Gradient Constraint Research Articles

Lagrange multipliers and non-constant gradient constrained problem

The aim of the paper is to study a gradient constrained problem associated with a linear operator. Two types of problems are investigated. The first one is the equivalence between a non-constant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a Hamilton-Jacobi equation in the viscosity sense. The equivalence result is obtained under a condition on the gradient constraint. The second problem is the existence of Lagrange multipliers. We prove that the non-constant gradient constrained problem admits a Lagrange multiplier, which is a Radon measure if the free term of the equation f∈Lp, p>1. If f is a positive constant, we regularize the result, namely we prove that the Lagrange multipliers belong to L2.

Optimized Graph cut Color Image Segmentation using Genetic Algorithm with Weighted Constraints (OGcut)

This paper proposes a Novel color image segmentation using Graph cut method by minimizing the weighted energy function. This method is applying a pair of optimal constraints namely: color constraint and gradient constraint. In the state-of-the-art methods, the background and foreground details are manually initialized and used for verifying the smoothness of the region. But in this proposed method, they are dynamically calculated from the input image. This feature of the proposed method can be used in color image segmentation where more number of unique segments exists in a single image. The genetic algorithm is applied to the graph obtained from the graph cut method. The crossover and mutation operators are applied on various subgraphs to populate the different segments.

Bank Monitoring Incentives Under Moral Hazard and Adverse Selection

In this paper, we extend the optimal securitization model of Pages [41] and Possamai and Pages [42] between an investor and a bank to a setting allowing both moral hazard and adverse selection. Following the recent approach to these problems of Cvitanic, Wan and Yang [12], we characterize explicitly and rigorously the so-called credible set of the continuation and temptation values of the bank, and obtain the value function of the investor as well as the optimal contracts through a recursive system of first-order variational inequalities with gradient constraints. We provide a detailed discussion of the properties of the optimal menu of contracts.

Optimal Singular Dividend Problem Under the Sparre Andersen Model

In this paper, we study the optimal dividend problem assuming that the underlying reserve process follows the Sparre Andersen model. In this model, there is no constant restriction on the dividend rates, i.e., the optimization problem is of singular type. In this case, the value function is no longer bounded and the associated Hamilton–Jacobi–Bellman equation is a variational inequality involving a first-order integro-differential operator and a gradient constraint. We prove the regularity properties for the value function by constructing strategies and show that the value function is a constrained viscosity solution of the associated Hamilton–Jacobi–Bellman equation. In addition, we prove that the value function is the upper semicontinuous envelope of the supremum for a class of subsolutions.

On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient

We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the $$\sigma $$ -gradient ( $$0<\sigma <1$$ ). We establish continuous dependence results with respect to the data, including the threshold of the fractional $$\sigma $$ -gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Using the approximation of the solutions with a family of quasilinear penalisation problems we show the existence of generalised Lagrange multipliers for the $$\sigma $$ -gradient constrained problem, extending previous results for the classical gradient case, i.e., with $$\sigma =1$$ .

Optimal dividend policies with random profitability

AbstractWe study an optimal dividend problem under a bankruptcy constraint. Firms face a trade‐off between potential bankruptcy and extraction of profits. In contrast to previous works, general cash flow drifts, including Ornstein–Uhlenbeck and CIR processes, are considered. We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as comparison between discontinuous sub‐ and supersolutions of the Hamilton–Jacobi–Bellman equation, and we provide an efficient and convergent numerical scheme for finding the solution. The value function is given by a nonlinear partial differential equation (PDE) with a gradient constraint from below in one direction. We find that the optimal strategy is both a barrier and a band strategy and that it includes voluntary liquidation in parts of the state space. Finally, we present and numerically study extensions of the model, including equity issuance and gambling for resurrection.

Improving GNSS PPP accuracy through WVR PWV augmentation

Using 5 months of observations at a Global Navigation Satellite System (GNSS) and a Water Vapor Radiometer (WVR) collocated station at Tongji University, Shanghai, a mid-latitude coastal city in China with high level of water vapor, we analyzed the precipitable water vapor (PWV) from different sources including WVR, GNSS, Numerical Weather Prediction model (NWP) and radiosonde (RS). The highest correlation coefficient of 99.8% between GNSS PWV and WVR PWV with a linear fitting root-mean-square (RMS) error of 1 mm was obtained. The WVR observations were further applied in GNSS Precise Point Positioning (PPP) to demonstrate its benefits compared to the traditional PPP where troposphere delay was estimated. Both the Global Positioning System (GPS) and Globalnaya Navigatsionnaya Sputnikovaya Sistema (GLONASS) observations were used, and the impact of estimating tropospheric gradients was also investigated. Experiments show that the WVR-constrained PPP improves the weekly repeatability, convergence time, and short-term precision in the vertical component for GPS + GLONASS and GPS-only PPP, in both static and kinematic cases. For the vertical component of daily static GPS + GLONASS PPP, the weekly repeatability of the daily static solutions was improved by ~ 5%; the convergence time was shortened by ~ 30–50%. The short-term static GPS + GLONASS PPP vertical precision was improved by 30–53% when the WVR PWV was used as a constraint and troposphere gradients were estimated. The kinematic GPS-only PPP solution showed 10–15% improvement in the vertical component when the WVR PWV was used as a constraint. However, the kinematic GPS + GLONASS PPP solution showed very limited improvement in the vertical precision when the WVR PWV was constrained. In general, the use of WVR PWV constraint did not improve the horizontal accuracy in either GPS-only or GPS + GLONASS PPP solutions, in either static or kinematic cases.

Microscopic imaging quality improvement through L0 gradient constraint model based on multi-fields of view analysis

The degradation of optical microscopic imaging is space-variant, and how to fast restore optical degraded image remains a special problem. Based on point spread function (PSF) estimation under each field of view (FOV), a L0 gradient-constrained image restoration method is proposed to solve optical degradation in microscopic imaging. Firstly, the whole scene is segmented into several different regions according to different FOV. The PSFs for each region are estimated from modulation transfer function (MTF) measured in advance. Secondly, a penalty function is designed using L0 gradient constraint to deblur the degraded images of each sub-FOV. Finally, a weighted stitching approach is used to stitch the restored images of multiple FOV (m-FOV). Experimental results indicate that the m-FOV analysis could well solve the problem of space-variant degradation. Compared with the other methods, both subjective and objective evaluation results prove that the L0 norm idea could rapidly and effectively restore the degraded image. The approach could be well applied to a real product.

Optimum ramp design in open pit mines

The problem studied in this paper is that of designing the optimal open pit haulage ramp that, for a given ramp width and a maximum ramp gradient, connects two points of the mine, minimising construction and operational costs. Because in-pit ramps require the removal of a considerable amount of non-valuable material (stripping), we discuss two different problems: high stripping (or in-pit) ramp design and low stripping (or ex-pit) road design. For the first situation, we present an integer programming model; in the second case, a shortest path approach is undertaken. In both cases, the models can be solved exactly, and include gradient and curvature constraints. The proposed formulations have been tested on real mine data, showing a significant reduction in cost compared to the previous mine design.

A modified search direction method for inequality constrained optimization problems using the singular-value decomposition of normalized response gradients

Motivated by the applications of the node-based shape optimization problems, where various response evaluations are often considered in constrained optimization, we propose a modified search direction method for optimization design updates. There exist numerous methods for solving constrained optimization problems. The methods of the class active-set strategy try to travel along the active constraints to solve constrained optimization problems. Contrary to the active-set methods, the algorithms of the class interior-point method reach an optimal solution by traversing the interior of the feasible domain. This characteristic is considered to be beneficial for shape optimization because the usual zig-zagging behavior when traveling along the active constraints is avoided. However, the interior-point methods require the solution of a Newton problem in each iteration step, which is generally considered to be difficult for the shape optimization problem. In the present work, we propose a modified search direction method that applies the singular-value decomposition method for the normalized objective and constraint gradients. The modified search direction is a descent direction of the objective function. Using this search direction for the design update in each iteration, we observe that the centrality conditions for the interior-point method are approached iteratively. We are able to achieve a local minimum by traversing the interior of the feasible domain by only using the gradient information of the objective and constraint functions. The results are shown first with analytical 2D problems, and then the results of shape optimization problems with a large number of design variables are discussed. To robustly deal with complex geometries, the Vertex Morphing method is used.

Image Super Resolution Based on Gradient Constrained POCS Method

A gradient constraint set based on edge detection is proposed to solve the problem of edge blurring in traditional projections onto convex sets (POCS) images reconstruction. Using the complete edge information of the original low-resolution(LR) image, the edge gradient constraint is applied to the reconstructed image, and the high-frequency information of the reconstructed image is retained to solve the problem of blurring the reconstructed image edge. And this paper uses the resolution test card to carry out simulation experiments, quantitative evaluation of the reconstructed image with respect to the LR images to improve the actual resolution of multiple. Simulation results show that the proposed method can reduce the edge blur of the reconstructed image, and the actual resolution is greatly improved compared with the traditional convex set projection method.

On an optimal control problem with gradient constraints

ABSTRACTUsually, control functions in control-constrained optimal control are chosen from a Lebesgue space. This choice, however, makes it impossible to postulate additional conditions on the control function's slope which is practically relevant in some situations. In order to overcome this disadvantage, a natural assumption would be to demand at least first-order Sobolev regularity for control functions. The present paper is devoted to the study of an elliptic optimal control problem whose control function is chosen from a Sobolev space and has to satisfy additional equality constraints on its weak gradient. Noting that the associated Karush–Kuhn–Tucker conditions do not provide a necessary optimality condition for the underlying optimal control problem in general, one cannot simply solve the problem of interest by considering the system of first-order optimality conditions. Instead a penalization procedure with strong convergence properties for the computational solution is suggested and its computational implementation is studied in detail. Particularly, some essential difficulties arising from the gradient constraints which do not appear in standard optimal control are discussed.

Homogenization of some quasi-linear elliptic equations with gradient constraints

We prove a homogenization formula for quasi-linear elliptic equations with gradient constraints on a disperse set, within the framework of monotonic operator theory and compensated compactness methods.

Data Rate Utility Analysis for Uplink Two-Hop Internet of Things Networks

We study the fundamental problem of spectrum allocation and device association in uplink two-hop Internet of Things (IoT) networks under two spectrum allocation schemes: 1) orthogonal spectrum partition (OSP) and 2) full spectrum reuse (FSR). We propose a novel analytical model to estimate the uplink data rate utility function, which takes into account power control fractional and spatial density of aggregators. We then compute the optimal aggregator association bias (for the FSR scheme) and the optimal joint spectrum partition ratio and optimal aggregator association bias (for the OSP scheme) using constraint gradient ascent optimization. Using the above obtained optimal values and the proposed model, we compare the performance of the optimized OSP and FSR schemes with the benchmark maximum-SIR-based association scheme and the minimum-distance association scheme in terms of the cumulative distribution function of device uplink data rate. By optimizing key network parameters, namely the spectrum partition ratio and aggregator association bias, we mitigate interference and enhance the mean uplink per-device data rate for both FSR and OSP. To the best of our knowledge, this paper is the first that proposes an analytical model to estimate the log utility of the uplink data rate of two-hop IoT networks.

MPB-CNN: a multi-scale parallel branch CNN for choroidal neovascularization segmentation in SD-OCT images

Choroidal neovascularization (CNV) generally appears in the advanced stage of age-related macular degeneration (AMD) and has more complex pathological characteristics comparing with most other retinal diseases. So far, few literatures focused on the CNV segmentation from a machine learning and computer vision perspective. Therefore, accurate CNV segmentation is a challenging problem in SD-OCT images. In this paper, a multi-scale parallel branch CNN (MPB-CNN) is proposed for CNV segmentation. First, three parallel branch networks are used for multi-scale feature extraction. In each branch, standard convolution is replaced with atrous convolution, in which wider and more powerful multi-scale information can be extracted due to the sparsity of convolutional kernels. To further improve the segmentation results, intra-branch connections are introduced to preserve signal transmission and inter-branch connections are introduced to interact multi-scale information. Then, feature maps from different branches are cascaded with low-level features computing by several convolutional layers. The combined feature is used as the input to acquire the final prediction map by stacking three convolutional layers. Besides, extra branch supervisions are applied at the end of each branch to guarantee the discrimination of feature representations from each branch and benefit the network optimization. Finally, gradient constraint is added to loss function to preserve the boundary of the CNV lesion. An effective cross validation is performed on 202 cubes from 12 patients based on patient independence with the mean dice value being 0.757 and the mean overlap ratio being 60.8%. Experiment results indicated that the proposed MPB-CNN can provide reliable segmentations for CNV from SD-OCT images.

Image Fusion Based on Kernel Estimation and Data Envelopment Analysis

This paper reports the improvement of the image quality during the fusion of remote sensing images by minimizing a novel energy function. First, by introducing a gradient constraint term in the energy function, the spatial information of the panchromatic image is transferred to the fused results. Second, the spectral information of the multispectral image is preserved by importing a kernel function to the data fitting term in the energy function. Finally, an objective parameter selection method based on data envelopment analysis (DEA) is proposed to integrate state-of-the-art image quality metrics. Visual perception measurement and selected fusion metrics are employed to evaluate the fusion performance. Experimental results show that the proposed method outperforms other established image fusion techniques.

Flight phase and altitude-dependent geometrical vertical flight plan optimization minimizing the total number of vertical plan segments

This paper presents a new method for the geometrical construction of an optimal vertical flight plan associated to a provided lateral flight plan defined as a succession of waypoints characterized by their along-the-track distance relative to the first waypoint and their constraints. The principal objective of the proposed method is the minimization of the total number of vertical flight plan segments, whose slope values closest match the values set for their corresponding flight phase and altitude. The main advantage of the proposed method is that it constructs the optimized vertical flight plan employing faster—and less-intensive computations than methods based solely on aircraft performance models. Also, the proposed algorithm has the advantage of generating ground-fixed predicted vertical flight plans which, when flown, are less sensitive to varying wind conditions, thus, smaller trajectory deviations than those computed using solely the model-based algorithms. Two implementations corresponding to different trade-offs between conflicting preferred gradient and minimal segment length constraints were compared. The results show that a vertical flight path segment’s construction and resulting configuration is dependent on the configuration of the vertical flight plan segments that precede it. The results also show that for a majority of the test cases, the resulting flight plans computed using the two implementations were identical. Moreover, even when the flight plans were not completely identical, many of the corresponding segments were identical.

Implicit surface reconstruction based on generalized radial basis functions interpolant with distinct constraints

• We propose a generalized radial basis function interpolant with difference constraints. • Various types of constraints are derived to satisfy the interpolation conditions . • An adaptive iterative algorithm is used for approximating gradient and tangent constraints. • The numerical results reveal the improved performance of our method. In this work, we present a new algorithm for reconstruction of implicit surfaces from a set of cloud points with normals (Hermite data), based on the generalized radial basis functions interpolant with various types of constraints. Our key contribution is a novel construction of difference constraints in the interpolant to satisfy certain linear functional data, named as the domain constraints, the difference constraints of the gradient and the difference constraints of the tangent. To avoid ambiguous constraints, we apply an iterative algorithm to determine the adaptive distance in the difference constraints. The extended interpolant can not only provide an exact interpolation to the function values, but also approximate the function's derivatives by constructing a difference operator along the gradient or tangent direction. We interpolate the sampling points using our method by constructing a signed distance field in the geometry domain. Compared to the Hermite–Birkhoff interpolation with RBFs, the interpolant we use has faster solution efficiency. The experimental results of several data sets show the reliability and performance of our method in a variety of practical scenarios.

Three-Dimensional Reconstruction Method for Machined Surface Topography Based on Gray Gradient Constraints

In the modern engineering field, recovering the machined surface topography is important for studying mechanical product function and surface characteristics by using the shape from shading (SFS)-based reconstruction method. However, due to the limitations of many constraints and oversmoothing, the existing SFS-based reconstruction methods are not suitable for machined surface topography. This paper presents a new three-dimensional (3D) reconstruction method of machined surface topography. By combining the basic principle of SFS and the analytic method, the analytic model of a surface gradient is established using the gray gradient as a constraint condition. By efficiently solving the effect of quantization errors and ambiguity of the gray scale on reconstruction accuracy using a wavelet denoising algorithm and image processing technology, the reconstruction algorithm is implemented for machined surface topography. Experimental results on synthetic images and machined surface topography images show that the proposed algorithm can accurately and efficiently recover the 3D shape of machined surface topography.

A Dimensional Reduction Algorithm and Software for Acyclically Dependent Constraints

For discrete equations of motion with acyclic equality constraints and within the context of the null-space method, an original Algorithm is introduced. By first permuting and then topologically ordering the degrees-of-freedom in the constraint gradient matrix, the saddle point problem can be solved with a sparse triangular system for the constraint equations. In this work, we show that saddle problems resulting from constrained (nonlinear) mechanical problems can always be set in this form, with constraint pivots being selected a priori. Given n discrete motion equations and m equality constraints, the original square sparse system is replaced by a sparse system and a sparse triangular solve with m2 coefficients and n – m right-hand sides. This triangular solve, which involves three sparse matrices (in existing literature only two of the three matrices are sparse), is here discussed in detail. Seven sparse operations are addressed (five standard and two nonstandard) in addition to some specific ad-hoc operations. Algorithms, source code and examples are presented in this work.