Dual Algorithm Research Articles

Proximal primal\u2013dual best approximation algorithm with memory

We propose a new modified primal–dual proximal best approximation method for solving convex not necessarily differentiable optimization problems. The novelty of the method relies on introducing memory by taking into account iterates computed in previous steps in the formulas defining current iterate. To this end we consider projections onto intersections of halfspaces generated on the basis of the current as well as the previous iterates. To calculate these projections we are using recently obtained closed-form expressions for projectors onto polyhedral sets. The resulting algorithm with memory inherits strong convergence properties of the original best approximation proximal primal–dual algorithm. Additionally, we compare our algorithm with the original (non-inertial) one with the help of the so called attraction property defined below. Extensive numerical experimental results on image reconstruction problems illustrate the advantages of including memory into the original algorithm.

Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs

In this work, we propose an approximating scheme based on the proximal point algorithm, for solving generalized fractional programs (GFP) by their continuous reformulation, also known to as partial dual counterparts of GFP. Bundle dual algorithms are then derived from this scheme. We prove the convergence and the rate of convergence of these algorithms. As for dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of \begin{document}$(P)$\end{document} , and a sequence of approximate solutions that converges to a solution of the dual problem. For certain classes of problems, the convergence is at least linear.

A bandwidth allocation and energy-optimal transmission rate scheduling scheme in multi-services wireless networks

The energy-saving of mobile devices during their application offloading process has always been the research hotspot in the field of mobile cloud computing (MCC). In this paper, we focus on the scenario where multiple mobile devices with MCC and non-MCC services coexist. A bandwidth allocation and the corresponding transmission rate scheduling schemes are proposed with the objectives of simultaneously maximizing the overall system throughput and minimizing the energy consumption of individual mobile device with MCC service. To allocate the bandwidth to all mobile devices, two different algorithms are proposed, i.e., 0–1 integer programming algorithm and Lagrange dual algorithm. The transmission rate scheduling scheme for mobile device with MCC service is presented based on reverse order iteration method. The numerical results suggest that energy consumed by individual mobile device with MCC service can be remarkably saved while the overall system throughput can also be maximized. Moreover, the results show that 0–1 integer programming algorithm can get greater system throughput but has higher computational complexity, which means the algorithm is more suitable for small-scale systems, whereas Lagrange dual algorithm can achieve a good balance between the performance and computational complexity.

Dual estimation based vocal tract shape computation

This paper presents a new method for direct estimation of vocal tract shape from the speech signal. The method computes cross-sectional areas of uniform-length cylindrical tubes comprising the vocal tract. Cross-sectional areas are calculated from reflection coefficients at tube junctions whose values depend on the areas of adjoining tubes. A new state space representation of the speech production system has been formulated in which reflection coefficients are parameters. The state space model has been constructed using state equations of the glottal flow signal and vocal tract formulated from Liljencrants–Fant model and concatenated tube model respectively. Dual extended Kalman filtering algorithm has been used for estimation of unknown parameters of the system. The estimated reflection coefficients are then used to compute cross-sectional areas of the vocal tract. The performance of proposed technique has been compared to an existing shape estimation method proposed by Wakita. For both synthesized and natural speech signals, the performance of proposed method has been found to be comparable to the existing one. Nevertheless, the Kalman filter algorithm used in proposed method has provisions to tune measurement noise covariance which can be adjusted based on the noise level in speech. Therefore, the performance of proposed method has been seen to be comparatively more robust to noise than the existing technique.

Primal–dual interior-point method for linear optimization based on a kernel function with trigonometric growth term

In this paper, we propose a large-update primal–dual interior-point algorithm for linear optimization problems based on a new kernel function with a trigonometric growth term. By simple analysis, we prove that in the large neighbourhood of the central path, the worst case iteration complexity of the new algorithm is bounded above by , which matches the currently best known iteration bound for large-update methods. Moreover, we show that, most of the so far proposed kernel functions can be rewritten as a kernel function with trigonometric growth term. Finally, numerical experiments on some test problems confirm that the new kernel function is well promising in practice in comparison with some existing kernel functions in the literature.

A multiframe super-resolution technique based on a nonlocal Bregman distance of bilateral total variation term

In this paper, we propose a variational multiframe super-resolution (SR) model based on a nonlocal regularization term using Bregman distances. Since the SR algorithms always skip the complex spatial interactions within images, we introduce a nonlocal form of the bilateral total variation (BTV) regularization term which can take into consideration these interactions and also efficiently preserve strong edges and contours of the reconstructed high resolution (HR) image for our model. In addition, to avoid contrast loss and smoothing gray values in the SR process, we introduce Bregman distances which produce a more consistent model. Moreover, to resolve the obtained SR algorithm, we use a first-order primal dual algorithm to ensure that the convergence to the desired HR image can be achieved in a fast way. As a result, the proposed algorithm shows improved performance compared to other SR methods.

Primal and dual algorithms for optimization over the efficient set

ABSTRACTOptimization over the efficient set of a multi-objective optimization problem is a mathematical model for the problem of selecting a most preferred solution that arises in multiple criteria decision-making to account for trade-offs between objectives within the set of efficient solutions. In this paper, we consider a particular case of this problem, namely that of optimizing a linear function over the image of the efficient set in objective space of a convex multi-objective optimization problem. We present both primal and dual algorithms for this task. The algorithms are based on recent algorithms for solving convex multi-objective optimization problems in objective space with suitable modifications to exploit specific properties of the problem of optimization over the efficient set. We first present the algorithms for the case that the underlying problem is a multi-objective linear programme. We then extend them to be able to solve problems with an underlying convex multi-objective optimization problem. We compare the new algorithms with several state of the art algorithms from the literature on a set of randomly generated instances to demonstrate that they are considerably faster than the competitors.

Optimization of Hydropower Dam Design Parameters for Adaptation to Climate Change using stochastic dual dynamic programming algorithm (SDDP)

Optimization of Hydropower Dam Design Parameters for Adaptation to Climate Change using stochastic dual dynamic programming algorithm (SDDP)

Optimization of Hydropower Dam Design Parameters for Adaptation to Climate Change using stochastic dual dynamic programming algorithm (SDDP)

Optimization of Hydropower Dam Design Parameters for Adaptation to Climate Change using stochastic dual dynamic programming algorithm (SDDP)

Vulnerability Identification of Urban Road Network under Unexpected Congestion

An original method is used to analyze the topology structure of a road network and two indexes of road network performance, namely, connectivity and network efficiency, are used to evaluate road network robustness in medium measure and thus quantify the variety of urban road network vulnerability caused by unexpected congestion. Then, a variety of road network impedances is set as the vulnerability identification index and a network traffic flow model with a link queuing capacity constraint is established, considering such constraint on unexpected congestion condition. Finally, a Lagrange dual algorithm is designed to solve the model, whose accuracy is subsequently tested by an example. Results show that the model based on the vulnerability index can accurately recognize road network vulnerability. Invalid links caused by unexpected congestion lead to changes in road network impedance, thereby directly affecting the robustness of the entire road network.

Locating transit hubs in a multi-modal transportation network: A cluster-based optimization approach

This paper develops a mathematical model with equilibrium constraints for integrated design of transit hub locations and link establishments in a multimodal network, grounded on the cluster-based framework to capture real-world geographical and administrative features as well as to improve the overall transit-oriented system performance. By explicitly capturing multimodal transfer activities and cost implications with virtual transfer nodes and arcs, this study models transit hub as a realistic transportation infrastructure rather than a single node in the network. A genetic-algorithm-based heuristic embedded with the Augmented Lagrangian Dual algorithm is developed to solve the problem to meta-optimality.

Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces

Proximal splitting algorithms for monotone inclusions (and convex optimization problems) in Hilbert spaces share the common feature to guarantee for the generated sequences in general weak convergence to a solution. In order to achieve strong convergence, one usually needs to impose more restrictive properties for the involved operators, like strong monotonicity (respectively, strong convexity for optimization problems). In this paper, we propose a modified Krasnosel'skiĭ–Mann algorithm in connection with the determination of a fixed point of a nonexpansive mapping and show strong convergence of the iteratively generated sequence to the minimal norm solution of the problem. Relying on this, we derive a forward–backward and a Douglas–Rachford algorithm, both endowed with Tikhonov regularization terms, which generate iterates that strongly converge to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators. Furthermore, we formulate strong convergent primal–dual algorithms of forward–backward and Douglas–Rachford-type for highly structured monotone inclusion problems involving parallel-sums and compositions with linear operators. The resulting iterative schemes are particularized to the solving of convex minimization problems. The theoretical results are illustrated by numerical experiments on the split feasibility problem in infinite dimensional spaces.

Primal–Dual Algorithms for Convex Optimization via Regret Minimization

We consider a large-scale convex program with functional constraints, where interior point methods are intractable due to the problem size. The effective solution techniques for these problems permit only simple operations at each iteration, and thus are based on primal–dual first-order methods such as the Arrow–Hurwicz–Uzawa subgradient method, which utilize only gradient computations and projections at each iteration. Such primal–dual algorithms admit the interpretation of solving the associated saddle point problem arising from the Lagrange dual. We revisit these methods through the lens of regret minimization from online learning and present a flexible framework. While it is well known that two regret-minimizing algorithms can be used to solve a convex–concave saddle point problem at the standard rate of $O$ ( $1/\sqrt {T}$ ), our framework for primal–dual algorithms allows us to exploit structural properties such as smoothness and/or strong convexity and achieve better convergence rates in favorable cases. In particular, for non-smooth problems with strongly convex objectives, our primal–dual framework equipped with an appropriate modification of Nesterov’s dual averaging algorithm achieves $O$ ( $1/T$ ) convergence rate.

A Load-Shedding Model Based on Sensitivity Analysis in on-Line Power System Operation Risk Assessment

The traditional load-shedding models usually use global optimization to get the load-shedding region, which will cause multiple variables, huge computing scale and other problems. This makes it hard to meet the requirements of timeliness in on-line power system operation risk assessment. In order to solve the problems of the present load-shedding models, a load-shedding model based on sensitivity analysis is proposed in this manuscript. By calculating the sensitivity of each branch on each bus, the collection of buses which have remarkable influence on reducing the power flow on over-load branches is obtained. In this way, global optimization is turned to local optimization, which can narrow the solution range. By comprehensively considering the importance of load bus and adjacency principle regarding the electrical coupling relationship, a load-shedding model is established to get the minimum value of the load reduction from different kinds of load buses, which is solved by the primal dual interior point algorithm. In the end, different cases on the IEEE 24-bus, IEEE 300-bus and other multi-node systems are simulated. The correctness and effectiveness of the proposed load-shedding model are demonstrated by the simulation results.

Method for improving optical flow estimation

A method is proposed that improves the robustness and accuracy of optical flow estimation in real complex scenes. The method overcomes the limitations incurred by illumination variations using a combination of the brightness constancy and gradient constancy. In addition, the method improves the reliability of optical flow estimation by applying both a bilateral filter and penalty function. Furthermore, it improves the calculations and applicability of the estimated optical flow by employing a dual algorithm and coarse-to-fine scheme. We verify the proposed method using scenes from the Middlebury optical flow database and a real complex scene. The results show that the proposed method is robust to illumination variations and improves both the accuracy of the optical flow estimation and the ability to extract target edges.

A New Primal–Dual Algorithm for Minimizing the Sum of Three Functions with a Linear Operator

In this paper, we propose a new primal–dual algorithm for minimizing $$f({\mathbf {x}})+g({\mathbf {x}})+h({\mathbf {A}}{\mathbf {x}})$$ , where f, g, and h are proper lower semi-continuous convex functions, f is differentiable with a Lipschitz continuous gradient, and $${\mathbf {A}}$$ is a bounded linear operator. The proposed algorithm has some famous primal–dual algorithms for minimizing the sum of two functions as special cases. E.g., it reduces to the Chambolle–Pock algorithm when $$f=0$$ and the proximal alternating predictor–corrector when $$g=0$$ . For the general convex case, we prove the convergence of this new algorithm in terms of the distance to a fixed point by showing that the iteration is a nonexpansive operator. In addition, we prove the O(1 / k) ergodic convergence rate in the primal–dual gap. With additional assumptions, we derive the linear convergence rate in terms of the distance to the fixed point. Comparing to other primal–dual algorithms for solving the same problem, this algorithm extends the range of acceptable parameters to ensure its convergence and has a smaller per-iteration cost. The numerical experiments show the efficiency of this algorithm.

RConf(PD): Automated resource configuration of complex services in the cloud

Optimal deployment of complex services in a virtualized environment is still an open problem. These services typically consist of a set of connected components, and each component may consist of multiple instances. Each instance can in turn be run in different virtual flavors, while the service constructed by the combination of these instances must satisfy a customer Service Level Objective (SLO).While there have been efforts to answer the questions of when to provision additional resources in a running service, and how many resources are needed, the question of what (i.e., which combination of instances) should be provisioned has not been investigated yet. In this work, we offer to service providers, the first system that automatically deploys component instances for complex services such that the resource utilization at the provider’s premises is maximized in the presence of customer constraints.Our system consists of two key technologies (RConf and RConfPD), both of which build on an analytical model based on robust queueing theory to accurately model arbitrary components. With the help of this model, RConf proposes an algorithm to ultimately find the optimal combination of component instances. Our real-world experiments show that, compared to greedy approaches, RConf provisions 20% less resources in the first place, and can reduce resource wastage on live resources by up to 50%. At the same time, RConfPD trades-off some of the optimality of RConf for a computational expense 1–2 orders of magnitude below that of RConf to provision time-sensitive services. Based on a primal–dual algorithm framework RConfPD relaxes the optimality constraints of RConf and removes dominated combinations to determine an approximation for the optimal solution. Our evaluation shows that RConfPD allows for fast decisions (in many cases <1ms), while maintaining 80%–99% of the solution quality of RConf.

A perishable food supply chain problem considering demand uncertainty and time deadline constraints: Modeling and application to a high-speed railway catering service

This paper attempts to optimize the flow patterns in a perishable food supply chain network for a high-speed rail catering service. The proposed variational inequality models describe the uncertain demand on trains using the Newsvendor model and impose time deadline constraints on paths considering flow-dependent lead time. The constraints are then reformulated based on the Dirac delta function so that they can be directly dualized. An Euler algorithm with an Augmented Lagrangian Dual algorithm is developed to solve the model. A case study using 246 trains in the Beijing-Shanghai high-speed corridor is applied to demonstrate the applicability of the method.

Interior Point Algorithms in Linear Optimization

This is a survey of the results concerning the development and study of the interior point algorithms. Some families of the direct and dual algorithms are considered. These algorithms entering the domain of feasible solutions take into account the objective function, which makes it possible to obtain the first feasible solution close to the optimal solution. The main results on the theoretical justification of algorithms are given. Recommendations are proposed concerning the advantages of individual variants of algorithms on the basis of the obtained theoretical results, available experimental studies, and experience of using algorithms in the models of energy engineering. Some numerically efficient version of the polynomial optimization algorithm in the cone of the central path is also presented.

Dual neighbourhood discrete artificial bee colony algorithm for the convex polygons packing problem with constraints of equilibrium

For the optimal packing problem of convex polygons in a circular container with performance constraint, a hybrid algorithm which combines an improved ripple exploratory heuristic algorithm (IREHA) with dual-neighbourhood discrete artificial bee colony algorithm (DDABC) is proposed. IREHA improves search efficiency of the original REHA. Because of the excellent performance of artificial bee colony algorithm, which can not be directly used to optimise the discrete packing order, DDABC based on discrete dual neighbourhood structure is designed to combine with IREHA to form hybrid algorithm. The numerical experiments show that the hybrid algorithm is very effective.