Optimal Smoothing Research Articles

General Hölder Smooth Convergence Rates Follow from Specialized Rates Assuming Growth Bounds

Often in the analysis of first-order methods for both smooth and nonsmooth optimization, assuming the existence of a growth/error bound or KL condition facilitates much stronger convergence analysis. Hence, separate analysis is typically needed for the general case and for the growth bounded cases. We give meta-theorems for deriving general convergence rates from those assuming a growth lower bound. Applying this simple but conceptually powerful tool to the proximal point, subgradient, bundle, dual averaging, gradient descent, Frank–Wolfe and universal accelerated methods immediately recovers their known convergence rates for general convex optimization problems from their specialized rates. New convergence results follow for bundle methods, dual averaging and Frank–Wolfe. Our results can lift any rate based on Hölder continuous gradients and Hölder growth bounds. Moreover, our theory provides simple proofs of optimal convergence lower bounds under Hölder growth from textbook examples without growth bounds.

Conditions for linear convergence of the gradient method for non-convex optimization

In this paper, we derive a new linear convergence rate for the gradient method with fixed step lengths for non-convex smooth optimization problems satisfying the Polyak-Łojasiewicz (PŁ) inequality. We establish that the PŁ inequality is a necessary and sufficient condition for linear convergence to the optimal value for this class of problems. We list some related classes of functions for which the gradient method may enjoy linear convergence rate. Moreover, we investigate their relationship with the PŁ inequality.

Software tool for processing and selecting objects in infrared images

The article describes the scientific problem of processing images obtained in the infrared range. Methods for filtering such images in order to reduce noise are presented. Methods for optimal smoothing are described. A software tool has been developed that allows you to select the desired objects on the processed images.

APPROACH TO SOLVING THE PROBLEM OF SECTOR FORMATION
 IN ORGANIZING THE WORK OF DRIVERS
 AT THE ENTERPRISES OF ROUTE PASSENGER TRANSPORT
 BASED ON PSEUDOBOOLEAN OPTIMIZATION

An approach to solving the optimization problem arising during formation of sectors at the enterprises of route passenger transport by the sectoral method is considered, which allows to increase the efficiency of the mode of using drivers' working hours and reduce unproductive costs of the enterprise. The presented approach and its implementation based on pseudo-boolean optimization with the use of smooth optimization methods, in particular, “inner point” method have found their application in organization of drivers' work.

Experimental study of noise level optimization in brain single-photon emission computed tomography images using non-local means approach with various reconstruction methods

The noise reduction algorithm using the non-local means (NLM) approach is very efficient in nuclear medicine imaging. In this study, the applicability of the NLM noise reduction algorithm in single-photon emission computed tomography (SPECT) images with a brain phantom and the optimization of the NLM algorithm by changing the smoothing factors according to various reconstruction methods are investigated. Brain phantom images were reconstructed using filtered back projection (FBP) and ordered subset expectation maximization (OSEM). The smoothing factor of the NLM noise reduction algorithm determined the optimal coefficient of variation (COV) and contrast-to-noise ratio (CNR) results at a value of 0.020 in the FBP and OSEM reconstruction methods. We confirmed that the FBP- and OSEM-based SPECT images using the algorithm applied with the optimal smoothing factor improved the COV and CNR by 66.94% and 8.00% on average, respectively, compared to those of the original image. In conclusion, an optimized smoothing factor was derived from the NLM approach-based algorithm in brain SPECT images and may be applicable to various nuclear medicine imaging techniques in the future.

Path Planning for Unmanned Delivery Robots Based on EWB-GWO Algorithm.

With the rise of robotics within various fields, there has been a significant development in the use of mobile robots. For mobile robots performing unmanned delivery tasks, autonomous robot navigation based on complex environments is particularly important. In this paper, an improved Gray Wolf Optimization (GWO)-based algorithm is proposed to realize the autonomous path planning of mobile robots in complex scenarios. First, the strategy for generating the initial wolf pack of the GWO algorithm is modified by introducing a two-dimensional Tent-Sine coupled chaotic mapping in this paper. This guarantees that the GWO algorithm generates the initial population diversity while improving the randomness between the two-dimensional state variables of the path nodes. Second, by introducing the opposition-based learning method based on the elite strategy, the adaptive nonlinear inertia weight strategy and random wandering law of the Butterfly Optimization Algorithm (BOA), this paper improves the defects of slow convergence speed, low accuracy, and imbalance between global exploration and local mining functions of the GWO algorithm in dealing with high-dimensional complex problems. In this paper, the improved algorithm is named as an EWB-GWO algorithm, where EWB is the abbreviation of three strategies. Finally, this paper enhances the rationalization of the initial population generation of the EWB-GWO algorithm based on the visual-field line detection technique of Bresenham's line algorithm, reduces the number of iterations of the EWB-GWO algorithm, and decreases the time complexity of the algorithm in dealing with the path planning problem. The simulation results show that the EWB-GWO algorithm is very competitive among metaheuristics of the same type. It also achieves optimal path length measures and smoothness metrics in the path planning experiments.

Kalman filter/smoother-based design and implementation of digital IIR filters

Recently, a unified framework was proposed for forward-backward filtering and penalized least-squares optimization. It was shown that forward-backward filtering can be presented as instances of penalized least-squares optimization. In other words, the output of a zero-phase digital infinite-impulse response (IIR) filter can be computed by solving a constrained optimization problem, in which the weight controlling the constraint is directly related to cutoff frequencies with closed-form equations. It was also shown that a zero-phase digital IIR filter can be formed as an optimal smoothing Wiener filter for a random process obtained from an autoregressive (AR) or AR-moving average (ARMA) model driven by input (innovation) noise in presence of an observation noise.In this paper, the problem of zero-phase digital IIR filtering is re-examined using Kalman filter/smoother. The paper shows that every zero-phase digital IIR filter can be viewed as a special case of an optimal smoothing Wiener filter. Based on the fact that the formulations of the optimum filter by Wiener and Kalman are equivalent in steady state, we present a Kalman filter/smoother framework to the design and implementation of digital IIR filters. As an example, the zero-phase digital Butterworth filter is designed using Kalman smoother and compared with the traditional design (forward filtering and backward smoothing) method.

Application of Improved PNN in Transformer Fault Diagnosis

A transformer is an important part of the power system. Existing transformer fault diagnosis methods are still limited by the accuracy and efficiency of the solution and excessively rely on manpower. In this paper, a novel neural network is designed to overcome this issue. Based on the traditional method of judging the ratio of dissolved gas in transformer internal insulation oil, a fast fault diagnosis model of a transformer was built with an improved probabilistic neural network (PNN). The particle swarm optimization (PSO) algorithm was used to find the global optimal smoothing factor and improve the fault diagnosis accuracy of PNN. The transformer fault diagnosis model based on improved PNN not only eliminates the influence of human subjective factors but also significantly improves the diagnosis speed and accuracy, meeting the requirements for real-time application in practical projects. The feasibility and effectiveness of the method proposed in this paper are illustrated by a case study of actual data. Through analysis and comparison, the diagnostic accuracy of the proposed method is 10% higher than that of the general BPNN and 5% higher than that of the traditional PNN on the premise of ensuring the efficiency of the solution.

Optimal smoothing factor with coarsening by a factor of three for the MAC scheme for the Stokes equations

Optimal smoothing factor with coarsening by a factor of three for the MAC scheme for the Stokes equations

Iteration Complexity of an Inner Accelerated Inexact Proximal Augmented Lagrangian Method Based on the Classical Lagrangian Function

This paper establishes the iteration complexity of an inner accelerated inexact proximal augmented Lagrangian (IAIPAL) method for solving linearly constrained smooth nonconvex composite optimization problems that is based on the classical augmented Lagrangian (AL) function. More specifically, each IAIPAL iteration consists of inexactly solving a proximal AL subproblem by an accelerated composite gradient (ACG) method followed by a classical Lagrange multiplier update. Under the assumption that the domain of the composite function is bounded and the problem has a Slater point, it is shown that IAIPAL generates an approximate stationary solution in ACG iterations where is a tolerance for both stationarity and feasibility. Moreover, the above bound is derived without assuming that the initial point is feasible. Finally, numerical results are presented to demonstrate the strong practical performance of IAIPAL.

Smoothing Analysis of Two Robust Multigrid Methods for Elliptic Optimal Control Problems

In this paper we study and compare two multigrid relaxation schemes with coarsening by two, three, and four for solving elliptic sparse optimal control problems with control constraints and combined and cost functional. First, we perform a detailed local Fourier analysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme for the unconstrained case with only cost functional, where the optimal smoothing factors are derived. This insightful analysis reveals that the optimal relaxation parameters depend on both the mesh step size and the regularization parameter , which was not investigated in literature. Second, we propose and analyze a new mass-based Braess--Sarazin relaxation (BSR) scheme, which is proven to provide smaller smoothing factors than the CJR scheme when for a small constant . Finally, these schemes are successfully extended to control-constrained cases through the semismooth Newton method, where the corresponding Jacobian systems are treated by the proposed multigrid schemes. The nonstandard coarsening by three or four with BSR is competitive with the standard coarsening by two. Numerical examples are presented to validate our theoretical outcomes. The proposed inexact BSR (IBSR) scheme, where two preconditioned conjugate gradients (PCG) iterations are applied to solve the Schur complement system, yields a better computational efficiency than the CJR scheme in the conducted numerical comparison.

A novel sequential optimality condition for smooth constrained optimization and algorithmic consequences

In the smooth constrained optimization setting, this work introduces the Domain Complementary Approximate Karush–Kuhn–Tucker (DCAKKT) condition, inspired by a sequential optimality condition recently devised for non-smooth constrained optimization problems. It is shown that the augmented Lagrangian method can generate limit points satisfying DCAKKT, and it is proved that such a condition is not related to previously established sequential optimality conditions. An essential characteristic of the DCAKKT is to capture the asymptotic potential increasing of the Lagrange multipliers using a single parameter. Besides that, DCAKKT points satisfy the Strong Approximate Gradient Projection (SAGP) condition. Due to the intrinsic features of DCAKKT, which combine strength and generality, this novel and genuine sequential optimality condition may shed some light upon the practical performance of algorithms that are yet to be devised.

Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems

Frank and Wolfe’s celebrated conditional gradient method is a well-known tool for solving smooth optimization problems for which minimizing a linear function over the feasible set is computationally cheap. However, when the objective function is nonsmooth, the method may fail to compute a stationary point. In this work, we show that the Frank–Wolfe algorithm can be employed to compute Clarke-stationary points for nonconvex and nonsmooth optimization problems consisting of minimizing upper-C 1,α functions over convex and compact sets. Furthermore, under more restrictive assumptions, we propose a new algorithm variant with stronger stationarity guarantees, namely directional stationarity and even local optimality.

Accelerating inexact successive quadratic approximation for regularized optimization through manifold identification

AbstractFor regularized optimization that minimizes the sum of a smooth term and a regularizer that promotes structured solutions, inexact proximal-Newton-type methods, or successive quadratic approximation (SQA) methods, are widely used for their superlinear convergence in terms of iterations. However, unlike the counter parts in smooth optimization, they suffer from lengthy running time in solving regularized subproblems because even approximate solutions cannot be computed easily, so their empirical time cost is not as impressive. In this work, we first show that for partly smooth regularizers, although general inexact solutions cannot identify the active manifold that makes the objective function smooth, approximate solutions generated by commonly-used subproblem solvers will identify this manifold, even with arbitrarily low solution precision. We then utilize this property to propose an improved SQA method, $$^{+}$$ + , that switches to efficient smooth optimization methods after this manifold is identified. We show that for a wide class of degenerate solutions, $$^{+}$$ + possesses superlinear convergence not only in iterations, but also in running time because the cost per iteration is bounded. In particular, our superlinear convergence result holds on problems satisfying a sharpness condition that is more general than that in existing literature. We also prove iterate convergence under a sharpness condition for inexact SQA, which is novel for this family of methods that could easily violate the classical relative-error condition frequently used in proving convergence under similar conditions. Experiments on real-world problems support that $$^{+}$$ + improves running time over some modern solvers for regularized optimization.

Percentile Bootstrap Interval on Univariate Local Polynomial Regression Prediction

This study offers a new technique for constructing percentile bootstrap intervals to predict the regression of univariate local polynomials. Bootstrap regression uses resampling derived from paired and residual bootstrap methods. The main objective of this study is to perform a comparative analysis between the two resampling methods by considering the nominal coverage probability. Resampling uses a nonparametric bootstrap technique with the return method, where each sample point has an equal chance of being selected. The principle of nonparametric bootstrapping uses the original sample data as a source of diversity in contrast to parametric bootstrapping, where the variety comes from generating a particular distribution. The simulation results show that the paired and residual bootstrap interval coverage probabilities are close to nominal coverage. The results showed no significant difference between paired bootstrap interval and percentile residual. Increasing the bootstrap sample size sufficiently large gives the scatterplot smoothness of the confidence interval. Applying the smoothing parameter by choice gives a second-order polynomial regression with a smoother distribution than the first-order polynomial regression. The scatterplot shows that the second-degree polynomial regression can capture the data curvature feature compared to the first-degree polynomial. The bands made from second-degree polynomials give a narrower width than first-degree polynomials. In contrast, applying optimal smoothing parameters to the model provides different conclusions by using smoothing parameters based on choice. In addition to the differences based on the scatterplot, the bootstrap estimates of the coverage probability are also other. Selecting smoothing parameters based on a particular value provides probability coverage with the paired bootstrap method for the first-degree local polynomial regression is 0.93, while the second-degree local polynomial is 0.96. The probability of coverage based on the residual bootstrap method for the first-degree local polynomial regression is 0.95, while the second-degree local polynomial is 0.96. The probability coverage based on the optimal parameters of the paired bootstrap method for the first-degree local polynomial regression is 0.945, while the second-degree local polynomial is 0.93. The residual bootstrap method gives the first-degree local polynomial regression of 0.95, while the second-degree local polynomial is 0.93. In general, both bootstrap methods work well for estimating prediction confidence intervals.

A Decentralized Primal-Dual Framework for Non-Convex Smooth Consensus Optimization

In this work, we introduce ADAPD, A DecentrAlized Primal-Dual algorithmic framework for solving non-convex and smooth consensus optimization problems over a network of distributed agents. The proposed framework relies on a novel problem formulation that elicits ADMM-type updates, where each agent first inexactly solves a local strongly convex subproblem with any method of its choice and then performs a neighbor communication to update a set of dual variables. We present two variants that allow for a single gradient step for the primal updates or multiple communications for the dual updates, to exploit the tradeoff between the per-iteration cost and the number of iterations. When multiple communications are performed, ADAPD can achieve theoretically optimal communication complexity results for non-convex and smooth consensus problems. Numerical experiments on several applications, including a deep-learning one, demonstrate the superiority of ADAPD over several popularly used decentralized methods.

A Smooth Bidirectional Evolutionary Structural Optimization of Vibrational Structures for Natural Frequency and Dynamic Compliance

A smooth bidirectional evolutionary structural optimization (SBESO), as a bidirectional version of SESO is proposed to solve the topological optimization of vibrating continuum structures for natural frequencies and dynamic compliance under the transient load. A weighted function is introduced to regulate the mass and stiffness matrix of an element, which has the inefficient element gradually removed from the design domain as if it were undergoing damage. Aiming at maximizing the natural frequency of a structure, the frequency optimization formulation is proposed using the SBESO technique. The effects of various weight functions including constant, linear and sine functions on structural optimization are compared. With the equivalent static load (ESL) method, the dynamic stiffness optimization of a structure is formulated by the SBESO technique. Numerical examples show that compared with the classic BESO method, the SBESO method can efficiently suppress the excessive element deletion by adjusting the element deletion rate and weight function. It is also found that the proposed SBESO technique can obtain an efficient configuration and smooth boundary and demonstrate the advantages over the classic BESO technique.

Effects of polishing press-on force on surface roughness and gloss of CAD-CAM composites

The aim of this study was to determine the effect of polishing press-on force on surface roughness and gloss of computer-aided design-computer-aided manufacturing (CAD-CAM) composites. The materials evaluated included a CAD-CAM ceramic, a polymer-infiltrated ceramic, and three filler-based CAD-CAM composites. The CAD-CAM blocks were sectioned, embedded in self-cured resin, finished with abrasive papers and ultrasonically cleaned. Specimens were subsequently polished using the Sof-Lex disk system with 0.5, 1.0, 1.5, and 2.0 N press-on force by means of a custom-made apparatus. Contour arithmetic mean deviation (Ra) and gloss value (GU) data were acquired with a profilometer and glossmeter, respectively, and analyzed using ANOVA/Bonferroni post hoc test and Pearson's correlation (α = 0.05). Representative samples of the various materials at baseline and after each polishing step were examined under scanning electron microscope. Mean Ra and GU values ranged from 0.096 ± 0.004 μm to 0.295 ± 0.045 μm and 13.4 ± 1.9 to 67.6 ± 11.3 correspondingly for the various material-force combinations. Surface roughness and gloss were found to be press-on force and material dependent. A moderately strong and negative correlation (rs = -0.69) existed between Ra and GU values. For optimal smoothness and gloss, ceramic and polymer-infiltrated ceramic CAD-CAM materials must be polished with a 2.0 N force, while filler-based CAD-CAM composites should, in general, be polished using a 1.0 to 1.5 N force.

Robot Pose Optimization and Joint Path Smoothing Method for Aero-engine Complex Casing Deep Cavity Measurement

Robot Pose Optimization and Joint Path Smoothing Method for Aero-engine Complex Casing Deep Cavity Measurement

Digital Marketing Ecosystem Perspective of Regional Featured Product in North Sulawesi Province, Indonesia

The industrial sector that develops along with technological developments certainly has an impact on the economy of a country. Indonesia is a country that has great potential for the development of the digital economy with a high level of digital media penetration and a large market share. Strengthening local products based on the digital economy seems to be one way to maintain the positive momentum for post-pandemic national economic improvement. This is done by encouraging the marketing digitalization ecosystem to SMEs. This article is constructed based on literature studies that have been carried out previously by mapping articles, research journals, and related data to be able to provide a real picture regarding the role of the digital marketing ecosystem in encouraging marketing penetration of regional superior products. Some of the aspects that are used as barometer like consumer ecosystem, producer ecosystem, and core communication. These three things are collaborated with the facilitation of stakeholders such as the government, suppliers, consumers and business owners, who work together to encourage the smooth running and optimization of digital marketing operations in North Sulawesi Province, Indonesia. However, several obstacles remain, namely not all regional superior products have been reached in digital marketing. As a result, the role of actors in the local digital marketing ecosystem still needs to be carried out with various empowerment efforts. Stakeholders like the government and banking must creating more synergy with local business owners to maximize regional superior products that have enormous potential.