Local Minimum Solutions Research Articles

Self potential data interpretation utilizing the particle swarm method for the finite 2D inclined dike: mineralized zones delineation

The self potential data interpretation is very important to delineate and trace the mineralized zones in several regions. We study how to interpret self potential anomalies due to a finite two-dimensional inclined dike using the particle swarm algorithm. However, the precise estimation of the model parameters during the inverse solution are unknown. Here, we show that the particle swarm algorithm is capable of estimating the unknown parameters with acceptable accuracy. The evaluated parameters are the polarization parameter, the depth, the inclination angle, the width, and the location of the source of the target. We found in controlled in free-noise synthetic case that the particle swarm algorithm has a remarkable capability of assessing the parameters. For a noisy case, the results also are very competitive. Furthermore, it is utilized for real mineralized zones examples from Germany and India. Our results demonstrate how the particle swarm algorithm overcomes in trapping in local minimum solutions (undesired) and go faster to the global solutions (desired). Finally, the target parameters estimated are matched with accessible geologic and geophysical information.

Topology Optimization for Multipatch Fused Deposition Modeling 3D Printing

This paper presents a hybrid topology optimization method for multipatch fused deposition modeling (FDM) 3D printing to address the process-induced material anisotropy. The ‘multipatch’ concept consists of each printing layer disintegrated into multiple patches with different zigzag-type filament deposition directions. The level set method was employed to represent and track the layer shape evolution; discrete material optimization (DMO) model was adopted to realize the material property interpolation among the patches. With this set-up, a concurrent optimization problem was formulated to simultaneously optimize the topological structure of the printing layer, the multipatch distribution, and the corresponding deposition directions. An asynchronous starting strategy is proposed to prevent the local minimum solutions caused by the concurrent optimization scheme. Several numerical examples were investigated to verify the effectiveness of the proposed method, while satisfactory optimization results have been derived.

Using a Self‐Clustering Algorithm and Type‐2 Fuzzy Controller for Multi‐robot Deployment and Navigation in Dynamic Environments

AbstractThis study proposes a novel method for multi‐robot deployment and navigation under dynamic environments. To automatically determine the location deployment of multiple robots, a grid‐based method and self‐clustering algorithm (SCA) were used to simplify the environmental information and automatically deploy robot locations. In the navigation process, a behavior selector automatically turns on towards goal mode or wall‐following mode (WFM) depending on environmental conditions. WFM control adopts an interval type‐2 fuzzy controller (IT2FC). The parameters of the IT2FC are adjusted by using the dynamic group whale optimization algorithm (DGWOA). The proposed DGWOA uses a dynamic group and Lévy flight strategy to overcome the problem of falling into a local minimum solution. Experimental results reveal that the proposed method can successfully complete navigation tasks under dynamic environments.

CANONICAL DUAL FINITE ELEMENT METHOD FOR SOLVING NONCONVEX MECHANICS AND TOPOLOGY OPTIMISATION PROBLEMS

Canonical duality theory (CDT) is a newly developed, potentially powerful methodological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m n and without a duality gap. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving general nonconvex variational problems. This thesis, first, presents a detailed study of large deformation problems in 2-D structural system. Based on the canonical duality theory, a canonical dual finite element method is applied to find a global minimization to the general nonconvex optimization problem using a new primal-dual semi-definite programming algorithm. Applications are illustrated by numerical examples with different structural designs and different external loads. Next, a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam is investigated. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using the canonical dual finite element method, a new primal-dual semi-definite programming algorithm is presented, which can be used to obtain all possible post-buckled solutions. In order to verify the triality theory, mixed meshes of different dual stress interpolation are applied to obtain the closed dimensions between discretized displacement and discretized stress. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to the external load, thickness of the beam, numerical precision, and the size of finite elements. Finally, a mathematically rigorous and computationally powerful method for solving 3-D topology optimization problems is demonstrated. This method is based on CDT developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard Knapsack problem in topology optimization can be solved deterministically in polynomial-time via a canonical penalty-duality (CPD) method to obtain precise global optimal 0-1 density distribution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Finally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly…

A cross-modal multimedia retrieval method using depth correlation mining in big data environment

Cross-media retrieval is a technology aimed at breaking through the shackles of single-mode retrieval technology, which is limited to the same multimedia form. It is also hoped to be able to search each other across the media form. Comprehensive processing of different multimedia morphological data is an urgent problem to be solved in cross-media retrieval area, in other words, the semantic relationship between potential features should be mined, which will improve their similarity. To solve the above problems, a deep correlation mining method is proposed, which trains different media features by deep learning, and then fuses the correlation between the trained features to solve the heterogeneity between different features, which will make the features of different multimedia data comparable. On this basis, Levenberg-Marquart method is applied to solve the problem that deep learning is easy to fall into local minimum solution in gradient training. Experiments on different databases show that the proposed method is effective in the field of cross-media retrieval. Compared with other advanced multimedia retrieval methods, the proposed method has achieved better retrieval results.

Multi-Objective Optimization of Off-Grid Hybrid Renewable Energy Systems in Buildings with Prior Design-Variable Screening

This work presents an optimization strategy and the cost-optimal design of an off-grid building served by an energy system involving solar technologies, thermal and electrochemical storages. Independently from the multi-objective method (e.g., utility function) and algorithm used (e.g., genetic algorithms), the optimization of this kind of systems is typically characterized by a high-dimensional variables space, computational effort and results uncertainty (e.g., local minimum solutions). Instead of focusing on advanced optimization tools to handle the design problem, the dimension of the full problem has been reduced, only considering the design variables with a high “effect” on the objective functions. An off-grid accommodation building is presented as test case: the original six-variable design problem consisting of about 300,000 possible configurations is reduced to a two-variable problem, after the analysis of 870 Monte Carlo simulations. The new problem includes only 220 possible design alternatives with a clear benefit for the multi-objective optimization algorithm. The energy-economy Pareto frontiers obtained by the original and the reduced problems overlap, showing the validity of the proposed methodology. The No-RES (no renewable energy sources) primary energy consumption can be reduced up to almost 0 kWh/(m2yr) and the net present value (NPV) after 20 years can reach 70 k€ depending on the number of photovoltaic panels and electrochemical storage size. The reduction of the problem also allows for a plain analysis of the results and the drawing of handy decision charts to help the investor/designer in finding the best design according to the specific investment availability and target performances. The configurations on the Pareto frontier are characterized by a notable electrical overproduction and a ratio between the two main design variables that goes from 4 to 8 h. A sensitivity analysis to the unitary price of the electrochemical storage reveals the robustness of the sizing criterion.

Necessary and sufficient conditions for achieving global optimal solutions in multiobjective quadratic fractional optimization problems

If $$x^*$$ is a local minimum solution, then there exists a ball of radius $$r>0$$ such that $$f(x)\ge f(x^*)$$ for all $$x\in B(x^*,r)$$ . The purpose of the current study is to identify the suitable $$B(x^*,r)$$ of the local optimal solution $$x^*$$ for a particular multiobjective optimization problem. We provide a way to calculate the largest radius of the ball centered at local Pareto solution in which this solution is optimal. In this process, we present the necessary and sufficient conditions for achieving a global Pareto optimal solution. The results of this investigation might be useful to determine stopping criteria in the algorithms development.

Progressive Representation Adaptation for Weakly Supervised Object Localization.

We address the problem of weakly supervised object localization where only image-level annotations are available for training object detectors. Numerous methods have been proposed to tackle this problem through mining object proposals. However, a substantial amount of noise in object proposals causes ambiguities for learning discriminative object models. Such approaches are sensitive to model initialization and often converge to undesirable local minimum solutions. In this paper, we propose to overcome these drawbacks by progressive representation adaptation with two main steps: 1) classification adaptation and 2) detection adaptation. In classification adaptation, we transfer a pre-trained network to a multi-label classification task for recognizing the presence of a certain object in an image. Through the classification adaptation step, the network learns discriminative representations that are specific to object categories of interest. In detection adaptation, we mine class-specific object proposals by exploiting two scoring strategies based on the adapted classification network. Class-specific proposal mining helps remove substantial noise from the background clutter and potential confusion from similar objects. We further refine these proposals using multiple instance learning and segmentation cues. Using these refined object bounding boxes, we fine-tune all the layer of the classification network and obtain a fully adapted detection network. We present detailed experimental validation on the PASCAL VOC and ILSVRC datasets. Experimental results demonstrate that our progressive representation adaptation algorithm performs favorably against the state-of-the-art methods.

Constrained Burn Optimization for the International Space Station

Long-term trajectory planning for the International Space Station (ISS) uses simple rules of thumb embedded in a manual framework to target translation burns. This approach is limited by coarse knowledge of the problem sensitivities, it lacks automation, and it is unable to optimize these burns to minimize total . These issues can be resolved by casting the ISS trajectory planning problem as a constrained burn optimization problem. Unfortunately, the six-month to two-year time frames for long-term ISS trajectory planning preclude the use of numeric partial derivatives for accurate constraint sensitivities. A framework is presented to solve the constrained burn optimization problem for the ISS in a robust, automated fashion. The cost function for this problem is total , with constraints imposed on longitude of ascending node and semimajor axis altitude. The framework uses analytic derivatives for the cost function and variational equations using state transition matrices for the constraints. A local minimum solution is found using this framework, assuming fixed dates for the burns. Burn dates are then randomly varied to find other local solutions, which are searched for a candidate global minimum solution. This process is shown to work on real-world ISS burn-planning problems.

Full waveform inversion of combined towed streamer and limited OBS seismic data: a theoretical study

In marine seismic oil exploration, full waveform inversion (FWI) of towed-streamer data is used to reconstruct velocity models. However, the FWI of towed-streamer data easily converges to a local minimum solution due to the lack of low-frequency content. In this paper, we propose a new FWI technique using towed-streamer data, its integrated data sets and limited OBS data. Both integrated towed-streamer seismic data and OBS data have low-frequency components. Therefore, at early iterations in the new FWI technique, the OBS data combined with the integrated towed-streamer data sets reconstruct an appropriate background model. And the towed-streamer seismic data play a major role in later iterations to improve the resolution of the model. The new FWI technique is tested on numerical examples. The results show that when starting models are not accurate enough, the models inverted using the new FWI technique are superior to those inverted using conventional FWI.

Joint anisotropic amplitude variation with offset inversion of PP and PS seismic data

With the increase in exploration target complexity, more parameters are required to describe subsurface properties, particularly for finely stratified reservoirs with vertical transverse isotropic (VTI) features. We have developed an anisotropic amplitude variation with offset (AVO) inversion method using joint PP and PS seismic data for VTI media. Dealing with local minimum solutions is critical when using anisotropic AVO inversion because more parameters are expected to be derived. To enhance the inversion results, we adopt a hierarchical inversion strategy to solve the local minimum solution problem in the Gauss-Newton method. We perform the isotropic and anisotropic AVO inversions in two stages; however, we only use the inversion results from the first stage to form search windows for constraining the inversion in the second stage. To improve the efficiency of our method, we built stop conditions using Euclidean distance similarities to control iteration of the anisotropic AVO inversion in noisy situations. In addition, we evaluate a time-aligned amplitude variation with angle gather generation approach for our anisotropic AVO inversion using anisotropic prestack time migration. We test the proposed method on synthetic data in ideal and noisy situations, and find that the anisotropic AVO inversion method yields reasonable inversion results. Moreover, we apply our method to field data to show that it can be used to successfully identify complex lithologic and fluid information regarding fine layers in reservoirs.

On SDP method for solving canonical dual problem in post buckling of large deformed elastic beam

This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality

Symmetric nonnegative matrix factorization (SymNMF) has important applications in data analytics problems such as document clustering, community detection and image segmentation. In this paper, we propose a novel nonconvex variable splitting method for solving SymNMF. The proposed algorithm is guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points of the nonconvex SymNMF problem. Furthermore, it achieves a global sublinear convergence rate. We also show that the algorithm can be efficiently implemented in parallel. Further, sufficient conditions are provided which guarantee the global and local optimality of the obtained solutions. Extensive numerical results performed on both synthetic and real data sets suggest that the proposed algorithm converges quickly to a local minimum solution.

Trajectory Tracking Controller of Mobile Robot under Time Variation Parameters based on Neural Networks and Stochastic Fractal Algorithm

This study suggests an adaptive Artificial Neural Network (ANN) controller that based on Stochastic Fractal Search algorithm (SFS), the purpose of the Adaptive Neural Controller (ANC) is to track a proposed velocities and path trajectory with the minimum required error, in the presence of mobile robot parameters time variation and dynamical system model uncertainties. The proposed ANC will consist of two sub-neural controllers; the Kinematic Neural feedback Controller (KNC) and the Dynamic Neural feedback Controller (DNC). The external feedback kinematic neural controller is responsible for generating velocity tracking signals that track the mobile robot linear and angular velocities depending on the robot posture error and the desired velocities, while the internal dynamic neural controller is used to enhance the mobile robot against parameters uncertainty, parameters time variation and disturbance noise. The stochastic fractal search algorithm is a Metaheuristic Optimization Algorithm (MOA) that has been used to optimize the Neural Networks (NNs) weight connections to has the behavior of an adaptive nonlinear trajectory tracking controller of a differential drive wheeled mobile robot. The proposed controller has the capability to prepare an appropriate dynamic control left and right torque signals to drive various mobile robot platforms using the same offline optimized weight connections. Metaheuristic optimization algorithms have been used due to theirs unique characteristics especially theirs free of derivative, ability to optimize discretely and continuous nonlinear functions and their ability to get rid of local minimum solution trapping.

Subspace Procrustes Analysis

Procrustes analysis (PA) has been a popular technique to align and build 2-D statistical models of shapes. Given a set of 2-D shapes PA is applied to remove rigid transformations. Later, a non-rigid 2-D model is computed by modeling the residual (e.g., PCA). Although PA has been widely used, it has several limitations for modeling 2-D shapes: occluded landmarks and missing data can result in local minima solutions, and there is no guarantee that the 2-D shapes provide a uniform sampling of the 3-D space of rotations for the object. To address previous issues, this paper proposes subspace PA (SPA). Given several instances of a 3-D object, SPA computes the mean and a 2-D subspace that can model rigid and non-rigid deformations of the 3-D object. We propose a discrete (DSPA) and continuous (CSPA) formulation for SPA, assuming that 3-D samples of an object are provided. DSPA extends the traditional PA, and produces unbiased 2-D models by uniformly sampling different views of the 3-D object. CSPA provides a continuous approach to uniformly sample the space of 3-D rotations, being more efficient in space and time. We illustrate the benefits of SPA in two different applications. First, SPA is used to learn 2-D face and body models from 3-D datasets. Experiments on the FaceWarehouse and CMU motion capture (MoCap) datasets show the benefits of our 2-D models against the state-of-the-art PA approaches and conventional 3-D models. Second, SPA learns an unbiased 2-D model from CMU MoCap dataset and it is used to estimate the human pose on the Leeds Sports dataset.

Could the Faustmann model have an interior minimum solution?

The growth of an even-aged stand usually follows a S-shaped pattern, implying that the growth function is convex when stand age is low and concave when stand age is high. Given such a growth function, the Faustmann model could in theory have multiple optima and hence an interior local minimum solution. To ensure that the rotation age at which the first derivative of the land expectation value equals zero is a maximum, it is often assumed that the growth function is concave in stand age. Yet there is no convincing argument for excluding the possibility of conducting the final harvest before the growth function changes to concave. We argue that under normal circumstances the Faustmann model does not have any interior minimum. It is neither necessary nor proper to assume that the growth function is concave in the vicinity of the optimal rotation age. When the interest rate is high, the optimal rotation may lie in the interval on which the growth function is convex, i.e. before volume or value growth culminates.

Large margin classification with indefinite similarities

Classification with indefinite similarities has attracted attention in the machine learning community. This is partly due to the fact that many similarity functions that arise in practice are not symmetric positive semidefinite, i.e. the Mercer condition is not satisfied, or the Mercer condition is difficult to verify. Examples of such indefinite similarities in machine learning applications are ample including, for instance, the BLAST similarity score between protein sequences, human-judged similarities between concepts and words, and the tangent distance or the shape matching distance in computer vision. Nevertheless, previous works on classification with indefinite similarities are not fully satisfactory. They have either introduced sources of inconsistency in handling past and future examples using kernel approximation, settled for local-minimum solutions using non-convex optimization, or produced non-sparse solutions by learning in Krein spaces. Despite the large volume of research devoted to this subject lately, we demonstrate in this paper how an old idea, namely the 1-norm support vector machine (SVM) proposed more than 15 years ago, has several advantages over more recent work. In particular, the 1-norm SVM method is conceptually simpler, which makes it easier to implement and maintain. It is competitive, if not superior to, all other methods in terms of predictive accuracy. Moreover, it produces solutions that are often sparser than more recent methods by several orders of magnitude. In addition, we provide various theoretical justifications by relating 1-norm SVM to well-established learning algorithms such as neural networks, SVM, and nearest neighbor classifiers. Finally, we conduct a thorough experimental evaluation, which reveals that the evidence in favor of 1-norm SVM is statistically significant.

Mean Field Theory in Doing Logic Programming Using Hopfield Network

Logic program and neural networks are two important perspectives in artificial intelligence. Logic describes connections among propositions. Moreover, logic must have descriptive symbolic tools to represent propositions. Meanwhile representation of neural networks on the other hand is in non-symbolic form. The objective in performing logic programming revolves around energy minimization is to reach the best global solutions. On the other hand, we usually gets local minima solutions also. In order to improve this, based on the Boltzmann machine concept, we will derive a learning algorithm in which time-consuming stochastic measurements of collerations are replaced by solutions to deterministic mean field theory (MFT) equations. The main idea of mean field algorithm is to replace the real unstable induced local field for each neuron in the network with its average local field value. Then, we build agent based modelling (ABM) by using Netlogo for this task.

Beyond the strictly orthodox/mainstream divide: Applying geodemographic analysis to a small nationwide sub-population

The use of geodemographic analysis has a long history, arguably stretching back to Charles Booth's Descriptive Map of London's Poverty, produced in 1886 and the published classification of areas has invariably been based on all residents. The work described in this paper, however, is novel in the use of geodemographic analysis to focus on a single minority group within a national census. This paper describes the development of a methodology which allows geodemographic analysis to be applied to unevenly distributed minority sub-populations, overcoming two particular issues: finding a suitable geographic base to ensure data reliability; and developing a methodology to avoid known weaknesses in certain clustering techniques, specifically distortion caused by outlier cases and generation of sub-optimal local minimum solutions. The approach, which includes a visual element to final classification selection, has then been applied to establish the degree to which the Jewish population in an area is similar in character to, or differs from, Jews living in other areas of England and Wales, using data from the 2011 census. That group has been selected because of the maturity of its presence in Britain — study of this group may point the way for examination of other, more recently arrived, sub-populations. Previous studies have generally assumed homogeneity amongst ‘mainstream’ Jews and have not considered spatial variation, separating out only strictly orthodox enclaves. This paper demonstrates that there are indeed distinct socio-economic and demographic differences between Jewish groups in different areas, not fully attributable to the underlying mainstream social geography, whilst also identifying a strong degree of spatial clustering; it also establishes the practicality of applying geodemographic analysis to minority groups.

Using Rigid-Body Mechanism Topologies to Design Shape-Changing Compliant Mechanisms

This paper uses rigid-body mechanism topologies to synthesize fully distributed compliant mechanisms that approximate a shape change defined by a set of morphing curves in different positions. For a shape-change problem, a rigid-body mechanism solution is generated first to provide the base topology. This base topology defines a preselected design space for the structural optimization in one of two ways so as to obtain a compliant mechanism solution that is typically superior to the local minimum solutions obtained from searching more expansive design spaces. In the first strategy, the dimensional synthesis directly determines the optimal size and shape of the distributed compliant mechanism having exactly the base topology. In the second strategy, an initial mesh network established from the base topology is used to generate different topologies (in addition to the base), and an improved design domain parameterization scheme ensures that only topologies with well-connected structures are evaluated. The deformation of each generated compliant mechanism is evaluated using geometrically nonlinear finite element analysis (FEA). A two-objective genetic algorithm (GA) is employed to find a group of viable designs that trade off minimizing shape matching error with minimizing maximum stress. The procedure's utility is demonstrated with three practical examples—the first two approximating open-curve profiles of an adaptive antenna and the third approximating closed-curve profiles of a morphing wing.