Iterative Descent Method Research Articles

Effect of the local branching strategy on the descent method: The case of the generalized multiple knapsack with setup

In this paper, we investigate the use of the local branching strategy into an iterative descent method for tackling the generalized multiple knapsack problem with setup. An instance of the problem is composed of a set of classes containing items, and a set of available knapsacks. Its objective consists in selecting the subsets of items belonging to the classes with a maximum objective value: each item is characterized with its profit and weight while a class is characterized with its cost such that a given item may be selected whenever its corresponding class is activated, and an item can be configured (setup) for a single knapsack. The proposed iterative method starts by solving a series of reduced subproblems built by adding a series of cardinality constraints. Next, the local branching strategy is iteratively employed for highlighting the efficiency of the method. Finally, the behavior of the method is evaluated on a set of instances extracted from the literature, where its achieved bounds are compared to those reached by the best methods available in the literature. A statistical analysis is also provided showing the high performance of the method. New bounds are discovered.

IMAT: The Iterative Medial Axis Transform

AbstractWe present the iterative medial axis transform (IMAT), an iterative descent method that constructs a medial axis transform (MAT) for a sparse, noisy, oriented point cloud sampled from an object's boundary. We first establish the equivalence between the traditional definition of the MAT of an object, i.e., the set of centres and corresponding radii of all balls maximally inscribed inside the object, with an alternative characterization matching the boundary enclosing the union of the balls with the object boundary. Based on this boundary equivalence characterization, a new MAT algorithm is proposed, in which an error function that reflects the difference between the two boundaries is minimized while restricting the number of balls to within some a priori specified upper limit. An iterative descent method with guaranteed local convergence is developed for the minimization that is also amenable to parallelization. Both quantitative and qualitative analyses of diverse 2D and 3D objects demonstrate the noise robustness, shape fidelity, and representation efficiency of the resulting MAT.

Iterative detail-preserving thin-cloud removal method for panchromatic remote sensing images

Optical remote sensing images are frequently affected by clouds, haze, and mist in the atmosphere. We introduce an iterative minimization light-cloud removal method designed for the specific quality improvement needs of military reconnaissance panchromatic remote sensing images. The proposed method is required to fulfill the military reconnaissance demands for improvements in the quality of panchromatic high-resolution images while guaranteeing high fidelity between the restored and observed images. A heuristic approach based on contrast enhancement is proposed to solve the thin-cloud removal problem. We design the target function of a minimization algorithm that contains a fidelity term, a contrast penalty term, and an information loss penalty term. By minimizing the target function with the iterative steepest descent method, a high-quality image can be restored from the observed satellite cloudy image, and the details are preserved by the penalty terms. The application of our iterative method to Gaofen-1 (GF-1) and Ziyuan-3 (ZY-3) satellite data shows that the iterative method was applicable to GF-1 and ZY-3 satellite and the data showed that for panchromatic remote sensing images, the proposed method could reduce satellite image degradation caused by haze and thin clouds while preserving the details in the observed images.

Iterative Descent Method for Generalized Leontief Model

In this article, we show that under certain condition the generalized Leontief model is solvable by iterative descent method based on infeasible interior point algorithm. We prove the convergence of the method from strictly positive starting point. A numerical example is presented to demonstrate the performance of the algorithm.

Performance Analysis of Parameter Estimator Based on Closed-Form Newton Method for Ultrawideband Positioning

Ultrawideband (UWB) is considered as a promising technology in indoor positioning due to its accurate time resolution and good penetration through objects. Since the functional model of UWB positioning is nonlinear, iterative algorithms are often considered for solving the nonlinear problem. With a rough initial value, we can obtain the optimal solution by the way of continuous iteration. As an iterative descent method of high efficiency, the Gauss–Newton method is widely used to estimate the position. However, in the UWB indoor positioning, since the location information of the user is rather limited, it is not easy to get a good initial value for iteration. Besides, the positioning system is prone to become ill-posed. These factors make the Gauss–Newton method not easy to converge to a global optimal solution or even diverge, especially under ill-conditioned positioning configuration. Furthermore, the linearization of the positioning functional model results in biased least-squares estimation. The Gauss–Newton method only includes the first-order Taylor expansion of distance equations. The bias comes from neglected higher order terms. In this study, the closed-form Newton method which considers the high order partial derivatives is introduced and proposed. The simulation and measurement experiments are implemented to analyze the performance of the closed-form Newton method in UWB positioning. Both the initial value factor and geometric configuration factor are discussed. Experimental results are given to demonstrate that the closed-form Newton method can converge to the global optimal solution with better convergence and higher efficiency regardless of whether the initial iteration value is reasonable or not. Meanwhile, the closed-form Newton method can improve the accuracy of positioning results, particularly when the anchors are not uniformly and symmetrically distributed, or the ranging error is relatively large. The study shows that closed-form of the Newton method has better convergence and positioning performance than the Gauss–Newton method, especially under ill-conditioned positioning configuration or relatively low measurement precision, even though it adds a little computational cost.

Bias Analysis of Parameter Estimator Based on Gauss-Newton Method Applied to Ultra-Wideband Positioning

Ultra-wideband (UWB) is considered as a promising technology in short-distance indoor wireless positioning due to its accurate time resolution and good penetration through objects. Since the functional model of UWB positioning is nonlinear, the optimal solution is generally estimated by the way of continuous iteration. As an iterative descent method of high efficiency, the Gauss–Newton method is widely used to estimate the position. The nonlinear distance equations are linearized, and the solution can be found iteratively. Therefore, the nonlinear least-squares solution is generally biased even if the observations are normally distributed. In outdoor satellite positioning, the ranging distances are long enough so that the bias caused by nonlinearity is very small. However, in UWB positioning, the relative ranging error is large, and the positioning system is prone to become ill-posed, hence the bias due to nonlinearity is not negligible. In this study, both the statistical factor and geometric factor for bias in the nonlinear least-squares estimator of UWB positioning are discussed. In order to assess whether the linearized model is sufficiently approximate for the positioning estimation, a hypothesis test criterion based on Mahalanobis distance is proposed. The simulation and measurement experiments are performed to analyze the factors affecting the bias in UWB positioning. Experimental results are given to demonstrate that the linearization is valid and the bias in UWB positioning estimation can be neglected for the relatively high measurement precision. Moreover, for a positioning configuration, when the anchors are evenly distributed, the amount of nonlinearity is orthogonal to the ranging space of the design matrix, the UWB positioning estimation tends to be unbiased. Meanwhile, the hypothesis test based on Mahalanobis distance is carried out to determine the validity of the linearized model. When the bias is large for UWB positioning, the bias estimate can be used to correct the estimator to guarantee the unbiasedness for UWB positioning. Furthermore, the correction of parameter estimator bias is more effective in the case of relatively low measurement precision or ill-conditioned configuration.

Toward Collision-Free and Efficient Coordination for Automated Vehicles at Unsignalized Intersection

With the significant advance of vehicle-to-everything (V2X) techniques, unsignalized intersection coordination has been widely recognized to facilitate the development of automated vehicles (AVs) for the intelligent transportation system. However, how to guarantee driving safety while improving the unsignalized intersection management efficiency is a challenging issue. In this article, we investigate the collision-free and efficient V2X-enabled AV scheduling problem at unsignalized intersections. First, by dividing the intersection zone into different collision sections (CSs), we formulate the intersection collision-free model into an absolute value programming (AVP) problem, which is proved to be NP-hard. We consider both nonplatoon and platoon traffic scenarios, and unlike previous algorithms, which require to control all the AVs at each scheduling step with computational intractability, our scheduling algorithm can assign a feasible time for each arriving AV with low complexity. Further, we propose an alternately iterative descent method (AIDM) to solve the AVP problem by assigning the optimal entering time for each arriving AV. Through extensive simulations with various traffic data generated by SUMO, we demonstrate that our proposed AIDM algorithm can significantly enhance the scheduling performance in terms of passing delay and scheduling throughput. Even though the AIDM algorithm achieves the same level of transportation performances with the state-of-the-art algorithm, it advances dramatically in computational complexity and communication overhead, which is easier to be implemented in practice.

A Compressive Sensing-Based Colocated MIMO Radar Power Allocation and Waveform Design

Compressive sensing (CS) is a widely used technique for (multiple) target detection in multiple input multiple output (MIMO) radars. In this paper, our goal is to enhance the quality of CS-based detection techniques for a colocated MIMO radar with given location of transmit and receive nodes. Our approach is to design the transmit waveforms based on the given antenna locations and optimally allocate the total power budget among the transmitters. The design criterion in this paper is the coherence of the resulting sensing matrix. Based on this criterion, we derive and solve a convex optimization problem for power allocation. For waveform design, however, the direct method studied is non-convex, and although iterative descent methods could be used to achieve suboptimal solutions, they might be unfeasible waveforms (e.g., waveforms with high peak to average power ratios). Here, we first show that the coherence measure depends only on the covariance matrix of the waveforms (rather than the waveforms themselves). Next, we introduce three different convex programs to achieve the covariance matrix. Finally, we transform the covariance matrix into realistic waveforms; although multiple solutions exist, a closed-form expression for all possible solutions is available. Specifically, we design the waveforms by applying practical constraints such as constant modulus. Simulation results confirm that the introduced designs improve the detection performance of a CS-MIMO radar.

A family of quasi-Newton methods for unconstrained optimization problems

ABSTRACTIn this paper, we present a class of approximating matrices as a function of a scalar parameter that includes the Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno methods as special cases. A powerful iterative descent method for finding a local minimum of a function of several variables is described. The new method maintains the positive definiteness of the approximating matrices. For a region in which the function depends quadratically on the variables, no more than n iterations are required, where n is the number of variables. A set of computational results that verifies the superiority of the new method are presented.

Peak‐to‐average power ratio reduction in multiple‐input multiple‐output orthogonal frequency‐division multiple access systems using geodesic descent method

In this study, the authors consider a peak-to-average power ratio (PAPR) reduction for orthogonal frequency-division multiplexing systems based on the decomposition of the set of subcarriers in subsets of subcarriers, denoted resource blocks, each one weighted by a different complex factor. They present a new iterative sphere-geodesic descent method for obtaining these weighting factors so as to minimise the PAPR of the transmitted signals. This method, which they term geodesic descent method, efficiently makes use of the Riemannian structure of the power constraint. The authors’ performance results show that the proposed technique provides good trade-off between the PAPR reduction and the bit error rate performance, for both uncoded and coded scenarios.

An industrial process optimization approach based on input and output statistical data analysis

A major concern in process industry is to improve the quality of final products. This is highly dependent on the raw materials composition and on the industrial processes settings. The main issue in operating these systems is to identify a correlation between the process settings and the quality of the final product. The aim of this work is to first build a model of the industrial process. Then we search for a set of parameters in order to minimize an objective function based on the quality of the final product. Since the number of parameters of these processes may be important (several hundred in some instances), we perform a Support Vector machines Regression (SVR) method as multiple regression to model the manufacturing process, based on the input (various settings) and output (product quality) data. The settings optimization using the regression function is done by a heuristic. It is based on an iterative descent method applied iteratively on each parameter. The proposed approach is used on a fluidized bed combustion boiler in the context of paper industry. The experiment confirms the efficiency of the approach.

A numerical minimization scheme for the complex Helmholtz equation

We use the work of Milton, Seppecher, and Bouchitte on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate the method with numerical experiments.

Margin adaptive optimization in multi-user MISO-OFDM systems under rate constraint

In this paper, we focus on the total transmission power minimization problem for downlink beamforming multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems while ensuring each user's QoS requirement. Although the linear integer programming (LIP) solution we formulate provides the performance upper bound of the margin adaptive (MA) optimization problem, it is hard to be implemented in practice due to its high computational complexity. By regarding each user's equivalent channel gain as approximate independent values and using iterative descent method, we present a heuristic MA resource allocation algorithm. Simulation results show that the proposed algorithm efficiently converges to the local optimum, which is very close to the performance of the optimal LIP solution. Compared with existing space division multiple access (SDMA) OFDM systems with or without adaptive resource allocation, the proposed algorithm achieves significant performance improvement by exploiting the frequency diversity and multi-user diversity in down-link multiple-input single-output (MISO) OFDM systems.

Minimizing and Stationary Sequences of Constrained Optimization Problems

We study some fundamental asymptotic properties associated with the well-posedness of constrained optimization problems. Emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic Karush--Kuhn--Tucker (KKT) optimality conditions. Unlike the subdifferential approach used by Auslender, Cominetti, and Crouzeix, we use a residual function approach that is closely tied to the theory of error bounds; this approach handles constraints explicitly and allows the effective treatment of infeasible, stationary sequences. The asymptotic KKT conditions provide an asymptotic optimality certificate for inequality constrained programs. Specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with Hölderian minima are discussed. An application of the results to the convergence of a family of Newton-type iterative descent methods involving singular quasi-Newton matrices for solving a constrained minimization problem is also presented.

Solving constraint satisfaction problems using hybrid evolutionary search

We combine the concept of evolutionary search with the systematic search concepts of arc revision and hill climbing to form a hybrid system that quickly finds solutions to static and dynamic constraint satisfaction problems (CSPs). Furthermore, we present the results of two experiments. In the first experiment, we show that our evolutionary hybrid outperforms a well-known hill climber, the iterative descent method (IDM), on a test suite of 750 randomly generated static CSPs. These results show the existence of a "mushy region" which contains a phase transition between CSPs that are based on constraint networks that have one or more solutions and those based on networks that have no solution. In the second experiment, we use a test suite of 250 additional randomly generated CSPs to compare two approaches for solving CSPs. In the first method, all the constraints of a CSP are known by the hybrid at run-time. We refer to this method as the static method for solving CSPs. In the second method, only half of the constraints of a CSPs are known at run-time. Each time that our hybrid system discovers a solution that satisfies all of the constraints of the current network, one additional constraint is added. This process of incrementally adding constraints is continued until all the constraints of a CSP are known by the algorithm or until the maximum number of individuals has been created. We refer to this second method as the dynamic method for solving CSPs. Our results show hybrid evolutionary search performs exceptionally well in the presence of dynamic (incremental) constraints, then also illuminate a potential hazard with solving dynamic CSPs.

Solving Randomly Generated Static and Dynamic Fuzzy Constraint Networks using Microevolutionary Hill-Climbing

ABSTRACTIn this article we combine the concept of microevolutionary hill-climbing search with the systematic search concept of arc revision to form a hybrid system that quickly finds solutions to static and dynamic Fuzzy Constraint Truth Optimization Problems (FCTOPs). Furthermore, we present the results of two experiments. In the first experiment, our microevolutionary hybrid system outperforms a modified version of a well known hill-climber, the Iterative Descent Method (IDM), on a test suite of 500 randomly generated static Fuzzy Constraint Networks.In the second experiment, we compare two methods for solving FCTOPs using a test suite of an additional 250 randomly generated FCTOPs. In the first method, all the constraints of a FCTOP are known by the hybrid system at run-time. We refer to this method as the static method for solving FCTOPs. In the second method, only half of the constraints of a FCTOP are known at run-time. Each time our hybrid system discovers a solution that satisfies all of the const...

On the general contact problem of an inflated nonlinear plane membrane

The nonaxisymmetric contact problem between an inflated membrane and a rigid indentor is considered. The membrane is assumed to be an initially thin plane sheet. The shape and the boundary of the contact region and the configuration of the deformed membrane under both inflation and indentation are found by employing the minimum potential energy principle subjected to an inequality constraint condition. A slack variable that converts the inequality constraint to an equality constraint condition is introduced. The coordinate functions that describe the deformed configurations of the membrane are assumed to be represented by a series of geometric admissible functions with unknown coefficients. The unknown coefficients that minimize the total potential energy are determined by Fletcher and Powell's[1] iterative descent method for finding the minimum of a function of multivariables.

A Note on the Application of Davidon's Method to Nonlinear Regression Problems

In the statistical literature, Booth et al. (Ref. 1) and Hartley (Ref. 2) developed modifications of the well-known Gauss-Newton method of iterative solution and applied it to estimate a set of parameters for nonlinear regression problems by least squares. To ensure convergence of his method, Hartley restricts its applicability to problems that satisfy his assumptions. Herein, we describe a method that removes two of Hartley's three assumptions, thereby making it applicable to those problems for which his method fails to converge. The method we propose is based upon an algorithm due to Davidon (Ref. 3). It is an iterative descent method whose rate of convergence is quadratic in the limit. It was later modified by Fletcher and Powell (Ref. 4), and has been used for locating an unconstrained local minimum of a function of several variables. Fletcher and Powell's account of Davidon's method has been found to be very useful when first derivatives of the function are available. However, in realistic situations it frequently is practically impossible to calculate first derivatives. Therefore, the method we describe is a modification of the basic Fletcher and Powell method due to McGill (Ref. 5) and Taylor (Ref. 6) and includes the case for which the gradient is not given analytically. Our concern is with the application and introduction of the modified Davidon method (MDM) to nonlinear regression problems of the type discussed by Hartley. We have omitted all proofs concerning stability and rate of convergence of the method and refer the reader to the excellent paper of Fletcher and Powell for such proofs. Included is a description of the MDM, a description of ways of terminating the iterations, and an illustration of the method with the numerical example Hartley used in his paper. We show the results for both the case of analytic and approximate gradient. Also included are results obtained in applying both methods to a second example, which involves the estimation of six parameters for an exponential regression function. For this example, it is shown that the method described in this note converges when Hartley's method fails to converge.

Power-spectrum identification in terms of rational models

A technique is described for the identification of unknown power-spectral densities from sampled data in terms of a rational function of z . The problem is reduced to the minimization of a function of K parameters, where K is the order of the numerator of the model. This criterion, called the minimum residual criterion, reduces to the maximum likelihood criterion when the observed signal is Gaussian. A computational technique is described for minimizing this function which uses filtering and correlation to obtain the gradient and an iterative descent method due to M. J. D. Powell for minimization. Some computational results are given in which the method is compared with all-pole and conventional spectrum estimation techniques.