is developed and a distributed algorithm is proposed accordingly. Simulation results show that the proposed approach achieves a near-optimal solution.

Linearized primal-dual methods for linear inverse problems with total variation regularization and finite element discretization

Linear inverse problems with total variation regularization can be reformulated as saddle-point problems; the primal and dual variables of such a saddle-point reformulation can be discretized in piecewise affine and constant finite element spaces, respectively. Thus, the well-developed primal-dual approach (a.k.a. the inexact Uzawa method) is conceptually applicable to such a regularized and discretized model. When the primal-dual approach is applied, the resulting subproblems may be highly nontrivial and it is necessary to discuss how to tackle them and thus make the primal-dual approach implementable. In this paper, we suggest linearizing the data-fidelity quadratic term of the hard subproblems so as to obtain easier ones. A linearized primal-dual method is thus proposed. Inspired by the fact that the linearized primal-dual method can be explained as an application of the proximal point algorithm, a relaxed version of the linearized primal-dual method, which can often accelerate the convergence numerically with the same order of computation, is also proposed. The global convergence and worst-case convergence rate measured by the iteration complexity are established for the new algorithms. Their efficiency is verified by some numerical results.

Strategy-making for a proactive distribution company in the real-time market with demand response

This paper proposes a methodology to optimize the trading strategies of a proactive distribution company (PDISCO) in the real-time market by mobilizing the demand response. Each distribution-level demand is considered as an elastic one. To capture the interrelation between the PDISCO and the real-time market, a bi-level model is presented for the PDISCO to render continuous offers and bids strategically. The upper-level problem expresses the PDISCO’s profit maximization, while the lower-level problem minimizes the operation cost of the transmission-level real-time market. To solve the proposed model, a primal-dual approach is used to translate this bi-level model into a single-level mathematical program with equilibrium constraints. Results of case studies are reported to show the effectiveness of the proposed model.

A primal–dual method for SVM training

Training support vector machines (SVM) consists of solving a convex quadratic problem (QP) with one linear equality and box constraints. In this paper, we solve this QP by a primal–dual approach that combines the adaptive method with an interior point method. To initialize the algorithm, a procedure of an interior point method is used to construct an initial support. The proposed approach provides an efficient implementation of a new algorithm that exploits the advantage of the adaptive method for training SVM problems. It is based on the principle of the support and the suboptimality estimate. Experimental results confirm the efficiency of our approach over state-of-the-art SVM algorithms such as SMO, LIBSVM and SVMLight for medium-sized problems.

Joint Congestion Control and Routing Optimization: An Efficient Second-Order Distributed Approach

Distributed joint congestion control and routing optimization has received a significant amount of attention recently. To date, however, most of the existing schemes follow a key idea called the back-pressure algorithm. Despite having many salient features, the first-order subgradient nature of the back-pressure based schemes results in slow convergence and poor delay performance. To overcome these limitations, in this paper, we make a first attempt at developing a second-order joint congestion control and routing optimization framework that offers utility-optimality, queue-stability, fast convergence, and low delay. Our contributions in this paper are three-fold: i) we propose a new second-order joint congestion control and routing framework based on a primal-dual interior-point approach; ii) we establish utility-optimality and queue-stability of the proposed second-order method; and iii) we show how to implement the proposed second-order method in a distributed fashion.

Trading strategies for distribution company with stochastic distributed energy resources

This paper proposes a methodology to address the trading strategies of a proactive distribution company (PDISCO) engaged in the transmission-level (TL) markets. A one-leader multi-follower bilevel model is presented to formulate the gaming framework between the PDISCO and markets. The lower-level (LL) problems include the TL day-ahead market and scenario-based real-time markets, respectively with the objectives of maximizing social welfare and minimizing operation cost. The upper-level (UL) problem is to maximize the PDISCO’s profit across these markets. The PDISCO’s strategic offers/bids interactively influence the outcomes of each market. Since the LL problems are linear and convex, while the UL problem is non-linear and non-convex, an equivalent primal–dual approach is used to reformulate this bilevel model to a solvable mathematical program with equilibrium constraints (MPEC). The effectiveness of the proposed model is verified by case studies.

A Combinatorial, Primal-Dual Approach to Semidefinite Programs

Semidefinite programs (SDPs) have been used in many recent approximation algorithms. We develop a general primal-dual approach to solve SDPs using a generalization of the well-known multiplicative weights update rule to symmetric matrices. For a number of problems, such as S parsest C ut and B alanced S eparator in undirected and directed weighted graphs, M in U n C ut and M in 2CNF D eletion , this yields combinatorial approximation algorithms that are significantly more efficient than interior point methods. The design of our primal-dual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones. Our ideas have proved useful in quantum computing, especially the recent result of Jain et al. [2011] that QIP = PSPACE.