Gradient Optimization Method Research Articles

On Some Works of Boris Teodorovich Polyak on the Convergence of Gradient Methods and Their Development

The paper presents a review of the current state of subgradient and accelerated convex optimization methods, including the cases with the presence of noise and access to various information about the objective function (function value, gradient, stochastic gradient, higher derivatives). For nonconvex problems, the Polyak–Lojasiewicz condition is considered and a review of the main results is given. The behavior of numerical methods in the presence of a sharp minimum is considered. The aim of this review is to show the influence of the works of B.T. Polyak (1935–2023) on gradient optimization methods and their surroundings on the modern development of numerical optimization methods.

A projection-free method for solving convex bilevel optimization problems

AbstractWhen faced with multiple minima of an inner-level convex optimization problem, the convex bilevel optimization problem selects an optimal solution which also minimizes an auxiliary outer-level convex objective of interest. Bilevel optimization requires a different approach compared to single-level optimization problems since the set of minimizers for the inner-level objective is not given explicitly. In this paper, we propose a new projection-free conditional gradient method for convex bilevel optimization which requires only a linear optimization oracle over the base domain. We establish $$O(t^{-1/2})$$ O ( t - 1 / 2 ) convergence rate guarantees for our method in terms of both inner- and outer-level objectives, and demonstrate how additional assumptions such as quadratic growth and strong convexity result in accelerated rates of up to $$O(t^{-1})$$ O ( t - 1 ) and $$O(t^{-2/3})$$ O ( t - 2 / 3 ) for inner- and outer-levels respectively. Lastly, we conduct a numerical study to demonstrate the performance of our method.

New hybrid conjugate gradient method for nonlinear optimization with application to image restoration problems

New hybrid conjugate gradient method for nonlinear optimization with application to image restoration problems

Grid Optimization of Free-Form Spatial Structures Considering the Mechanical Properties

In recent years, the application of free-form surface spatial grid structures in large public buildings has become increasingly common. The layouts of grids are important factors that affect both the mechanical performance and aesthetic appeal of such structures. To achieve a triangular grid with good mechanical performance and uniformity on free-form surfaces, this study proposes a new method called the “strain energy gradient optimization method”. The grid topology is optimized to maximize the overall stiffness, by analyzing the sensitivity of nodal coordinates to the overall strain energy. The results indicate that the overall strain energy of the optimized grid has decreased, indicating an improvement in the structural stiffness. Specifically, compared to the initial grid, the optimized grid has a 30% decrease in strain energy and a 43.3% decrease in maximum nodal displacement. To optimize the smoothness of the grid, the study further applies the Laplacian grid smoothing method. Compared to the mechanically adjusted grid, the structural mechanical performance does not significantly change after smoothing, while the geometric indicators are noticeably improved, with smoother lines and regular shapes. On the other hand, compared to the initial grid, the smoothed grid has a 21.4% decrease in strain energy and a 28.3% decrease in maximum nodal displacement.

Deep bidirectional hierarchical matrix factorization model for hyperspectral unmixing

Existing deep nonnegative matrix factorization-based approaches treat shallow and deep layers equally or with similar strategies, neglecting the heterogeneous physical structures between the shallow and progressively deeper layers, thus failing to explore the latent different sub-manifold structures in the hyperspectral image. In this paper, we propose a deep nonnegative matrix factorization model with bidirectional constraints to achieve hyperspectral unmixing. The sub-manifold structures in hyperspectral image are fully exploited by filtering and penalizing the shallow abundance layer with a denoised regularizer and a manifold regularizer. In contrast to the shallow abundance layer, the remaining layers are constrained by an extremely common regularizer to avoid over-denoising and maintain fidelity. In this way, the fine cues between different substances are exploited to a large extent. Additionally, the overall reconstruction error can be well controlled because the performance of the designed feedback mechanism can be fine-tuned by the inverse hierarchical constraints. Finally, we employ Nesterov's optimal gradient method to solve the optimization problem effectively. Experiment results are conducted on both synthetic datasets and real datasets, and all results show that the proposed method is superior to recent canonical unmixing methods.

New Formula on the Conjugate Gradient Method for Unconstrained Optimization and its Application

In this paper, we have presented a new formula by applying the conjugacy condition for the second-order Taylor expansion. In comparison with classic conjugate gradient methods, the new formula uses both available gradient and function value information. The formula's global convergence findings are described. Numerical results demonstrate that this strategy is effective and its application.

Two modified conjugate gradient methods for unconstrained optimization

Conjugate gradient (CG) methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems. Motivated by the construction of some modern CG methods, we propose two modified CG methods, named DHS* and DPRP*. Under the strong Wolfe line search (SWLS), the two presented methods are proven to be sufficient descent and globally convergent. Preliminary numerical results show that the DHS* and DPRP* methods are effective for the given test problems.

A feasible smoothing accelerated projected gradient method for nonsmooth convex optimization

Smoothing accelerated gradient methods achieve faster convergence rates than that of the subgradient method for some nonsmooth convex optimization problems. However, Nesterov's extrapolation may require gradients at infeasible points, and thus they cannot be applied to some structural optimization problems. We introduce a variant of smoothing accelerated projected gradient methods where every variable is feasible. The O(k−1log⁡k) convergence rate is obtained using the Lyapunov function. We conduct a numerical experiment on the robust compliance optimization of a truss structure.

Deep reinforcement learning-based low-latency task offloading for mobile-edge computing networks

Wireless Mobile Edge Computing (MEC) is an emerging solution to facilitate users’ computing. However, there are still some drawbacks, such as slow learning speed and high task latency. To overcome these difficulties, a deep reinforcement learning (DRL)-based task offloading (DRTO) framework is proposed, taking into account the time-varying nature of wireless channels and the unpredictability of task arrivals in multi-user MEC systems. Many deep neural networks (DNNs) are also used as scalable solutions to quickly determine the best offloading strategy by combining the perceptual capabilities of deep learning with the decision making capabilities of reinforcement learning (RL). Additionally, the concept of a minimum upper bound on the target Q-value is utilized to address the value overestimation problem that occurs during offload policy updates. This approach aims to minimize the latency of offload decisions by considering different channel environments, variable latency constraints among many users, and limited computational resources. Simulation results show that the algorithm reduces computation time and latency while achieving near-optimal performance compared to traditional Deep Deterministic Policy Gradient (DDPG) and Q-learning optimization methods. In addition, DRTO achieves more than 99.6% computational speedup compared to current almost ideal benchmarking techniques. Thus, the DRTO computational offload approach offers great potential for reducing overall system latency and improving the user experience.

Nonlinear conjugate gradient methods for unconstrained set optimization problems whose objective functions have finite cardinality

In this paper, we propose nonlinear conjugate gradient methods for unconstrained set optimization problems in which the objective function is given by a finite number of continuously differentiable vector-valued functions. First, we provide a general algorithm for the conjugate gradient method using Wolfe line search but without imposing an explicit restriction on the conjugate parameter. Later, we study two variants of the algorithm, namely Fletcher-Reeves and conjugate descent, using two different choices of the conjugate parameter but with the same line search rule. In the general algorithm, the direction of movement at each iterate is identified by finding out a descent direction of a vector optimization problem. This vector optimization problem is identified with the help of the concept of partition set at the current iterate. As this vector optimization problem is different at different iterations, the conventional conjugate gradient method for vector optimization cannot be straightly extended to solve the set optimization problem under consideration. The well-definedness of the methods is provided. Further, we prove the Zoutendijk-type condition, which assists in proving the global convergence of the methods even without a regular condition of the stationary points. No convexity assumption is assumed on the objective function to prove the convergence. Lastly, some numerical examples are illustrated to exhibit the performance of the proposed method. We compare the performance of the proposed conjugate gradient methods with the existing steepest descent method. It is found that the proposed method commonly outperforms the existing steepest descent method for set optimization.

Surrogate gradient methods for data-driven foundry energy consumption optimization

In many industrial applications, data-driven models are more and more commonly employed as an alternative to classical analytical descriptions or simulations. In particular, such models are often used to predict the outcome of an industrial process with respect to specific quality characteristics from both observed process parameters and control variables. A major step in proceeding from purely predictive to prescriptive analytics, i.e., towards leveraging data-driven models for process optimization, consists of, for given process parameters, determining control variable values such that the output quality improves according to the process model. This task naturally leads to a constrained optimization problem for data-driven prediction algorithms. In many cases, however, the best available models suffer from a lack of regularity: methods such as gradient boosting or random forests are generally non-differentiable and might even exhibit discontinuities. The optimization of these models would therefore require the use of derivative-free techniques. Here, we discuss the use of alternative, independently trained differentiable machine learning models as a surrogate during the optimization procedure. While these alternatives are generally less accurate representations of the actual process, the possibility of employing derivative-based optimization methods provides major advantages in terms of computational performance. Using classical benchmarks as well as a real-world dataset obtained from an industrial environment, we demonstrate that these advantages can outweigh the additional model error, especially in real-time applications.

A modified PRP conjugate gradient method for unconstrained optimization and nonlinear equations

A modified Polak Ribiere Polyak(PRP) conjugate gradient(CG) method is proposed for solving unconstrained optimization problems. The search direction generated by this method satisfies sufficient descent condition at each iteration and this method inherits one remarkable property of the standard PRP method. Under the standard Armijo line search, the global convergence and the linearly convergent rate of the presented method is established. Some numerical results are given to show the effectiveness of the proposed method by comparing with some existing CG methods.

Understanding world models through multi-step pruning policy via reinforcement learning

In model-based reinforcement learning, the conventional approach to addressing world model bias is to use gradient optimization methods. However, using a singular policy from gradient optimization methods in response to a world model bias inevitably results in an inherently biased policy. This is because of constraints on the imperfect and dynamic data of state-action pairs. The gap between the world model and the real environment can never be completely eliminated. This article introduces a novel approach that explores a variety of policies instead of focusing on either world model bias or singular policy bias. Specifically, we introduce the Multi-Step Pruning Policy (MSPP), which aims to reduce redundant actions and compress the action and state spaces. This approach encourages a different perspective within the same world model. To achieve this, we use multiple pruning policies in parallel and integrate their outputs using the cross-entropy method. Additionally, we provide a convergence analysis of the pruning policy theory in tabular form and an updated parameter theoretical framework. In the experimental section, the newly proposed MSPP method demonstrates a comprehensive understanding of the world model and outperforms existing state-of-the-art model-based reinforcement learning baseline techniques.

On the convergence analysis of a proximal gradient method for multiobjective optimization

On the convergence analysis of a proximal gradient method for multiobjective optimization

You only compress once: Towards effective and elastic BERT compression via exploit–explore stochastic nature gradient

Despite superior performance on various natural language processing tasks, pre-trained models such as BERT are challenged by deploying on resource-constraint devices. Most existing model compression approaches require re-compression or fine-tuning across diverse constraints to accommodate various hardware deployments. This practically limits the further application of model compression. Moreover, the ineffective training and searching process of existing elastic compression paradigms (Wang et al., 2020; Cai et al., 2020) prevents the direct migration to BERT compression. Motivated by the necessity of efficient inference across various constraints on BERT, we propose a novel approach, YOCO-BERT, to achieve compress once and deploy everywhere. Specifically, we first construct a huge search space with 1013 architectures, which covers nearly all configurations in BERT model. Then, we propose a novel stochastic nature gradient optimization method to guide the generation of optimal candidate architecture which could keep a balanced trade-off between explorations and exploitation. When a certain resource constraint is given, a lightweight distribution optimization approach is utilized to obtain the optimal network for target deployment without fine-tuning. Compared with state-of-the-art algorithms, YOCO-BERT provides more compact models, yet achieving 2.1%–4.5% average accuracy improvement on the GLUE benchmark. Besides, YOCO-BERT is also more effective, e.g., the training complexity is O(1) for N different devices. Codes available https://github.com/MAC-AutoML/YOCO-BERT.

Discrete Adjoint Optimization Method for Low-Boom Aircraft Design Using Equivalent Area Distribution

This paper introduces a low-boom aircraft optimization design method guided by equivalent area distribution, which effectively improves the intuitiveness and refinement of inverse design. A gradient optimization method based on discrete adjoint equations is proposed to achieve the fast solution of the gradient information of target equivalent area distribution relative to design variables and to drive the aerodynamic shape update to the optimal solution. An optimization experiment is carried out based on a self-developed supersonic civil aircraft configuration with engines. The results show that the equivalent area distribution adjoint equation can accurately solve the gradient information. After optimization, the sonic boom level of the aircraft was reduced by 13.2 PLdB, and the drag coefficient was reduced by 60.75 counts. Moreover, the equivalent area distribution adjoint optimization method has outstanding advantages, such as high sensitivity and fast convergence speed, and can take both the low sonic boom and the low drag force of the aircraft into account, providing a powerful tool for the comprehensive optimization design of supersonic civil aircraft by considering sonic boom and aerodynamic force.

A geometric integration approach to smooth optimization: foundations of the discrete gradient method

Abstract Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimization problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we present a thorough analysis of the discrete gradient method for optimization that provides a solid theoretical foundation. We show that the discrete gradient method is well-posed by proving the existence of iterates for any positive time step, as well as uniqueness in some cases, and propose an efficient method for solving the associated discrete gradient equation. Moreover, we establish an $\text{O}(1/k)$ convergence rate for convex objectives and prove linear convergence if instead the Polyak–Łojasiewicz inequality is satisfied. The analysis is carried out for three discrete gradients—the Gonzalez discrete gradient, the mean value discrete gradient, and the Itoh–Abe discrete gradient—as well as for a randomised Itoh–Abe method. Our theoretical results are illustrated with a variety of numerical experiments, and we furthermore demonstrate that the methods are robust with respect to stiffness.

On the Class of Wei-Yao-Liu Conjugate Gradient Methods for Vector Optimization

On the Class of Wei-Yao-Liu Conjugate Gradient Methods for Vector Optimization

Ambient noise imaging for municipal solid waste landfill structure detection based on the common-midpoint two-station analysis with distributed acoustic sensing

SUMMARY Distributed acoustic sensing (DAS) enables high-density sampling of seismic wavefields at low cost compared to conventional geophones. This capability facilitates structural detection of a municipal solid waste (MSW) landfill, which is important for protecting the surrounding ecosystem. However, processing the vast amount of data from DAS array for ambient noise imaging can be computationally intensive. To address this, we employed the common-midpoint two-station (CMP-TS) analysis to enhance the efficiency of ambient noise imaging in the MSW landfill. CMP-TS analysis involves selecting pairs of traces at equal distances on both sides with the subarray midpoint as symmetry, which reduces the number of DAS array recordings for cross-correlation calculations. After positioning the DAS arrays linearly on top of the MSW landfill to automatically collect ambient noise, we used the CMP-TS analysis in the cross-correlation calculations to speed up the measurement of dispersion. The S-wave velocity structure of the study region was obtained quickly by inverting the extracted dispersion curves using the gradient optimization method. Ambient noise imaging based on CMP-TS analysis with DAS was applied to a test of an area-type MSW landfill. The resulting S-wave velocity section revealed a discontinuous low-velocity zone, validated by the high-density resistivity method. This low-velocity zone was interpreted as containing leachate from waste decomposition, and its discontinuity may be caused by excessive differences in the waste residues settling rates under compaction. Employing CMP-TS analysis in ambient noise data collected by DAS offers more cost-effective monitoring and a reliable basis for environmental pollution prevention and control.

A modified Fletcher-Reeves conjugate gradient method for unconstrained optimization with applications in image restoration

A modified Fletcher-Reeves conjugate gradient method for unconstrained optimization with applications in image restoration