Nonlinear Optimization Method Research Articles

An efficient nonlinear interval uncertain optimization method using Legendre polynomial chaos expansion

Many specialized numerical techniques have been developed on interval uncertainty optimization. However, the conventional nested optimization strategy is time-consuming. The improved Taylor-based interval optimization avoids the laborious inner optimization, nonetheless, it is intrusive and presents significant computational challenges in the large-level uncertain optimization problems. To end these, this paper proposes a Legendre polynomial expansion method combined with the subinterval technique to evaluate the range enclosure of an interval function, where the expansion coefficients are computed through the collocation method. The detailed comparisons demonstrate that such Legendre-based polynomial expansion provides much more confidence in interval bounds evaluation than the Taylor expansion, and is capable of maintaining very sharp solution enclosures even when the uncertainty level is increased. More exciting, it is highly efficient. Subsequent application of the Legendre polynomial expansion in the interval bound estimation of the uncertain objective function and constraints, and combining an appropriate outer optimizer, result in a novel nonlinear interval uncertain optimization method for complicated engineering systems, whereby the time-consuming optimization nesting is avoided. A beam design example and an electromagnetic buffer optimization example illustrate that the proposed interval optimization method has fine computational precision and high computational efficiency, especially is capable of attacking the increase of uncertainty level. Moreover, the new method does not require the derivative information of the uncertain objective and constraints, such that applies to any engineering uncertain optimization problems, including the complex engineering problems which are usually referred to as “black-box”.

An estimation–Based optimal scheduling model for settable renewable penetration level in energy hubs

Several recent studies suggest that power–to–gas (PtG) facilities can be operated to absorb and convert surplus renewable energy into hydrogen or synthetic natural gas, which is then supplied to the natural gas grid, forming an energy hub (EH). This paper argues and demonstrates that there is a trade off between maximizing the renewable penetration level (PRL) into the EH and minimizing the cost associated with it. The paper presents a new optimal scheduling model allowing variable and settable PRLs into the EH to minimize the system operating cost. The RPL, as determined by the grid operator, is taken in as a setting to the scheduling model. A new algorithm is presented by which the relationship between the RPL and scheduling setpoints of PtG and gas–fired generation (GfG) facilities are estimated. Accordingly, the optimal scheduling setpoints of PtG and GfG facilities are decided by the model, so that the required RPL can be met. The problem is formulated using the mixed integer nonlinear optimization method where the estimation algorithm is developed using the least error square technique. The developed problem is then solved using the combined interior point nonlinear programming and Newton trust region techniques. Numerical studies are conducted on a test system using historical operating data to validate the efficacy of the model. The system operating parameters are analyzed, and it is demonstrated that it is imperative to optimally manage the desired RPL taking into consideration the arbitrage revenue shortfall resulted from RPL enhancement in an EH.

An inverse source identification by nonlinear optimization in a two-dimensional hyperbolic problem

This study deals with the identification of source function from final time state observation in a two-dimensional hyperbolic problem. The solution to the direct problem is obtained by the weak solution approach and finite element method. In the part of the inverse problem, the trust-region method and Levenberg–Marquardt method, which are nonlinear least-squares optimization methods, are used for the identification of source function. The findings are presented with numerical examples.

A variational-based homogenization model for plastic shakedown analysis of porous materials with a large range of porosity

In this paper, a new homogenization model is developed for the determination of macroscopic shakedown limit state for porous media subjected to general cyclic loads. Based on the variational principle over the stabilized cycle, a new macroscopic fatigue criterion is established, beyond which the collapse of material will occur due to fatigue. Unlike the classical Melan’s theory by statical approaches relying on the complex construction of a time-independent residual stress field, the formulation of the new criterion is derived from a variational-based kinematical approach which allows overcoming this difficulty. Dirac’s measure is adopted to simplify the volume integral with the assumption of vanishing plastic strain increment over a stabilized cycle. The established criteria exhibit directly the dependence of the plastic shakedown limit load on the invariants of the macroscopic stress tensor, Poisson’s ratio of the solid matrix and the porosity. The macroscopic safety domain is defined by the intersection of limit surface of this new fatigue criteria and that corresponds to the incremental collapse which is reached monotonically. The theoretical predictions of shakedown limits by the new criterion are compared with the numerical results obtained from non-linear optimization method and with those given by some representative existing criteria. It is shown that the new criterion significantly improves the accuracy of plastic shakedown limit prediction for porous materials with large values of porosity.

An Improved Localization Method for the Transition between Autonomous Underwater Vehicle Homing and Docking.

Docking technology for autonomous underwater vehicles (AUVs) involves energy supply, data exchange and navigation, and plays an important role to extend the endurance of the AUVs. The navigation method used in the transition between AUV homing and docking influences subsequent tasks. How to improve the accuracy of the navigation in this stage is important. However, when using ultra-short baseline (USBL), outliers and slow localization updating rates could possibly cause localization errors. Optical navigation methods using underwater lights and cameras are easily affected by the ambient light. All these may reduce the rate of successful docking. In this paper, research on an improved localization method based on multi-sensor information fusion is carried out. To improve the localization performance of AUVs under motion mutation and light variation conditions, an improved underwater simultaneous localization and mapping algorithm based on ORB features (IU-ORBSALM) is proposed. A nonlinear optimization method is proposed to optimize the scale of monocular visual odometry in IU-ORBSLAM and the AUV pose. Localization tests and five docking missions are executed in a swimming pool. The localization results indicate that the localization accuracy and update rate are both improved. The 100% successful docking rate achieved verifies the feasibility of the proposed localization method.

Geometrically Nonlinear Topology Optimization of Continuum Structures Based on an Independent Continuous Mapping Method

A geometrically nonlinear topology optimization method for continuum structures is proposed based on the independent continuous mapping method. The stress constraint problem is studied due to the importance of structural strength in engineering applications. First, a topology optimization model is established for a lightweight structure with element stress as constraints. Second, the stress globalization method is adopted to convert local stress constraints into strain energy constraints, which overcomes the difficulties caused by local stress constraints, such as model establishment, sensitivity analysis, and massive solution calculations. Third, the sensitivity of the objective function and constraint function is analyzed, and the method of moving asymptotes is employed to solve the optimization model. In addition, the additive hyperelasticity technique is utilized to solve the numerical instability induced by structures undergoing large deformation. Numerical examples are given to validate the feasibility of the proposed method. The method provides a significant reference for geometrically nonlinear optimization design.

Removal of cadmium ions from water using coaxially electrospun PAN/ZnO-encapsulated PVDF nanofiber membranes

Electrospun nanofiber membranes have been widely used as heavy metal ion adsorbents. However, they are usually mechanically weak, which reduces the chances of employing such materials in environments that involve stresses and strains. In this work, core–shell nanofiber membranes were directly produced by coaxial electrospinning of polyvinylidene fluoride (PVDF) nanofibers as core nanofibers to provide mechanical support, and zinc oxide (ZnO) nanoparticles loaded PAN as shell nanofibers to enhance the adsorption of cadmium ions. The effect of pH, time and concentration on the adsorption properties was investigated. The incorporation of ZnO nanoparticles to the membrane matrix led to a significant enhancement of 75% in adsorption capacity compared to the membrane without ZnO nanoparticles (i.e., 350 mg/g for PVDF-PAN/ZnO compared to 200 mg/g for PVDF-PAN). The nonlinear optimization method showed that the experimental results are better fitted with the pseudo-second-order kinetic model and Langmuir isotherm equation. In addition, the tensile breaking strength values for PAN, PVDF, PVDF-PAN and PVDF-PAN/ZnO nanofiber membranes were 1.3 ± 0.3, 8.2 ± 0.3, 8.4 ± 0.1 and 3.1 ± 0.3 MPa, respectively, which suggests the enhancement of mechanical properties with the addition of PVDF as core nanofibers.

Biped Walking Based on Stiffness Optimization and Hierarchical Quadratic Programming.

The spring-loaded inverted pendulum model is similar to human walking in terms of the center of mass (CoM) trajectory and the ground reaction force. It is thus widely used in humanoid robot motion planning. A method that uses a velocity feedback controller to adjust the landing point of a robot leg is inaccurate in the presence of disturbances and a nonlinear optimization method with multiple variables is complicated and thus unsuitable for real-time control. In this paper, to achieve real-time optimization, a CoM-velocity feedback controller is used to calculate the virtual landing point. We construct a touchdown return map based on a virtual landing point and use nonlinear least squares to optimize spring stiffness. For robot whole-body control, hierarchical quadratic programming optimization is used to achieve strict task priority. The dynamic equation is given the highest priority and inverse dynamics are directly used to solve it, reducing the number of optimizations. Simulation and experimental results show that a force-controlled biped robot with the proposed method can stably walk on unknown uneven ground with a maximum obstacle height of 5 cm. The robot can recover from a 5 Nm disturbance during walking without falling.

Optimizing carbon tax rates and revenue recycling schemes: Model development, and a case study for the Bohai Bay area, China

Carbon taxation has long been proposed to mitigate carbon emissions. Carbon tax rate determination and tax revenue recycling are two key steps to achieve a double dividend under carbon taxation. However, few research has sought to optimize the allocation proportion of tax revenue to various categories of taxpayers (households, enterprises, sectors, etc.), or to optimize the two steps synchronously based on nonlinear optimization methods. In this study, a computable general equilibrium model was established to explore the influence of carbon tax rate and tax revenue recycling shares on the economy and on carbon emissions. Meanwhile, a nonlinear optimization model was proposed, for optimizing both steps of carbon taxation synchronously, as well as for promoting GDP and CO2 emission reduction. The Bohai Bay area, a typical area with enormous carbon emissions in China, was adopted as the study case for this research. The results showed that the optimized taxation scheme could lead to lower carbon emissions and greater economic growth, i.e., a strong double dividend was obtained. The optimized taxation scheme could lead to both cleaner air and cleaner energy and industrial structures while still promoting economic growth.

A spectral-in-time Newton–Krylov method for nonlinear PDE-constrained optimization

Abstract We devise a method for nonlinear time-dependent partial-differential-equation-constrained optimization problems that uses a spectral-in-time representation of the residual, combined with a Newton–Krylov method to drive the residual to zero. We also propose a preconditioner to accelerate this scheme. Numerical results indicate that this method can achieve fast and accurate solution of nonlinear problems for a range of mesh sizes and problem parameters. The numbers of outer Newton and inner Krylov iterations required to reach the attainable accuracy of a spatial discretization are robust with respect to the number of collocation points in time and also do not change substantially when other problem parameters are varied.

Reliability optimization design of deformation of CNC lathe spindle considering thermal effect

Angular contact ball bearings are widely used in the field of rotating machinery due to their obvious advantages such as relatively good positioning accuracy, high speed rotating performance and low cost, which have already become the most important transmission components. The heterogeneous thermal deformation caused by the high speed effect of bearing parts will lead to excessive noise and even gluing, which can further significantly reduce the machining accuracy of machine tool. Therefore, it is vital to improve the deformation resisting capability and operational reliability of the whole system. For this purpose, this paper presents a reliability model for computerized numerical control (CNC) lathe spindle by considering thermal effect. A five-degree-of-freedom quasi-static model considering thermal deformation is firstly proposed to calculate contact load and contact angle. Then the transient thermal network method is used to solve the temperature value of multi-node spindle-bearing system, and the validity of the proposed model is verified by experiments at different speeds. Next the modified first-order and second-moment method (FOSM) is used to calculate the reliability and reliability sensitivity of CNC lathe spindle deformation model considering thermal effect. Finally, the constrained nonlinear optimization method for the reliability model is proposed and applied to CNC lathe spindle. The results show that the reliability of the optimized model is significantly improved and the reliability robustness is enhanced.

Reconstruction of strains with a non-smooth distribution using optical fiber Bragg grating

The paper presents a reconstruction of strains with a non-smooth distribution affecting the optical fiber Bragg grating based on its reflection intensity spectrum. The reconstruction was performed for a grating strained along 0.25, 0.5 and 0.75 of its length. The transition matrix method and the Nelder-Mead nonlinear optimization method were used for this purpose. The results of simulations and reconstruction of the strain profile indicate good consistency with the original strain profiles. Reconstruction errors of these profiles are within the single digit percentage range. For a grating strained along 0.5 or 0.75 of its length, the error does not exceed 0.34% when the reconstruction is based on a simulated reflection spectrum; the use of a measured spectrum results in an error of up to 2.2%.

THE FINANCIAL COMPONENT OF THE COMMUNICATION MANAGEMENT STRATEGY IN HIGHER EDUCATION INSTITUTIONS

Abstract. The article defines the role of the financial component of the communication management strategy in HEIs based on the construction of the algorithm for optimizing the distribution of costs for communication processes and the construction of the financial mechanism of communication management. Communication processes in HEIs have a number of specific features due to the fact that a large number of people and groups of people with different levels of education, level of intelligence, age, upbringing and interests take part in the communication process. That is, effective communication management of HEIs requires from the management of institutions to develop a management strategy that would satisfy the interests of all groups of communication process participants. It has been emphasized that the communication management strategy in HEIs should be developed for the long term taking into account the conditions of the changing external environment, and have the goal to continuously transmit information to employers, applicants, their parents, and the whole society in order to make the services of institutions attractive to the target audience and have feedback which would allow monitoring the quality of communication processes and have an economic effect from their implementation. It has been noted that one of the main components of the communication management strategy in HEIs is finance. An algorithm has been built to optimize the costs of managing communication in HEIs based on the use of the classical method of mathematical nonlinear programming problems optimization. A financial mechanism for managing communication in HEIs has been built, which is a system of measures aimed at achieving a certain level of communication processes management while maintaining a certain level of financial stability and independence. It has been concluded that the financial component of the communication management strategy in HEIs is one of its main elements, which allows constant increasing the level of institutions competitiveness and the quality of education as well as scientific and technical potential. Keywords: financial component, communication, management strategy, costs, algorithm, mechanism, higher education institutions. Formulas: 11; fig.: 2; tabl.:1; bibl.: 15.

Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

The focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

A robust visual odometry based on RGB-D camera in dynamic indoor environments

To solve the accurate positioning problem of mobile robots, simultaneous localization and mapping (SLAM) or visual odometry (VO) based on visual information are widely used. However, most visual SLAM or VO cannot meet the accuracy requirements in dynamic indoor environments. This paper proposes a robust visual odometry based on deep learning to eliminate feature points matching error. However, when a camera and dynamic objects are in relative motion, the frames of camera will produce ghosting, especially in high-dynamic environments, which bring additional positioning error; in view of this problem, a novel method based on the average optical flow value of the dynamic region is proposed to identify feature points of the ghosting, and then the feature points of the ghosting and dynamic region are removed. After the remaining feature points are matched, we use a non-linear optimization method to calculate the pose. The proposed algorithm is tested on TUM RGB-D dataset, and the results show that our VO improves the positioning accuracy than other robust SLAM or VO and is strongly robust especially in high-dynamic environments.

Nonlinear optimal control and synchronization for chaotic electronic circuits

To achieve control and synchronization of chaotic electronic circuits, a nonlinear optimal (H-infinity) control method is developed and is tested on Chua’s circuit. Although this electronic circuit is deterministic, for specific values of its parameters its phase diagrams may change in a random-like manner, thus exhibiting a chaotic behavior. In the article’s control approach, an approximate linearization procedure is applied first to the dynamic model of the circuit. The linearization takes place around a temporary operating point which is recomputed at each iteration of the control method. It actually uses Taylor series expansion and the computation of the system’s Jacobian matrices. At a next stage, an H-infinity feedback controller is developed for the approximately linearized model of the circuit. This controller is obtained after solving an algebraic Riccati equation at each time step of the control method. To prove the stability properties of the control scheme and the elimination of the synchronization error, Lyapunov analysis is used. The proposed control scheme is demonstrated to satisfy the H-infinity tracking performance condition, and this indicates elevated robustness against model uncertainty and external perturbations. Moreover, the global asymptotic properties of the control method are proven. Finally, under the proposed nonlinear optimal control approach, it is shown that different Chua’s circuits can get synchronized and that chaotic behavior can be replicated by them.

Rates of superlinear convergence for classical quasi-Newton methods

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form (frac{n L^2}{mu ^2 k})^{k/2} and (frac{n L}{mu k})^{k/2} respectively, where k is the iteration counter, n is the dimension of the problem, mu is the strong convexity parameter, and L is the Lipschitz constant of the gradient.

Research on identification of unbalance parameters of rotor with multi-plane using improved particle swarm optimization

Based on Bentley RK4 test rig, the method of nonlinear particle swarm optimization (PSO) is used to solve inverse problem instead of previous methods, which reduces identification complexity and avoids the ill posed problem caused by the singular equation in the process of solving. This paper realized model-based standard particle swarm optimization (SPSO) algorithm after modelling of the finite element model of rotor system and creating the optimization variables and individual fitness function, and then established asynchronous adaptive particle swarm optimization with optimizing speed weight and search scope. In order to further improve algorithm accuracy, chaos weighted particle swarm optimization and double chaos particle swarm optimization (DCPSO) are established. The simulation results of four methods show that DCPSO can realize accurate identification of unbalanced parameters with an average error of 2.86%. The effectiveness and rationality of this algorithm are verified from the speed of 240–3000 r/min. Particularly, at 2040 r/min, the percentage of amplitude decrease in two measuring points is 94.07% and 95.93%, which has achieved excellent vibration suppression effect.

The semi-obnoxious minisum circle location problem with Euclidean norm

‎The objective of the classical version of the minisum circle location problem is finding a circle $C$ in the plane such that the sum of the weighted distances from the circumference of $C$ to a set of given points is minimized‎, ‎where every point has a positive weight‎. ‎In this paper‎, ‎we investigate the semi-obnoxious case‎, ‎where every existing facility has either a positive or negative weight‎. ‎The distances are measured by the Euclidean norm‎. ‎Therefore‎, ‎the problem has a nonlinear objective function and global nonlinear optimization methods are required to solve this problem‎. ‎Some properties of the semi-obnoxious minisum circle location problem with Euclidean norm are discussed‎. ‎Then a cuckoo optimization algorithm is presented for finding the solution of this problem‎.

Research on Aerosol Retrieval Algorithm of Multi-parameter Optimization Model

Aerosols can have an important impact on human society. Remote sensing method plays an important role in the aerosol retrieval. The aerosol retrieval SARA algorithm isn’t required a lot of prior knowledge and look-up table comparing with the traditional aerosol retrieval algorithms. However, the retrieval results of SARA algorithm are affected by the acquisition time differences and instability of MODIS data and AERONET data. In order to improve SARA algorithm, this paper studies the nonlinear optimization methods to construct the three-parameter model and the five-parameter model. And used the model to optimize the input parameters to improve the accuracy of aerosol retrieval. The correlation of five-parameter optimization results with AERONET sites Beijing and XiangHe is 0.7454 and 0.701 respectively, which is better than that of three-parameter optimization model’s 0.6012, 0.6479, and both are higher than SARA’s 0.3844 and 0.526.