Incremental Subgradient Method Research Articles

The Application of Lagrange Relaxation on Taxi Dispatchments During Evening Rush Hours

The city transportation status is directly related to the taxi supply arrangements during evening rush hours. The operation times, operation nature, operation requirements, vehicle resource types and quantities will have big influence on taxi drivers’ dispatchments, it is comparatively difficult to dispatch reasonably. This paper uses Lagrange Relaxation to solve the problem of taxi supply and demand during evening rush hours. NP-Hardness is the difficult point of dispatchments, thus Lagrange method is used to result an optimal solution on taxi dispatchments within reasonable calculating time, while the bottom limits of the approximate optimal solution and the approximate solution are resulted as well. This method can effectively solve the problem of the balance of taxi supply and demand during evening rush hours. Based on this, the dual problem is solved by using incremental subgradient method, while an improved strategy is proposed to rise the convergence rate.

Path-based incremental target level algorithm on Riemannian manifolds

ABSTRACTIn this paper, we propose and analyse a path-based incremental target level algorithm for minimizing a constrained convex optimization on complete Riemannian manifolds with lower bounded sectional curvature, where the object function consists of the sum of a large number of component functions. This algorithm extends, to the context of Riemannian manifolds, an incremental subgradient method emplying a version of dynamic stepsize rule. Some convergence results and iteration-complexity bounds of the algorithm are established.

Incremental quasi-subgradient methods for minimizing the sum of quasi-convex functions

The sum of ratios problem has a variety of important applications in economics and management science, but it is difficult to globally solve this problem. In this paper, we consider the minimization problem of a sum of a number of nondifferentiable quasi-convex component functions over a closed and convex set, which includes the sum of ratios problem as a special case. The sum of quasi-convex component functions is not necessarily to be quasi-convex, and so, this study goes beyond quasi-convex optimization. Exploiting the structure of the sum-minimization problem, we propose a new incremental subgradient method for this problem and investigate its convergence properties to a global optimal solution when using the constant, diminishing or dynamic stepsize rules and under a homogeneous assumption and the H\"{o}lder condition of order $p$. To economize on the computation cost of subgradients of a large number of component functions, we further propose a randomized incremental subgradient method, in which only one component function is randomly selected to construct the subgradient direction at each iteration. The convergence properties are obtained in terms of function values and distances of iterates from the optimal solution set with probability 1. The proposed incremental subgradient methods are applied to solve the sum of ratios problem, as well as the multiple Cobb-Douglas productions efficiency problem, and the numerical results show that the proposed methods are efficient for solving the large sum of ratios problem.

An Incremental Subgradient Method on Riemannian Manifolds

In this paper, we propose and analyze an incremental subgradient method with a diminishing stepsize rule for a convex optimization problem on a Riemannian manifold, where the object function consisted of the sum of a large number of component functions. This type of function is useful in lots of fields. We establish an important inequality about the sequence generated by the method, provided the sectional curvature of the manifold is nonnegative. Using the inequality, we prove Proposition 3.1, and then we obtain some convergence results of the incremental subgradient method.

Ex-Post Max-Min Fairness of Generalized AGV Mechanisms

We generalize the standard Arrow-d'Aspremont-Gerard-Varet (AGV) mechanism to balance the net payoffs received by all agents, while maintaining Bayesian incentive compatibility, ex-post efficiency, and ex-post budget balance of the standard AGV mechanism. In a private-value setting with independent agents' types and the principal's cost, we formulate a convex optimization problem to find the mechanism (that achieves ex-post max-min fairness) over a set of parameterized generalized AGV mechanisms, through maximizing the expected value of the minimum ex-post net payoff. We reformulate the convex program as a linear program that can be effectively solved when the number of agents is small. When the number of agents is large, we propose to solve the formulated convex program through the incremental subgradient method. Numerical results on two action models show that the proposed mechanism significantly outperforms the standard AGV mechanism in terms of the expected minimum ex-post payoff.

String-averaging incremental subgradients for constrained convex optimization with applications to reconstruction of tomographic images

We present a method for non-smooth convex minimization which is based on subgradient directions and string-averaging techniques. In this approach, the set of available data is split into sequences (strings) and a given iterate is processed independently along each string, possibly in parallel, by an incremental subgradient method (ISM). The end-points of all strings are averaged to form the next iterate. The method is useful to solve sparse and large-scale non-smooth convex optimization problems, such as those arising in tomographic imaging. A convergence analysis is provided under realistic, standard conditions. Numerical tests are performed in a tomographic image reconstruction application, showing good performance for the convergence speed when measured as the decrease ratio of the objective function, in comparison to classical ISM.

Incremental subgradient method for nonsmooth convex optimization with fixed point constraints

This paper proposes an incremental subgradient method for solving the problem of minimizing the sum of nondifferentiable, convex objective functions over the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. The proposed algorithm can work in nonsmooth optimization over constraint sets onto which projections cannot be always implemented, whereas the conventional incremental subgradient method can be applied only when a constraint set is simple in the sense that the projection onto it can be easily implemented. We first study its convergence for a constant step size. The analysis indicates that there is a possibility that the algorithm with a small constant step size approximates a solution to the problem. Next, we study its convergence for a diminishing step size and show that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. Moreover, we show the whole sequence generated by the algorithm with a diminishing step size strongly converges to the solution to the problem under certain assumptions. We also give examples of real applied problems which satisfy the assumptions in the convergence theorems and numerical examples to support the convergence analyses.

Proximal point algorithms for nonsmooth convex optimization with fixed point constraints

The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem. With the focus on fixed point approximation techniques, two algorithms are devised for solving the problem. One blends an incremental subgradient method, which is a useful algorithm for nonsmooth convex optimization, with a Halpern-type fixed point iteration algorithm. The other is based on an incremental subgradient method and the Krasnosel’skiĭ–Mann fixed point algorithm. It is shown that any weak sequential cluster point of the sequence generated by the Halpern-type algorithm belongs to the solution set of the problem and that there exists a weak sequential cluster point of the sequence generated by the Krasnosel’skiĭ–Mann-type algorithm, which also belongs to the solution set. Numerical comparisons of the two proposed algorithms with existing subgradient methods for concrete nonsmooth convex optimization show that the proposed algorithms achieve faster convergence.

Two Conclusions of Points Produced by Incremental Sub-Gradient Method for the Non-Differentiable Optimal Problem in Hilbert Space

The paper mainly research incremental sub-gradient method for the non-differentiable optimal problem in Hilbert space. And prove two conclusions about the sequence generated by this method.

Two Conclusions of Points Produced by Incremental Sub-Gradient Method for the Non-Differentiable Optimal Problem in Hilbert Space

The paper mainly research incremental sub-gradient method for the non-differentiable optimal problem in Hilbert space. And prove two conclusions about the sequence generated by this method.

Convergence of the Surrogate Lagrangian Relaxation Method

Studies have shown that the surrogate subgradient method, to optimize non-smooth dual functions within the Lagrangian relaxation framework, can lead to significant computational improvements as compared to the subgradient method. The key idea is to obtain surrogate subgradient directions that form acute angles toward the optimal multipliers without fully minimizing the relaxed problem. The major difficulty of the method is its convergence, since the convergence proof and the practical implementation require the knowledge of the optimal dual value. Adaptive estimations of the optimal dual value may lead to divergence and the loss of the lower bound property for surrogate dual values. The main contribution of this paper is on the development of the surrogate Lagrangian relaxation method and its convergence proof to the optimal multipliers, without the knowledge of the optimal dual value and without fully optimizing the relaxed problem. Moreover, for practical implementations, a stepsizing formula that guarantees convergence without requiring the optimal dual value has been constructively developed. The key idea is to select stepsizes in a way that distances between Lagrange multipliers at consecutive iterations decrease, and as a result, Lagrange multipliers converge to a unique limit. At the same time, stepsizes are kept sufficiently large so that the algorithm does not terminate prematurely. At convergence, the lower-bound property of the surrogate dual is guaranteed. Testing results demonstrate that non-smooth dual functions can be efficiently optimized, and the new method leads to faster convergence as compared to other methods available for optimizing non-smooth dual functions, namely, the simple subgradient method, the subgradient-level method, and the incremental subgradient method.

New Decentralized Positioning Schemes for Wireless Sensor Networks Based on Recursive Least-Squares Optimization

This letter presents a recursive least-squares (RLS) optimization process to solve the positioning problem for wireless sensor networks, where a recursive-in-time cost function is first defined and then an iterative decentralized algorithm is derived. It is shown that the RLS scheme is equivalent to the incremental subgradient method with an appropriate variable step size for each iteration. With this, we further replace each gradient value by its "sign" to form a reduced-complexity RLS (RCRLS) scheme. Simulation results show that RCRLS has some performance degradation as compared to RLS, but both of them outperform previous related methods.

Acceleration Method Combining Broadcast and Incremental Distributed Optimization Algorithms

This paper considers a networked system consisting of an operator, which manages the system, and a finite number of subnetworks with all users, and studies the problem of minimizing the sum of the operator's and all users' objective functions over the intersection of the operator's and all users' constraint sets. When users in each subnetwork can communicate with each other, they can implement an incremental subgradient method that uses the transmitted information from their neighbor users. Since the operator can communicate with users in the subnetworks, it can implement a broadcast distributed algorithm that uses all available information in the subnetworks. We present an iterative method combining broadcast and incremental distributed optimization algorithms. Our method has faster convergence and a wider range of application compared with conventional distributed algorithms. We also prove that under certain assumptions our method converges to the solution to the problem in the sense of the strong topology of a Hilbert space. Moreover, we numerically compare our method with the conventional distributed algorithms in the case of a data storage system. The numerical results demonstrate the effectiveness and fast convergence of our method.

A Direct Splitting Method for Nonsmooth Variational Inequalities

We propose a direct splitting method for solving a nonsmooth variational inequality in Hilbert spaces. The weak convergence is established when the operator is the sum of two point-to-set and monotone operators. The proposed method is a natural extension of the incremental subgradient method for nondifferentiable optimization, which strongly explores the structure of the operator using projected subgradient-like techniques. The advantage of our method is that any nontrivial subproblem must be solved, like the evaluation of the resolvent operator. The necessity to compute proximal iterations is the main difficulty of other schemes for solving this kind of problem.

A Randomized Incremental Subgradient Method for Distributed Optimization in Networked Systems

We present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to Nedić and Bertsekas [SIAM J. Optim., 12 (2001), pp. 109–138]. Our novel algorithm is particularly suitable for distributed implementation and execution, and possible applications include distributed optimization, e.g., parameter estimation in networks of tiny wireless sensors. The stochastic component in the algorithm is described by a Markov chain, which can be constructed in a distributed fashion using only local information. We provide a detailed convergence analysis of the proposed algorithm and compare it with existing, both deterministic and randomized, incremental subgradient methods.

The effect of deterministic noise in subgradient methods

In this paper, we study the influence of noise on subgradient methods for convex constrained optimization. The noise may be due to various sources, and is manifested in inexact computation of the subgradients and function values. Assuming that the noise is deterministic and bounded, we discuss the convergence properties for two cases: the case where the constraint set is compact, and the case where this set need not be compact but the objective function has a sharp set of minima (for example the function is polyhedral). In both cases, using several different stepsize rules, we prove convergence to the optimal value within some tolerance that is given explicitly in terms of the errors. In the first case, the tolerance is nonzero, but in the second case, the optimal value can be obtained exactly, provided the size of the error in the subgradient computation is below some threshold. We then extend these results to objective functions that are the sum of a large number of convex functions, in which case an incremental subgradient method can be used.

Incremental Stochastic Subgradient Algorithms for Convex Optimization

This paper studies the effect of stochastic errors on two constrained incremental subgradient algorithms. The incremental subgradient algorithms are viewed as decentralized network optimization algorithms as applied to minimize a sum of functions, when each component function is known only to a particular agent of a distributed network. First, the standard cyclic incremental subgradient algorithm is studied. In this, the agents form a ring structure and pass the iterate in a cycle. When there are stochastic errors in the subgradient evaluations, sufficient conditions on the moments of the stochastic errors are obtained that guarantee almost sure convergence when a diminishing step-size is used. In addition, almost sure bounds on the algorithm's performance with a constant step-size are also obtained. Next, the Markov randomized incremental subgradient method is studied. This is a noncyclic version of the incremental algorithm where the sequence of computing agents is modeled as a time nonhomogeneous Markov chain. Such a model is appropriate for mobile networks, as the network topology changes across time in these networks. Convergence results and error bounds for the Markov randomized method in the presence of stochastic errors for diminishing and constant step-sizes are obtained.

A Study On Distributed Model Predictive Consensus

We investigate convergence properties of a proposed distributed model predictive control (DMPC) scheme, where agents negotiate to compute an optimal consensus point using an incremental subgradient method based on primal decomposition as described in Johansson et al. [2006, 2007]. The objective of the distributed control strategy is to agree upon and achieve an optimal common output value for a group of agents in the presence of constraints on the agent dynamics using local predictive controllers. Stability analysis using a receding horizon implementation of the distributed optimal consensus scheme is performed. Conditions are given under which convergence can be obtained even if the negotiations do not reach full consensus.