Inexact Method Research Articles

Customized Douglas-Rachford splitting methods for structured inverse variational inequality problems

Recently, structured inverse variational inequality (SIVI) problems have attracted much attention. In this paper, we propose new splitting methods to solve SIVI problems by employing the idea of the classical Douglas-Rachford splitting method (DRSM). In particular, the proposed methods can be regarded as a novel application of the DRSM to SIVI problems by decoupling the linear equality constraint, leading to smaller and easier subproblems. The main computational tasks per iteration are the evaluations of certain resolvent operators, which are much cheaper than those methods without taking advantage of the problem structures. To make the methods more implementable in the general cases where the resolvent operator is evaluated in an iterative scheme, we further propose to solve the subproblems in an approximate manner. Under quite mild conditions, global convergence, sublinear rate of convergence, and linear rate of convergence results are established for both the exact and the inexact methods. Finally, we present preliminary numerical results to illustrate the performance of the proposed methods.

Methane Emissions Quantification and Resulting Methane Emissions Reduction in the Permian Basin Enabled by Automated Unmanned Systems

Summary Traditional methods for monitoring emissions from production operations have typically used optical gas imaging cameras or Method 21 systems, based on an intermittent basis to determine and document methane gas leaks, which are then subsequently identified for repair under the US Code of Federal Regulations (2017). These optical gas imaging emissions monitoring surveys can have a subjective bias, are highly conditional on the skill of the camera operator, and are an inexact method of measuring the quantity of the leak rate. With a renewed industry emphasis on methane emissions measurement and reduction, this paper describes a case study using a high-sensitivity sensor technology (laser absorption spectrometry) specifically targeting methane emissions, the unique capabilities engendered by its deployment on unmanned aerial systems (UAS), the leveraging of automation in field-operation and data analysis, and the system’s successful utilizationin enabling emissions limitations over several production sites in the Permian Basin. The use of automation enabled categorization of the leak type and intensity, and triage according to leak rate, facilitating prompt remedial action and directly limiting emissions. By automating the comprehensive flight paths specific to equipment groups (e.g., compressors, tanks, and flares), targeted repeat surveys confirmed that specific leaks were fixed, emphasizing a general downward trend in overall site- and asset-level emissions. These surveys were completed in 22.5 minutes, on average, at each of the five sites. Additionally, the use of high-resolution UAS-generated orthomosaic maps enabled the direct placement of emissions data into the context of the operations at the time of the survey, facilitating the generation of automated actionable reports, helping direct repair teams, and resulting in effective and necessary fixes. Furthermore, the campaign validated that following the set up of the initial survey, subsequent regular, repeat surveys could be commissioned at the “push of a button,” yielding reliable, actionable emissions data, with a direct impact on both environmental (6% reduction in emissions) and financial impact.

Inexact proximal point algorithm for quasiconvex optimization problems on Hadamard manifolds

ABSTRACT In this paper, by using the inexact scalarization proximal methods, we solve quasiconvex multiobjective optimization problems on Hadamard manifolds. Under some assumptions on the multifunction of problem and vector fields, our methods are proved to be convergent to a Pareto critical point of the problem. For the convex case, the sequences generated by the methods converge to a weak Pareto solution.

Discriminant Subspace Low-Rank Representation Algorithm for Electroencephalography-Based Alzheimer's Disease Recognition.

Alzheimer’s disease (AD) is a chronic progressive neurodegenerative disease that often occurs in the elderly. Electroencephalography (EEG) signals have a strong correlation with neuropsychological test results and brain structural changes. It has become an effective aid in the early diagnosis of AD by exploiting abnormal brain activity. Because the original EEG has the characteristics of weak amplitude, strong background noise and randomness, the research on intelligent AD recognition based on machine learning is still in the exploratory stage. This paper proposes the discriminant subspace low-rank representation (DSLRR) algorithm for EEG-based AD and mild cognitive impairment (MCI) recognition. The subspace learning and low-rank representation are flexibly integrated into a feature representation model. On the one hand, based on the low-rank representation, the graph discriminant embedding is introduced to constrain the representation coefficients, so that the robust representation coefficients can preserve the local manifold structure of the EEG data. On the other hand, the least squares regression, principle component analysis, and global graph embedding are introduced into the subspace learning, to make the model more discriminative. The objective function of DSLRR is solved by the inexact augmented Lagrange multiplier method. The experimental results show that the DSLRR algorithm has good classification performance, which is helpful for in-depth research on AD and MCI recognition.

A Note on the Optimal Convergence Rate of Descent Methods with Fixed Step Sizes for Smooth Strongly Convex Functions

Based on a result by Taylor et al. (J Optim Theory Appl 178(2):455–476, 2018) on the attainable convergence rate of gradient descent for smooth and strongly convex functions in terms of function values, an elementary convergence analysis for general descent methods with fixed step sizes is presented. It covers general variable metric methods, gradient-related search directions under angle and scaling conditions, as well as inexact gradient methods. In all cases, optimal rates are obtained.

Proximal point methods with possible unbounded errors for monotone operators in Hadamard spaces

In this paper, we investigate and analyse the strong convergence of the sequence generated by an inexact proximal point method with possible unbounded errors for finding zeros of monotone operators in Hadamard spaces. We show that the boundedness of the generated sequence is equivalent to the zero set of the operator to be nonempty. In this case, we prove the strong convergence of the generated sequence to a zero of the operator. We also provide some applications of our main results and give a numerical example to show the performance of the proposed algorithm.

An inexact successive quadratic approximation method for a class of difference-of-convex optimization problems

An inexact successive quadratic approximation method for a class of difference-of-convex optimization problems

Exact and inexact Douglas–Rachford splitting methods for solving large-scale sparse absolute value equations

AbstractExact and inexact Douglas–Rachford splitting methods are developed to solve the large-scale sparse absolute value equation (AVE) $Ax - |x| =b$, where $A\in \mathbb {R}^{n\times n}$ and $b\in \mathbb {R}^n$. The inexact method adopts a relative error tolerance and, therefore, in the inner iterative processes, the LSQR method is employed to find a qualified approximate solution of each subproblem, resulting in a lower cost for each iteration. When $\|A^{-1}\|\le 1$ and the solution set of the AVE is nonempty, the algorithms are globally and linearly convergent. When $\|A^{-1}\|= 1$ and the solution set of the AVE is empty, the sequence generated by the exact algorithm diverges to infinity on a trivial example. Numerical examples are presented to demonstrate the viability and robustness of the proposed methods.

A Class of Inexact Secant Algorithms with Line Search Filter Method for Nonlinear Programming

We propose a class of inexact secant methods in association with the line search filter technique for solving nonlinear equality constrained optimization. Compared with other filter methods that combine the line search method applied in most large-scale optimization problems, the inexact line search filter algorithm is more flexible and realizable. In this paper, we focus on the analysis of the local superlinear convergence rate of the algorithms, while their global convergence properties can be obtained by making an analogy with our previous work. These methods have been implemented in a Matlab code, and detailed numerical results indicate that the proposed algorithms are efficient for 43 problems from the CUTEr test set.

Inexact methods for sequential fully implicit (SFI) reservoir simulation

The sequential fully implicit (SFI) scheme was introduced (Jenny et al. J. Comput. Phys. 217(2), 627–641 2006) for solving coupled flow and transport problems. Each time step for SFI consists of an outer loop, in which there are inner Newton loops to implicitly and sequentially solve the pressure and transport sub-problems. In standard SFI, the sub-problems are usually solved with tight tolerances at every outer iteration. This can result in wasted computations that contribute little progress towards the coupled solution. The issue is known as ‘over-solving’. Our objective is to minimize the cost of inner solvers while maintaining the convergence rate of SFI. We first extended a nonlinear-acceleration (NA) framework (Jiang and Tchelepi, Comput. Methods Appl. Mech. Eng. 352, 246–275, 2019) to multi-component compositional models, for ensuring robust outer-loop convergence. We then developed inexact-type methods that alleviate ‘over-solving’. It is found that there is no need for one sub-problem to strive for perfection, while the coupled (outer) residual remains high due to the other sub-problem. The new SFI solver was tested using several complex cases. The problems involve multi-phase and EoS-based compositional fluid systems. We compared different strategies such as fixed relaxations on absolute and relative tolerances for the inner solvers, as well as an adaptive approach. The results show that the basic SFI method is quite inefficient. Away from a coupled solution, additional accuracy achieved in inner solvers is wasted, contributing to little or no reduction of the overall outer residual. By comparison, the adaptive inexact method provides relative tolerances adequate for the current convergence state of the sub-problems. We show across a wide range of flow conditions that the new solver can effectively resolve the over-solving issue, and thus greatly improve the overall efficiency.

Inexact line search method in full waveform inversion

Inexact line search method in full waveform inversion

A New Method to Solve Fuzzy Interval Flexible Linear Programming Using a Multi-Objective Approach

Recently fuzzy interval flexible linear programs have attracted many interests. These models are an extension of the classical linear programming which deal with crisp parameters. However, in most of the real-world applications, the nature of the parameters of the decision-making problems is generally imprecise. Such uncertainties can lead to increased complexities in the related optimisation efforts. Simply ignoring these uncertainties is considered undesired as it may result in inferior or wrong decisions. Therefore, inexact linear programming methods are desired under uncertainty. In this paper, we concentrate a fuzzy flexible linear programming model with flexible constraints and the interval objective function and then propose a new solving approach based on solving an associated multi-objective model. Finally, numerical example is included to illustrate the mentioned solving process.

Inexact variable metric method for convex-constrained optimization problems

This paper is concerned with the inexact variable metric method for solving convex-constrained optimization problems. At each iteration of this method, the search direction is obtained by inexactly minimizing a strictly convex quadratic function over the closed convex feasible set. Here, we propose a new inexactness criterion for the search direction subproblems. Under mild assumptions, we prove that any accumulation point of the sequence generated by the new method is a stationary point of the problem under consideration. In order to illustrate the practical advantages of the new approach, we report some numerical experiments. In particular, we present an application where our concept of the inexact solutions is quite appealing.

Inexact rational Krylov method for evolution equations

Linear and nonlinear evolution equations have been formulated to address problems in various fields of science and technology. Recently, methods using an exponential integrator for solving evolution equations, where matrix functions must be computed repeatedly, have been investigated and refined. In this paper, we propose a new method for computing these matrix functions which is called an inexact rational Krylov method. This is a more efficient version of the rational Krylov method with appropriate shifts, which was proposed by Hashimoto and Nodera (ANZIAM J 58:C149–C161, 2016). The advantage of the inexact rational Krylov method is that it computes linear equations that appear in the rational Krylov method efficiently while guaranteeing the accuracy of the final solution.

New inertial proximal gradient methods for unconstrained convex optimization problems

The proximal gradient method is a highly powerful tool for solving the composite convex optimization problem. In this paper, firstly, we propose inexact inertial acceleration methods based on the viscosity approximation and proximal scaled gradient algorithm to accelerate the convergence of the algorithm. Under reasonable parameters, we prove that our algorithms strongly converge to some solution of the problem, which is the unique solution of a variational inequality problem. Secondly, we propose an inexact alternated inertial proximal point algorithm. Under suitable conditions, the weak convergence theorem is proved. Finally, numerical results illustrate the performances of our algorithms and present a comparison with related algorithms. Our results improve and extend the corresponding results reported by many authors recently.

On inexact projected gradient methods for solving variable vector optimization problems

Variable order structures model situations in which the comparison between two points depends on a point-to-cone map. In this paper, inexact projected gradient methods for solving smooth constrained vector optimization problems on variable ordered spaces are presented. It is shown that every accumulation point of the generated sequences satisfies the first-order necessary optimality condition. Moreover, under suitable convexity assumptions for the objective function, it is proved that all accumulation points of any generated sequences are weakly efficient points. The convergence results are also derived in the particular case in which the problem is unconstrained and even if inexact directions are taken as descent directions. Furthermore, we investigate the application of the proposed method to optimization models where the domain of the variable order map coincides with the image of the objective function. In this case, similar concepts and convergence results are presented. Finally, some computational experiments designed to illustrate the behavior of the proposed inexact methods versus the exact ones (in terms of CPU time) are performed.

A multi-level ADMM algorithm for elliptic PDE-constrained optimization problems

In this paper, elliptic PDE-constrained optimization problems with box constraints on the control are considered. To numerically solve the problems, we apply the ‘optimize-discretize-optimize’ strategy. Specifically, the alternating direction method of multipliers (ADMM) algorithm is applied in function space first, and then, the standard piecewise linear finite-element approach is employed to discretize the subproblems in each iteration. Finally, some efficient numerical methods are applied to solve the discretized subproblems based on their structures. Motivated by the idea of the multi-level strategy, instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, the subproblems in each iteration are solved by appropriate inexact methods. Based on the strategies above, an efficient convergent multi-level ADMM (mADMM) algorithm is proposed. We present the convergence analysis and the iteration complexity results o(1/k) for the mADMM algorithm. Some numerical experiments are done and the numerical results show the high efficiency of the mADMM algorithm.

Modeling and Engineering Constrained Shortest Path Algorithms for Battery Electric Vehicles

We study the problem of computing constrained shortest paths for battery electric vehicles. Because battery capacities are limited, fastest routes are often infeasible. Instead, users are interested in fast routes on which the energy consumption does not exceed the battery capacity. For that, drivers can deliberately reduce speed to save energy. Hence, route planning should provide both path and speed recommendations. To tackle the resulting [Formula: see text]-hard optimization problem, previous work trades correctness or accuracy of the underlying model for practical running times. We present a novel framework to compute optimal constrained shortest paths (without charging stops) for electric vehicles that uses more realistic physical models, while taking speed adaptation into account. Careful algorithm engineering makes the approach practical even on large, realistic road networks: We compute optimal solutions in less than a second for typical battery capacities, matching the performance of previous inexact methods. For even faster query times, the approach can easily be extended with heuristics that provide high quality solutions within milliseconds.

Exact line search method for using the L1-norm misfit function in full waveform inversion

Full waveform inversion (FWI) is a non-linear inverse problem that can be sensitive to noise. The tolerance of the noise-interference characteristics depends on the types of misfit functions. To date, different misfit functions, such as the least-squares norm (L2), the least-absolute-value norm (L1), and combinations of the two (e.g., the Huber and hybrid criteria), have been applied to FWI. The L2 norm is highly sensitive to non-Gaussian errors in the data and gives rise to high-amplitude artifacts in reconstructed models. For non-Gaussian noise data, the L1 norm and the Huber and hybrid criteria always reliably reconstruct models. However, the Huber and hybrid criteria require tedious error investigations to estimate their threshold criterion. Thus, the L1 norm is adopted here to improve the anti-noise ability of the FWI. The step length is closely related to the misfit function, and an optimal step-length estimation method can rapidly make the FWI algorithm reach the global minimum, with a reduced number of iterations and fewer extra forward modeling simulations during each iteration. The step length can usually be obtained using the exact or inexact line search method. Generally, the exact line search method is faster than the inexact one. Therefore, we derived an exact line search method for the L1 norm in the FWI process. Its effectiveness was tested using noise-free data from Overthrust and the SEG/EAGE salt models. The results demonstrate that this method can recover high-resolution velocity models with low computational costs. Numerical tests using the synthetic Overthrust model contaminated by strong noise were used to further validate the robustness of this exact line search method.

Fast algorithm with theoretical guarantees for constrained low-tubal-rank tensor recovery in hyperspectral images denoising

Hyperspectral images (HSIs) are unavoidably degraded by mixed noise, including Gaussian noise and sparse noise. In this paper, we consider a constrained tubal rank and sparsity model (CTSD) to tackle the HSIs mixed noise removal, which characterizes the clean HSIs via the low-tubal-rank constraint and the sparse noise via the l0 and l∞ norm constraints, respectively. Due to the strong non-convexity, the CTSD model is challenging to solve. To tackle the CTSD, we develop the proximal alternating minimization (PAM) algorithm via the exact tensor singular value decomposition (t-SVD) and establish the global convergence under mild assumptions. Since the t-SVD is computationally expensive, especially for large scale images, we further design an efficient inexact PAM algorithm via an inexact t-SVD. The inexact PAM enjoys two advantages: (1) The computational complexity for SVDs of the inexact PAM (O(rn1n2n3)) is about twofold faster than that of the exact PAM (O(min(n1,n2)n1n2n3)) for r≪min(n1,n2); (2) The accuracy of the inexact PAM is theoretically guaranteed. Extensive experiments on HSIs denoising demonstrate that the exact and inexact methods both outperform comparing methods in quantitative evaluation metrics and visual effects, and the inexact PAM can compromise between the accuracy and efficiency for large scale HSIs.