Convergence Properties Of Algorithms Research Articles

Strong global convergence properties of algorithms for nonlinear symmetric cone programming

Strong global convergence properties of algorithms for nonlinear symmetric cone programming

Optimization of household medical waste recycling logistics routes: Considering contamination risks.

The escalating generation of household medical waste, a byproduct of industrialization and global population growth, has rendered its transportation and logistics management a critical societal concern. This study delves into the optimization of routes for vehicles within the household medical waste logistics network, a response to the imperative of managing this waste effectively. The potential for environmental and public health hazards due to improper waste disposal is acknowledged, prompting the incorporation of contamination risk, influenced by transport duration, waste volume, and wind velocity, into the analysis. To enhance the realism of the simulation, traffic congestion is integrated into the vehicle speed function, reflecting the urban roads' variability. Subsequently, a Bi-objective mixed-integer programming model is formulated to concurrently minimize total operational costs and environmental pollution risks. The complexity inherent in the optimization problem has motivated the development of the Adaptive Hybrid Artificial Fish Swarming Algorithm with Non-Dominated Sorting (AH-NSAFSA). This algorithm employs a sophisticated approach, amalgamating congestion distance and individual ranking to discern optimal solutions from the population. It incorporates a decay function to facilitate an adaptive iterative process, enhancing the algorithm's convergence properties. Furthermore, it leverages the concept of crossover-induced elimination to preserve the genetic diversity and overall robustness of the solution set. The empirical evaluation of AH-NSAFSA is conducted using a test set derived from the Solomon dataset, demonstrating the algorithm's capability to generate feasible non-dominated solutions for household medical waste recycling path planning. Comparative analysis with the Non-dominated Sorted Artificial Fish Swarm Algorithm (NSAFSA) and Non-dominated Sorted Genetic Algorithm II (NSGA-II) across metrics such as MID, SM, NOS, and CT reveals that AH-NSAFSA excels in MID, SM, and NOS, and surpasses NSAFSA in CT, albeit slightly underperforming relative to NSGA-II. The study's holistic approach to waste recycling route planning, which integrates cost-effectiveness with pollution risk and traffic congestion considerations, offers substantial support for enterprises in formulating sustainable green development strategies. AH-NSAFSA offers an eco-efficient, holistic approach to medical waste recycling, advancing sustainable management practices.

The Analysis of Optimization Algorithms: A Dissipativity Approach

Optimization problems in engineering and applied mathematics are typically solved in an iterative fashion, by systematically adjusting the variables of interest until an adequate solution is found. The iterative algorithms that govern these systematic adjustments can be viewed as a control system. In control systems, the output in measured and the input is adjusted using feedback to drive the error to zero. Similarly, in iterative algorithms, the optimization objective is evaluated and the candidate solution is adjusted to drive it toward the optimal point. Choosing an algorithm that works well for a variety of optimization problems is akin to robust controller design. Just as dissipativity theory can be used to analyze the stability properties of control systems, it can also be used to analyze the convergence properties of iterative algorithms. By defining an appropriate notion of "energy" that dissipates with every iteration of the algorithm, the convergence properties of the algorithm can be characterized. This article formalizes the connection between iterative algorithms and control systems and shows through examples how dissipativity theory can be used to analyze the performance of many classes of optimization algorithms. This control-theoretic viewpoint enables the selection and tuning of optimization algorithms to be performed in an automated and systematic way.

The case for model-driven interpretability of delay-based congestion control protocols

Analyzing and interpreting the exact behavior of new delay-based congestion control protocols with complex non-linear control loops is exceptionally difficult in highly variable networks such as cellular networks. This paper proposes a Model-Driven Interpretability (MDI) congestion control framework, which derives a model version of a delay-based protocol by simplifying a congestion control protocol's response into a guided random walk over a two-dimensional Markov model. We demonstrate the case for the MDI framework by using MDI to analyze and interpret the behavior of two delay-based protocols over cellular channels: Verus and Copa. Our results show a successful approximation of throughput and delay characteristics of the protocols' model versions across variable network conditions. The learned model of a protocol provides key insights into an algorithm's convergence properties.

A Least-Squares Temporal Difference based method for solving resource allocation problems

Value function approximation has a central role in Approximate Dynamic Programming (ADP) to overcome the so-called curse of dimensionality associated to real stochastic processes. In this regard, we propose a novel Least-Squares Temporal Difference (LSTD) based method: the “Multi-trajectory Greedy LSTD” (MG-LSTD). It is an exploration-enhanced recursive LSTD algorithm with the policy improvement embedded within the LSTD iterations. It makes use of multi-trajectories Monte Carlo simulations in order to enhance the system state space exploration.This method is applied for solving resource allocation problems modeled via a constrained Stochastic Dynamic Programming (SDP) based framework. In particular, such problems are formulated as a set of parallel Birth–Death Processes (BDPs). Some operational scenarios are defined and solved to show the effectiveness of the proposed approach. Finally, we provide some experimental evidence on the MG-LSTD algorithm convergence properties in function of its key-parameters.

On Maintaining Linear Convergence of Distributed Learning and Optimization Under Limited Communication

In distributed optimization and machine learning, multiple nodes coordinate to solve large problems. To do this, the nodes need to compress important algorithm information to bits so that it can be communicated over a digital channel. The communication time of these algorithms follows a complex interplay between a) the algorithm's convergence properties, b) the compression scheme, and c) the transmission rate offered by the digital channel. We explore these relationships for a general class of linearly convergent distributed algorithms. In particular, we illustrate how to design quantizers for these algorithms that compress the communicated information to a few bits while still preserving the linear convergence. Moreover, we characterize the communication time of these algorithms as a function of the available transmission rate. We illustrate our results on learning algorithms using different communication structures, such as decentralized algorithms where a single master coordinates information from many workers and fully distributed algorithms where only neighbours in a communication graph can communicate. We conclude that a co-design of machine learning and communication protocols are mandatory to flourish machine learning over networks.

Stochastic sub-gradient algorithm for distributed optimization with random sleep scheme

In this paper, we consider a distributed convex optimization problem of a multi-agent system with the global objective function as the sum of agents’ individual objective functions. To solve such an optimization problem, we propose a distributed stochastic sub-gradient algorithm with random sleep scheme. In the random sleep scheme, each agent independently and randomly decides whether to inquire the sub-gradient information of its local objective function at each iteration. The algorithm not only generalizes distributed algorithms with variable working nodes and multi-step consensus-based algorithms, but also extends some existing randomized convex set intersection results. We investigate the algorithm convergence properties under two types of stepsizes: the randomized diminishing stepsize that is heterogeneous and calculated by individual agent, and the fixed stepsize that is homogeneous. Then we prove that the estimates of the agents reach consensus almost surely and in mean, and the consensus point is the optimal solution with probability 1, both under randomized stepsize. Moreover, we analyze the algorithm error bound under fixed homogeneous stepsize, and also show how the errors depend on the fixed stepsize and update rates.

PARALLEL OPTIMIZATION ALGORITHM FOR SMOOTH CONVEX OPTIMIZATION OVER FIXED POINT SETS OF QUASI-NONEXPANSIVE MAPPINGS

Smooth convex optimization problems are solved over fixed point sets of quasi-nonexpansive mappings by using a distributed optimization technique. This is done for a networked system with an operator, who manages the system, and a finite number of users, by solving the problem of minimizing the sum of the operator's and users' differentiable, convex objective functions over the intersection of the operator's and users' fixed point sets of quasi-nonexpansive mappings. Under the assumption that the operator can communicate with all users, a parallel optimization algorithm can be devised that enables the operator to find a solution to the problem without using all user objective functions and quasi-nonexpansive mappings. This algorithm does not use proximity operators, in contrast to conventional parallel proximal algorithms. Moreover, it can optimize over fixed point sets of quasi-nonexpansive mappings, in contrast to conventional fixed point algorithms. Investigation of the algorithm's convergence properties for a constant step-size rule reveals that, with a small constant step size, it approximates the solution to the problem. Consideration of the case in which the step-size sequence is diminishing demonstrates that the algorithm converges to the problem solution. Application of the algorithm to network bandwidth allocation based on an operational policy is shown to make the network more stable and reliable.

Markov Chain Analyses of Random Local Search and Evolutionary Algorithm

Theoretical studies of evolutionary algorithms (EAs) have been developed by researchers whose main interests are convergence properties of algorithms. In this paper, we report the computational complexity of an algorithm that is a variant of (1+1) EA, called Random Local Search (RLS). While a standard EA uses a mutation of flipping each bit in a parent string, RLS flips exactly one bit at each step. It has been noted the close resemblance of RLS with the coupon collector problem (CCP). CCP has a long history of probabilistic research, and many interesting results are obtained. This study makes use of such results with some modifications. We also show some useful results representing the evolution process of (1+1) EA.

Richardson-Lucy Deblurring for Scenes under a Projective Motion Path

This paper addresses how to model and correct image blur that arises when a camera undergoes ego motion while observing a distant scene. In particular, we discuss how the blurred image can be modeled as an integration of the clear scene under a sequence of planar projective transformations (i.e., homographies) that describe the camera's path. This projective motion path blur model is more effective at modeling the spatially varying motion blur exhibited by ego motion than conventional methods based on space-invariant blur kernels. To correct the blurred image, we describe how to modify the Richardson-Lucy (RL) algorithm to incorporate this new blur model. In addition, we show that our projective motion RL algorithm can incorporate state-of-the-art regularization priors to improve the deblurred results. The projective motion path blur model, along with the modified RL algorithm, is detailed, together with experimental results demonstrating its overall effectiveness. Statistical analysis on the algorithm's convergence properties and robustness to noise is also provided.

Convergence of Algorithms in Optimization and Solutions of Nonlinear Equations

It is desirable that an algorithm in unconstrained optimization converges when the guessed initial position is anywhere in a large region containing a minimum point. Furthermore, it is useful to have a measure of the rate of convergence which can easily be computed at every point along a trajectory to a minimum point. The Lyapunov function method provides a powerful tool to study convergence of iterative equations for computing a minimum point of a nonlinear unconstrained function or a solution of a system of nonlinear equations. It is surprising that this popular and powerful tool in the study of dynamical systems is not used directly to analyze the convergence properties of algorithms in optimization. We describe the Lyapunov function method and demonstrate how it can be used to study convergence of algorithms in optimization and in solutions of nonlinear equations. We develop an index which can measure the rate of convergence at all points along a trajectory to a minimum point and not just at points in a small neighborhood of a minimum point. Furthermore this index can be computed when the calculations are being carried out.

Resolution method for mixed integer bi-level linear problems based on decomposition technique

In this article, we propose a new algorithm for the resolution of mixed integer bi-level linear problem (MIBLP). The algorithm is based on the decomposition of the initial problem into the restricted master problem (RMP) and a series of problems named slave problems (SP). The proposed approach is based on Benders decomposition method where in each iteration a set of variables are fixed which are controlled by the upper level optimization problem. The RMP is a relaxation of the MIBLP and the SP represents a restriction of the MIBLP. The RMP interacts in each iteration with the current SP by the addition of cuts produced using Lagrangian information from the current SP. The lower and upper bound provided from the RMP and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is within a small difference ?. In the case of MIBLP Karush---Kuhn---Tucker (KKT) optimality conditions could not be used directly to the inner problem in order to transform the bi-level problem into a single level problem. The proposed decomposition technique, however, allows the use of KKT conditions and transforms the MIBLP into two single level problems. The algorithm, which is a new method for the resolution of MIBLP, is illustrated through a modified numerical example from the literature. Additional examples from the literature are presented to highlight the algorithm convergence properties.

A convergent dynamic window approach to obstacle avoidance

The dynamic window approach (DWA) is a well-known navigation scheme developed by Fox et al. and extended by Brock and Khatib. It is safe by construction, and has been shown to perform very efficiently in experimental setups. However, one can construct examples where the proposed scheme fails to attain the goal configuration. What has been lacking is a theoretical treatment of the algorithm's convergence properties. Here we present such a treatment by merging the ideas of the DWA with the convergent, but less performance-oriented, scheme suggested by Rimon and Koditschek. Viewing the DWA as a model predictive control (MPC) method and using the control Lyapunov function (CLF) framework of Rimon and Koditschek, we draw inspiration from an MPC/CLF framework put forth by Primbs to propose a version of the DWA that is tractable and convergent.

A PROVABLY CONVERGENT DYNAMIC WINDOW APPROACH TO OBSTACLE AVOIDANCE

The dynamic window approach is a well known navigation scheme developed in Fox et al. (1997) and extended in Brock and Khatib (1999). It is safe by construction and has been shown to perform very efficiently in experimental setups. However, one can construct examples where the proposed scheme fails to attain the goal configuration. What has been lacking is a theoretical treatment of the algorithm's convergence properties. Here we present such a treatment. Furthermore, we highlight the similarity between the Dynamic Window Approach and the Control Lyapunov Function and Receding Horizon Control synthesis put forth by Primbs et al. (1999). Inspired by these similarities we propose a version of the Dynamic Window Approach that is provably convergent.

Using localizing learning to improve supervised learning algorithms

Slow learning of neural-network function approximators can frequently be attributed to interference, which occurs when learning in one area of the input space causes unlearning in another area. To mitigate the effect of unlearning, this paper develops an algorithm that adjusts the weights of an arbitrary, nonlinearly parameterized network such that the potential for future interference during learning is reduced. This is accomplished by the reduction of a biobjective cost function that combines the approximation error and a term that measures interference. An analysis of the algorithm's convergence properties shows that learning with this algorithm reduces future unlearning. The algorithm can be used either during online learning or can be used to condition a network to have immunity from interference during a future learning stage. A simple example demonstrates how interference manifests itself in a network and how less interference can lead to more efficient learning. Simulations demonstrate how this new learning algorithm speeds up the training in various situations due to the extra cost function term.

A Feasible Directions Algorithm for Optimal Control Problems with State and Control Constraints: Convergence Analysis

In this paper we describe an optimization algorithm for the computation of solutions to optimal control problems with control, state, and terminal constraints. Inequality and equality constraints are dealt with by means of feasible directions and exact penalty approaches, respectively. We establish a general convergence property of the algorithm which makes no reference to the existence of accumulation points; in this analysis the compactness of the space of relaxed controls is used only to guarantee boundedness of the sequence of penalty parameters. We also demonstrate that relaxed accumulation points of sequences generated by the algorithm satisfy standard first-order necessary conditions of optimality. The algorithm contains a number of computation saving features, including an $\varepsilon$-active strategy for dealing with the infinite dimensional inequality constraints. Our convergence analysis provides techniques for studying the convergence properties of related optimization algorithms in which direction-finding subproblems involve the approximation of directional derivatives of the Chebyshev functional associated with state constraints. A companion paper provides details of implementation and numerical examples.

Global and Superlinear Convergence of a Restricted Class of Self-Scaling Methods with Inexact Line Searches, for Convex Functions

This paper studies the convergence properties of algorithms belonging to the class of self-scaling (SS) quasi-Newton methods for unconstrained optimization. This class depends on two parameters, say t_k and τ_k, for which the choice τ_k=1 gives the Broyden family of unscaled methods, where t_k=1 corresponds to the well known DFP method. We propose simple conditions on these parameters that give rise to global convergence with inexact line searches, for convex objective functions. The q-superlinear convergence is achieved if further restrictions on the scaling parameter are introduced. These convergence results are an extension of the known results for the unscaled methods. Because the scaling parameter is heavily restricted, we consider a subclass of SS methods which satisfies the required conditions. Although convergence for the unscaled methods with t_k ≥ 1 is still an open question, we show that the global and superlinear convergence for SS methods is possible and present, in particular, a new SS-DFP method.

Superlinear convergence of infeasible-interior-point methods for linear programming

At present the interior-point methods of choice are primal—dual infeasible-interior-point methods, where the iterates are kept positive, but allowed to be infeasible. In practice, these methods have demonstrated superior computational performance. From a theoretical point of view, however, they have not been as thoroughly studied as their counterparts — feasible-interior-point methods, where the iterates are required to be strictly feasible. Recently, Kojima et al., Zhang, Mizuno and Potra studied the global convergence of algorithms in the primal—dual infeasible-interior-point framework. In this paper, we continue to study this framework, and in particular we study the local convergence properties of algorithms in this framework. We construct parameter selections that lead toQ-superlinear convergence for a merit function andR-superlinear convergence for the iteration sequence, both at rate 1 +τ whereτ can be arbitrarily close to one.

Convergence properties of the generalized fuzzy c-means clustering algorithms

Fuzzy c-means (FCM) clustering algorithms have been widely used to solve clustering problems. Yang and Yu [1] extended these to optimization procedures with respect to any probability distribution. They showed that the optimal cluster centers are the fixed points of these generalized FCM clustering algorithms. The convergence properties of algorithms are the important theoretical issue. In this paper, we present convergence properties of the generalized FCM clustering algorithms. These are global convergence, local convergence, and its rate of convergence.

Convergence properties of Gauss-Newton iterative algorithms in nonlinear image restoration

Nonlinear image restoration is a complicated problem that is receiving increasing attention. Since every image formation system involves a built-in nonlinearity, nonlinear image restoration finds applications in a wide variety of research areas. Iterative algorithms have been well established in the corresponding linear restoration problem. In this paper, a generalized analysis regarding the convergence properties of nonlinear iterative algorithms is introduced. Moreover, the applications of the iterative Gauss-Newton (GN) algorithm in nonlinear image restoration are considered. The convergence properties of a general class of nonlinear iterative algorithms are rigorously studied through the Global Convergence Theorem (GCT). The derivation of the convergence properties is based on the eigen-analysis, rather than on the norm analysis. This approach offers a global picture of the evolution and the convergence properties of an iterative algorithm. Moreover, the generalized convergence-analysis introduced may be interpreted as a link towards the integration of minimization and projection algorithms. The iterative GN algorithm for the solution of the least-squares optimization problem is introduced. The computational complexity of this algorithm is enormous, making its implementation very difficult in practical applications. Structural modifications are introduced, which drastically reduce the computational complexity while preserving the convergence rate of the GN algorithm. With the structural modifications, the GN algorithm becomes particularly useful in nonlinear optimization problems. The convergence properties of the algorithms introduced are readily derived, on the basis of the generalized analysis through the GCT. The application of these algorithms on practical problems, is demonstrated through several examples.