Gradient Search Algorithm Research Articles

The basic research of phase retrieval algorithm

This paper introduces two algorithms of the phase retrieval, Gerchberg–Saxton algorithm and gradient search algorithm. We respectively get the function of the object of gradient search algorithm about the generalized pupil, wavefront and the zernike coefficients of the partial derivatives when double-frame images and their defocus as the input. The relationship between GS algorithm and the gradient search algorithm are revealed. This paper designs the simulation experiment with GS algorithm and gradient search algorithm when single-frame images and double-frame images are used as input. The experiment results show that the gradient search algorithm is superior to GS algorithm for a single-frame image as input. Both GS algorithm and gradient search algorithm can primely work out wavefront for double-frame images of different defocus as the input, but the convergence rate of gradient search algorithm is evidently better than GS algorithm.

Distributed Seeking of Time-Varying Nash Equilibrium for Non-Cooperative Games

In this note, we address a Nash equilibrium seeking problem for non-cooperative games. In contrast to previous works on Nash equilibrium seeking, the Nash equilibrium under consideration can be time-varying. A non-model-based seeking scheme is proposed to achieve time-varying Nash equilibrium seeking, where each player updates its strategy by employing an extremum seeking method. The proposed Nash seeking scheme consists of a gradient estimation algorithm and a gradient search algorithm, which can be designed in a modular fashion. For symmetric quadratic games, the proposed Nash equilibrium seeking method enables the estimated strategy to globally asymptotically converge to the Nash equilibrium. For general quadratic games that are not necessarily symmetric, the estimated strategy converges to a neighborhood of the Nash equilibrium. For more general non-quadratic games that may admit multiple equilibria, local convergence to the Nash equilibrium is proven.

Maximal capacity nonorthogonal pulse shape modulation

To improve spectral utilization of communication system, a novel nonorthogonal pulse shape modulation (NPSM) based on prolate spheroidal wave function (PSWF) is proposed. The modulation employs nonorthogonal PSWF pulses to transmit information and it shows a higher capacity than traditional orthogonal modulations. The NPSM capacity under the constraint of finite input alphabet, which is determined by parameters of PSWF pulse, is derived. An optimization model for maximal capacity of NPSM is constructed and an exhaustive self-adapting gradient search algorithm for the model is proposed. A practical NPSM scheme with the maximal capacity is obtained by this search algorithm and it is proved to be superior to orthogonal signaling in the capacity. Our theoretical analysis is validated by numerical simulations and practical tests, and the results show that NPSM outperforms orthogonal modulations in the capacity and has a lower Peak-to-Average Power Ratio.

Reduced-rank gradient-based algorithms for generalized coupled Sylvester matrix equations and its applications

In this paper, by constructing an objective function and using the gradient search, full-rank and reduced-rank gradient-based algorithms are suggested for solving generalized coupled Sylvester matrix equations. It is proved that the reduced-rank iterative algorithm is convergent for proper initial iterative values. By analyzing the spectral radius of the related matrices, the convergence properties are studied and the optimal convergence factor of the reduced-rank algorithm is determined. The relationship between the reduced-rank algorithm and the full-rank algorithm is discussed. Consequently, the computation load can be reduced greatly for solving a class of matrix equation. A numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper.

Sum-Rate-Optimal Precoding for Multi-Cell Large-Scale MIMO Uplink Based on Statistical CSI

We investigate the sum-rate-optimal precoding for the uplink of the multi-cell large-scale MIMO systems. Specifically, we focus on transmitter precoder design based only on the statistical channel state information (CSI). We first consider the partial cooperation system, where only the CSI is shared among cells, and its base station (BS) decodes its in-cell users by treating out-cell signals as colored Gaussian noise. We derive the ergodic sum rate and its deterministic approximation for the regime where both the number of total user antennas and the number of BS antennas are large. We obtain the necessary conditions to maximize the deterministic approximation of the sum rate under the transmit power constraint, based on which a gradient search algorithm is proposed to find the local optimal precoders. Furthermore, a super cell system is also considered, where users from all cells are jointly decoded. The deterministic approximation to the sum rate and optimal precoder design for such systems are given. Numerical experiments show that the proposed precoders achieve significant performance gains over systems with no precoding. Compared with the existing precoding methods that require perfect CSI, the proposed precoders achieve similar sum rates and require only the statistical CSI.

Data‐based two‐degree‐of‐freedom iterative control approach to constrained non‐linear systems

This study proposes a data-based model-free approach to reference trajectory tracking in two-degree-of-freedom non-linear control system (CS) structures. This model-free control approach tunes both the feedback controller parameters and the reference input sequence accounting for control saturation and control rate constraints. The controller is iteratively tuned in a non-linear framework that employs a gradient descent search approach. The model-free gradient estimates are obtained by a perturbation-based approach. The reference input tuning is carried out in a linear framework using an iterative learning control-based approach, and it also includes a model-free gradient search algorithm where the gradient estimates are obtained by a similar perturbation-based approach. The number of real-world experiments is significantly reduced by the use of simulated models identified as neural networks. A digitally simulated case study concerning the angular position control of a non-linear aerodynamic twin-rotor system shows that the author's approach can effectively improve the CS performance.

Parameter estimation of analog circuits based on the fractional wavelet method**Project supported by the Key Research Project of Sichuan Provincial Department of Education, China (No. 13ZA0186).

Aiming at the problem of parameter estimation in analog circuits, a new approach is proposed. The approach is based on the fractional wavelet to derive the Volterra series model of the circuit under test (CUT). By the gradient search algorithm used in the Volterra model, the unknown parameters in the CUT are estimated and the Volterra model is identified. The simulations show that the parameter estimation results of the proposed method in the paper are better than those of other parameter estimation methods.

A Gradient Search Algorithm for the Maximal Visible Area Polygon Problem

This paper provides a gradient search algorithm for finding the maximal visible area polygon (VAP) viewed by an interior point in a simple polygon P. The algorithm is based on a natural partition of P into convex sets, such that each element of the partition is associated with a unique analytical form of the area function. We call this partition a back diagonal partition of P. Our maximal VAP algorithm converges in a finite number of steps, and is polynomial with a complexity of , for a simple polygon P with n vertices, and r reflex vertices. We use the maximal VAP algorithm as a basis for a greedy heuristic for the well known guardhouse problem with a computation complexity of .

Adaptive memetic computing for evolutionary multiobjective optimization.

Inspired by biological evolution, a plethora of algorithms with evolutionary features have been proposed. These algorithms have strengths in certain aspects, thus yielding better optimization performance in a particular problem. However, in a wide range of problems, none of them are superior to one another. Synergetic combination of these algorithms is one of the potential ways to ameliorate their search ability. Based on this idea, this paper proposes an adaptive memetic computing as the synergy of a genetic algorithm, differential evolution, and estimation of distribution algorithm. The ratio of the number of fitter solutions produced by the algorithms in a generation defines their adaptability features in the next generation. Subsequently, a subset of solutions undergoes local search using the evolutionary gradient search algorithm. This memetic technique is then implemented in two prominent frameworks of multiobjective optimization: the domination- and decomposition-based frameworks. The performance of the adaptive memetic algorithms is validated in a wide range of test problems with different characteristics and difficulties.

SU‐E‐T‐295: Simultaneous Beam Sampling and Aperture Shape Optimization for Station Parameter Optimized Radiation Therapy (SPORT)

Purpose:Station Parameter Optimized Radiation Therapy (SPORT) was recently proposed to fully utilize the technical capability of emerging digital LINACs, in which the station parameters of a delivery system, (such as aperture shape and weight, couch position/angle, gantry/collimator angle) are optimized altogether. SPORT promises to deliver unprecedented radiation dose distributions efficiently, yet there does not exist any optimization algorithm to implement it. The purpose of this work is to propose an optimization algorithm to simultaneously optimize the beam sampling and aperture shapes.Methods:We build a mathematical model whose variables are beam angles (including non‐coplanar and/or even nonisocentric beams) and aperture shapes. To solve the resulting large scale optimization problem, we devise an exact, convergent and fast optimization algorithm by integrating three advanced optimization techniques named column generation, gradient method, and pattern search. Column generation is used to find a good set of aperture shapes as an initial solution by adding apertures sequentially. Then we apply the gradient method to iteratively improve the current solution by reshaping the aperture shapes and updating the beam angles toward the gradient. Algorithm continues by pattern search method to explore the part of the search space that cannot be reached by the gradient method.Results:The proposed technique is applied to a series of patient cases and significantly improves the plan quality. In a head‐and‐neck case, for example, the left parotid gland mean‐dose, brainstem max‐dose, spinal cord max‐dose, and mandible mean‐dose are reduced by 10%, 7%, 24% and 12% respectively, compared to the conventional VMAT plan while maintaining the same PTV coverage.Conclusion:Combined use of column generation, gradient search and pattern search algorithms provide an effective way to optimize simultaneously the large collection of station parameters and significantly improves quality of resultant treatment plans as compared with conventional VMAT or IMRT treatments.

Adaptive gradient search for optimal sidelobe design of hopped‐frequency waveform

Linear step frequency waveforms can achieve large time-bandwidth product by uniformly inter-pulse frequency stepping, but have limited anti-jamming ability. Hopped-frequency waveforms overcome the limitation by introducing irregularly or randomly stepped frequency. However, a challenge in hopped-frequency waveform design is that, given fixed pulse number and time-bandwidth product, the distribution of its autocorrelation sidelobe varies significantly under different frequency combinations. In this study, the authors propose an adaptive gradient search algorithm for optimal sidelobe design of hopped-frequency waveform. The method is iterative and applies a negative gradient adjustment adaptively according to sidelobe discrepancy between the current output of the matched filtering and the ideal expectation. As a result, it can be used to find optimal frequency combinations that produce hopped-frequency pulses with a prescribed sidelobe specification. Experimental evidence suggests that the proposed procedure gives promising results in terms of sidelobe level and it terminates in remarkably little time.

State Sensitivity Evaluation Within UD Based Array Covariance Filters

This technical note addresses the UD factorization based Kalman filtering (KF) algorithms. Using this important class of numerically stable KF schemes, we extend its functionality and develop an elegant and simple method for computation of sensitivities of the system state to unknown parameters required in a variety of applications. For instance, it can be used for efficient calculations in sensitivity analysis and in gradient-search optimization algorithms for the maximum likelihood estimation. The new theory presented in this technical note is a solution to the problem formulated by Bierman in , which has been open since 1990s. As in the cited paper, our method avoids the standard approach based on the conventional KF (and its derivatives with respect to unknown system parameters) with its inherent numerical instabilities and, hence, improves the robustness of computations against roundoff errors.

Voltage and Reactive Power Load Flow Optimization in the Power System Using Fuzzy Logic

The paper presents issues related to voltage control in the power system. An original method of reactive power flow optimization is considered, which leads to improved voltages in the power system and reduced active power losses. The optimization method is based on a procedure that employs fuzzy logic and is supported by a gradient search algorithm. The method has been implemented in PLANS software and verified.

SU-E-T-462: Energy Modulated Photon Radiotherapy: A Monte Carlo Investigation

Purpose: To investigate the feasibility of energy modulated photon radiation therapy (EMXRT) using Monte Carlo simulations of photon beams of various energies. Methods: An EMXRT plan was generated for a prostate cancer patient treated with 10-MV 5-field intensity modulated radiotherapy (IMRT) optimized using Eclipse treatment planning system. Based on beam characterization of a Varian Clinac with EGS4/BEAM Monte Carlo code, multiple source models were first generated to represent the photon beams with energy of 1, 2, 3, 4, 5, 6, 10 and 15 MV, respectively. Optimal beam energy was then selected based on the effective path-length in the patient via ray-tracing for each beam angle. An inverse planning system based on gradient search algorithm was used to optimize photon beam intensity of various beam energies with pre-simulated Monte Carlo pencil beam dose distributions in patient anatomy. Finally, both EMXRT and IMRT plans were simulated with EGS4/MCSIM for 3D dose distributions in patient and compared. Results: Based on different path-lengths, 5, 6 and 10 MV photons were chosen in our EMXRT plan with lower energy photons used for shorter path-lengths. DVH analysis indicated that the target coverage in EMXRT plan was less uniform with Dmax and Dmin at 111.6% and 92.6% of prescription dose, respectively, as compared to 106.2% and 97.8% in regular IMRT plan. On the other hand, critical structures were much more spared in EMXRT plan than in IMRT plan, with 32% dose reduction in rectum and 40% reduction in bladder, respectively. The integral dose to the surrounding normal tissues was significantly reduced by 56% in EMXRT plan. Conclusion: This work demonstrates the potential of this Monte Carlo based tool for investigating the feasibility and desirability of EMXRT. Further investigation on more complex tumor sites such as head and neck cancer are now in progress.

Automatic leaking carrier canceller adjustment techniques

In this contribution, four automatic adjustment algorithms for leakage carrier cancellation in radio frequency identification (RFID) readers are compared: full search, gradient search, fast and direct I/Q algorithms. Further, we propose two enhanced adjustment procedures. First, we analytically calculate the performance of the fast adjustment algorithm in the presence of noise and derive its theoretical bias. We compare the theoretical results with the numerical results from accompanying simulations. Further, we evaluate the performance of these algorithms based on real-world measurements acquired with our RFID testbed. Finally, we propose and discuss the merits of two enhanced adjustment procedures based on the fast adjustment algorithm. The fast adjustment procedure with bipolar probing signals achieves the isolation gain of the (much slower) gradient search algorithm at the expense of a mean penalty of 0.48 dB. We observe that the fast adjustment aided gradient algorithm requires 72% less steps than the gradient search algorithm in our measurements.

An orthogonal subspace minimization method for finding multiple solutions to the defocusing nonlinear Schrödinger equation with symmetry

AbstractAn orthogonal subspace minimization method is developed for finding multiple (eigen) solutions to the defocusing nonlinear Schrödinger equation with symmetry. As such solutions are unstable, gradient search algorithms are very sensitive to numerical errors, will easily break symmetry, and will lead to unwanted solutions. Instead of enforcing a symmetry by the Haar projection, the authors use the knowledge of previously found solutions to build a support for the minimization search. With this support, numerical errors can be partitioned into two components, sensitive versus insensitive to the negative gradient search. Only the sensitive part is removed by an orthogonal projection. Analysis and numerical examples are presented to illustrate the method. Numerical solutions with some interesting phenomena are captured and visualized by their solution profile and contour plots. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Optimization of Transferable Site–Site Potentials Using a Combination of Stochastic and Gradient Search Algorithms

Discontinuous molecular dynamics (DMD) simulation and thermodynamic perturbation theory (TPT) have been used to study thermodynamic properties for organic compounds. The aim is to infer transferable intermolecular potential models based on correlating the vapor pressure and liquid density. The combination of DMD/TPT generates a straightforward global optimization problem for the attractive potential, instead of facing an iterative optimization–simulation type problem. This global optimization problem is then formulated as a black-box optimization problem and solved using a combination of random recursive search (RRS) and Levenberg–Marquardt (LM) optimization. RRS is suitable for black-box optimization problems since its algorithm is robust to the effect of random noises in the objective function and is advantageous in optimizing the objective function with negligible parameters. LM is efficient local to an optimum with a smooth response surface. The local response surface is shown to be smooth but very flat along valleys with a high degree of coupling between the potential parameters. The algorithm is demonstrated with discretized versions of the Lennard-Jones (LJ) potential and a linear step potential using a database of 231 hydrocarbons, alcohols, aldehydes, amines, amides, esters, ethers, ketones, phenols, sulfides, and thiols. A correspondence is established between the discretized LJ potential and the TraPPE model, demonstrating the manner of improving density estimates and a way of expediting improvement of continuous transferable potentials.

Genetic Algorithm to Optimize Layer Parameters in Light Weight Deflectometer Backcalculation

AbstractThe light weight deflectometer (LWD) is a portable, nondestructive testing device that can estimate pavement layer parameters, namely moduli. Conventional backcalculation of layer parameters from LWD deflections is formulated as an inverse problem where predicted vertical deflections are matched to observed vertical deflections using a gradient search algorithm. In this paper, we present an LWD backcalculation scheme to recover layer parameters, including top-layer thickness, of a two-layer earthwork system. Our approach resolves the problem using a dynamic finite-element (FE) model for the forward calculation of LWD deflection data and implements a genetic algorithm (GA) as the inverse solver. The objective function we minimize is formulated as a measure of the data misfit between predicted and observed data, normalized by the peak deflections, and it includes 180 data points from the dynamic deflection time history. The objective function contains multiple local minima that can potentially trap ...

Finding minimum energy configurations for constrained beam buckling problems using the Viterbi algorithm

In this work, we present a novel technique to find approximate minimum energy configurations for thin elastic bodies using an instance of dynamic programming called the Viterbi algorithm. This method can be used to find approximate solutions for large deformation constrained buckling problems as well as problems where the strain energy function is non-convex. The approach does not require any gradient computations and could be considered a direct search method. The key idea is to consider a discretized version of the set of all possible configurations and use a computationally efficient search technique to find the minimum energy configuration. We illustrate the application of this method to a laterally constrained beam buckling problem where the presence of unilateral constraints together with the non-convexity of the energy function poses challenges for conventional schemes. The method can also be used as a means for generating “very good” starting points for other conventional gradient search algorithms. These uses, along with comparisons with a direct application of a gradient search and simulated annealing, are demonstrated in this work.

A gradient-based method to guard against system degradation in robust parameter design

Robust parameter design (RPD) can be implemented in systems in which a user, through the setting of the levels of controllable factors, seeks to minimise the variability of a system response influenced by uncontrollable factors while obtaining a consistent and reliable system response over time. RPD is conducted under the assumption that the relationship between the system response and controllable and uncontrollable variables does not change over time. Since performance degradation in real-world systems will almost certainly occur; this assumption will inevitably be violated. We propose a methodology to find new levels of controllable factor settings that will be robust to moderate system degradation while remaining robust to noise variables within the system. A gradient-search algorithm is presented for this enhanced RPD analysis utilising quadratic regression.