Unconstrained Optimization Problems Research Articles

Solving the homogeneous Bethe-Salpeter equation with a quantum annealer

The homogeneous Bethe-Salpeter equation (hBSE), describing a bound system in a genuinely relativistic quantum-field theory framework, was solved for the first time by using a D-Wave quantum annealer. After applying standard techniques of discretization, the hBSE, in ladder approximation, can be formally transformed in a generalized eigenvalue problem (GEVP), with two square matrices: one symmetric and the other nonsymmetric. The latter matrix poses the challenge of obtaining a suitable formal approach for investigating the GEVP by means of a quantum annealer, i.e., to recast it as a quadratic unconstrained binary optimization problem. A broad numerical analysis of the proposed algorithms, applied to matrices of dimension up to 64, was carried out by using both the simulated-annealing package and the D-Wave . The numerical results very nicely compare with those obtained with standard classical algorithms, and also show interesting scalability features. Published by the American Physical Society 2024

Design, Analysis, and Validation of a Passive Parallel Continuum Ankle Exoskeleton for Support and Walking Assistance

Abstract In this paper, we propose a novel passive parallel continuum ankle exoskeleton that can provide assistive torque during ankle plantar flexion. Due to the flexible branches arranged in compliance with ankle motion and shape, the compact design can also offer some vertical support. The proposed parallel mechanism consists of two types of branches. The first type is a pre-bent flexible rod, mainly used to apply assistive force/torque during ankle plantar flexion. The second type of branch consists of a bounded sphere joint, flexible rod, and bounded sphere joint (BFB), which is mainly used for support. We formulate the kinetostatic model of the BFB branch as a series of parallelizable unconstrained optimization problems to ensure efficient solvability. After that, we derive the kinetostatic model of the proposed mechanism. After calibration, the wrench error of the kinetostatic model is 9.07%. Simulation analysis based on the calibrated model shows that the designed mechanism has high supporting stiffness and low rotational stiffness. The assistive torque caused by the nonlinear rotational stiffness in the sagittal plane is similar to that of passive clutch-like mechanisms. These properties can still be maintained when the joint center changes within a small range. Besides, a walking experiment was conducted, and the results show that the proposed design can reduce gastrocnemius activity.

Quantum-Annealing-Inspired Algorithms for Track Reconstruction at High-Energy Colliders

Charged particle reconstruction or track reconstruction is one of the most crucial components of pattern recognition in high-energy collider physics. It is known to entail enormous consumption of computing resources, especially when the particle multiplicity is high, which will be the conditions at future colliders, such as the High Luminosity Large Hadron Collider and Super Proton–Proton Collider. Track reconstruction can be formulated as a quadratic unconstrained binary optimization (QUBO) problem, for which various quantum algorithms have been investigated and evaluated with both a quantum simulator and hardware. Simulated bifurcation algorithms are a set of quantum-annealing-inspired algorithms, known to be serious competitors to other Ising machines. In this study, we show that simulated bifurcation algorithms can be employed to solve the particle tracking problem. The simulated bifurcation algorithms run on classical computers and are suitable for parallel processing and usage of graphical processing units, and they can handle significantly large amounts of data at high speed. These algorithms exhibit reconstruction efficiency and purity comparable to or sometimes improved over those of simulated annealing, but the running time can be reduced by as much as four orders of magnitude. These results suggest that QUBO models together with quantum-annealing-inspired algorithms are valuable for current and future particle tracking problems.

SDO: A novel sled dog-inspired optimizer for solving engineering problems

A new bio-inspired meta-heuristic algorithm, named Sled Dog Optimizer (SDO), is proposed in this paper. The inspiration for SDO is primarily drawn from the various behavioral patterns of sled dogs. It focuses on constructing mathematical models by simulating the process of dog sledding, training and retiring behaviors. The mutual constraints of multiple steps and the variation of several parameters make the exploitation and exploration capabilities of SDO well balanced. To test the ability of SDO, this paper sets up several different sets of experiments to be analyzed from a number of different aspects. First, SDO is qualitatively analyzed through several performance metrics, which include search history, search trajectory and population diversity. Second, SDO is compared with several novel and excellent metaheuristics at CEC 2017 as well as CEC 2020. The results show that SDO has extremely good performance in solving unconstrained optimization problems. SDO is then further applied to the CEC 2020 real-world problem test set to test its ability to solve real-world optimization problems with constraints. Finally, this paper extends and proposes a new path planning model to which SDO is applied. In these problems with constraints, SDO performs well and has a promising application. SDO demonstrates outstanding control of exploration and exploitation capabilities in all experiments. It can be said that SDO promotes the progress of meta-heuristic algorithms and provides another new powerful technology for complex optimization problems in the real world.

A new filled function method for solving constrained global optimization problems

ABSTRACT Filled function methods have been considered as effective algorithms for solving global optimization problems. However, their effectiveness is greatly affected by the selection of parameters, the noncontinuous or non-differentiable properties of the constructed filled function. In addition, many of the constructed filled functions are only for unconstrained optimization problems, and they are unable to solve constrained optimization problems. In this paper, a new filled function is constructed for solving constrained global optimization problems. The new filled function has only one parameter which needs to be adjusted, and, when the objective functions and constrained functions are all continuously differentiable functions, the constructed filled function is also a continuously differentiable function. Then, the classical local optimization methods can be used to find a better minimum of the proposed filled function and a few parameter adjustments are needed. At last, a new filled function algorithm for constrained global optimization is developed based on the proposed filled function. The new algorithm is applied to several test examples. The results of the numerical experiments show that the new filled function algorithm is effective and efficient.

Missing data recovery based on temporal smoothness and time-varying similarity for wireless sensor network

Wireless Sensor Networks (WSN) play a vital role in the Internet of Things (IoT) and show great potential in monitoring applications. However, due to harsh environmental conditions and unreliable communication links, WSN often encounter partial data loss during data collection, which inevitably affects the quality of service. To address this challenge, researchers have employed matrix completion techniques to recover missing data by exploiting the low-rank features in the data, but its accuracy is not satisfactory. This paper argues that the spatiotemporal characteristics of the data underlie its low-rank nature, enabling a more accurate capture of the intrinsic patterns within the data. Drawing on this insight, we propose a missing data recovery algorithm based on Temporal Smoothness and Time-Varying Similarity (TSTVS). Unlike traditional low-rank methods, the TSTVS algorithm directly utilizes the structural features of data in the spatiotemporal domain to establish a missing data completion model. Subsequently, the model is converted into an unconstrained optimization problem using the penalty function method, and the gradient descent method is applied to solve it, reconstructing the complete data matrix. Finally, simulation experiments were conducted on three real-world monitoring datasets, comparing the TSTVS with three low-rank methods, Efficient Data Collection Approach (EDCA), Matrix factorization with Smoothness constraints (MFS) and Data Recovery Based on Low Rank and Short-Term Stability(DRLRSS). The experimental results indicate that the proposed TSTVS algorithm consistently outperforms the three low-rank based algorithms in terms of recovery accuracy across different datasets and missing rate scenarios.

An integrated coupled oscillator network to solve optimization problems

Solving combinatorial optimization problems is essential in scientific, technological, and engineering applications, but can be very time and energy-consuming using classical algorithms executed on digital processors. Oscillator-based Ising machines offer a promising alternative by exploiting the analog coupling between electrical oscillators to solve such optimization problems more efficiently. Here we present the design and the capabilities of our scalable approach to solve Ising and quadratic unconstrained binary optimization problems. This approach includes routable oscillator connections to simplify the time-consuming embedding of the problem into the oscillator network. Our manufactured silicon chip, featuring 1440 oscillators implemented in a 28 nm technology, demonstrates the ability to solve optimization problems in 950 ns while consuming typically 319 μW per node. A frequency, phase, and delay calibration ensures robustness against manufacturing variations. The system is evaluated with multiple sets of benchmark problems to analyze the sensitivity for parameters such as the coupling strength or frequency.

Non-ergodic convergence rate of an inertial accelerated primal–dual algorithm for saddle point problems

In this paper, we design an inertial accelerated primal–dual algorithm to address the convex–concave saddle point problem, which is formulated as minxmaxyf(x)+〈Kx,y〉−g(y). Remarkably, both functions f and g exhibit a composite structure, combining “nonsmooth” ＋ “smooth” components. Under the assumption of partially strong convexity in the sense that f is convex and g is strongly convex, we introduce a novel inertial accelerated primal–dual algorithm based on Nesterov’s extrapolation. This algorithm can be reduced to two classical accelerated forward–backward methods for unconstrained optimization problem. We show that the proposed algorithm achieves a non-ergodic O(1/k2) convergence rate, where k represents the number of iterations. Several numerical experiments validate the efficiency of our proposed algorithm.

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.

Neural quantile optimization for edge–cloud networking

We seek the best traffic allocation scheme for the edge–cloud networking subject to SD-WAN architecture and burstable billing. First, we formulate a family of quantile-based integer programming problems for a fixed network topology with random parameters describing the traffic demands. Then, to overcome the difficulty caused by the discrete feature, we generalize the Gumbel-softmax reparameterization method to induce an unconstrained continuous optimization problem as a regularized continuation of the discrete problem. Finally, we introduce the Gumbel-softmax sampling neural network to solve optimization problems via unsupervised learning. The neural network structure reflects the edge–cloud networking topology and is trained to minimize the expectation of the cost function for unconstrained continuous optimization problems. The trained network works as an efficient traffic allocation scheme sampler, outperforming the random strategy in feasibility and cost value. Besides testing the quality of the output allocation scheme, we examine the generalization property of the network by increasing the time steps and the number of users. We also feed the solution to existing integer optimization solvers as initial conditions and verify the warm-starts can accelerate the short-time iteration process. The framework is general, and the decoupled feature of the random neural networks is adequate for practical implementations.

An improved preconditioned conjugate gradient method for unconstrained optimization problem with application in Robot arm control

AbstractThis work suggests improved conjugate gradient methods for enhancing the efficiency and robustness of the classical conjugate gradient methods. The study modifies the diagonal of the inverse Hessian approximation of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi‐Newton update in order to build a preconditioner for nonlinear conjugate gradient (NCG) methods applied to large‐scale unconstrained optimization problems. Damping techniques were embedded into the algorithm to impose the positive definiteness of the diagonal approximation. This made the methods easy to use and presented a viable way to increase the effectiveness of unconstrained optimization techniques. Experimental findings from a collection of benchmark problems demonstrate the efficiency and robustness of the proposed method when compared to five other existing NCG algorithms. Moreover, the successful application of the algorithm to manipulate robotic planar motion control systems with 3 degrees of freedom has been demonstrated, highlighting the practicality of the proposed approach.

A New Variant of the Conjugate Descent Method for Solving Unconstrained Optimization Problems and Applications

Unconstrained optimization problems have a long history in computational mathematics and have been identified as being among the crucial problems in the fields of applied sciences, engineering, and management sciences. In this paper, a new variant of the conjugate descent method for solving unconstrained optimization problems is introduced. The proposed algorithm can be seen as a modification of the popular conjugate descent (CD) algorithm of Fletcher. The algorithm of the proposed method is well-defined, and the sequence of the directions of search is shown to be sufficiently descending. The convergence result of the proposed method is discussed under the common standard conditions. The proposed algorithm together with some existing ones in the literature is implemented to solve a collection of benchmark test problems. Numerical experiments conducted show the performance of the proposed method is very encouraging. Furthermore, an additional efficiency evaluation is carried out on problems arising from signal processing and it works well.

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.

On the emerging potential of quantum annealing hardware for combinatorial optimization

Over the past decade, the usefulness of quantum annealing hardware for combinatorial optimization has been the subject of much debate. Thus far, experimental benchmarking studies have indicated that quantum annealing hardware does not provide an irrefutable performance gain over state-of-the-art optimization methods. However, as this hardware continues to evolve, each new iteration brings improved performance and warrants further benchmarking. To that end, this work conducts an optimization performance assessment of D-Wave Systems’ Advantage Performance Update computer, which can natively solve sparse unconstrained quadratic optimization problems with over 5,000 binary decision variables and 40,000 quadratic terms. We demonstrate that classes of contrived problems exist where this quantum annealer can provide run time benefits over a collection of established classical solution methods that represent the current state-of-the-art for benchmarking quantum annealing hardware. Although this work does not present strong evidence of an irrefutable performance benefit for this emerging optimization technology, it does exhibit encouraging progress, signaling the potential impacts on practical optimization tasks in the future.

An interior penalty function method for solving fuzzy nonlinear programming problems

In this article, we investigate fuzzy interior penalty function method for solving fuzzy nonlinear programming problems (FNLPP) based on a new fuzzy arith-metic, unconstrained optimization, and fuzzy ranking on the parametric form of triangular fuzzy numbers (TFN). The main objective of this paper is to solve constrained fuzzy nonlinear programming problems using interior penalty func-tions (IPF) by converting it into unconstrained optimization problems. We prove an important lemma and a convergence theorem for the interior penalty functions method. Interior penalty function techniques favor sites near the boundary of the feasible region in the interior. We present a numerical example of the suggested method and compare the results to those produced by existing methods.

A new spectral conjugate subgradient method with application in computed tomography image reconstruction

A new spectral conjugate subgradient method is presented to solve nonsmooth unconstrained optimization problems. The method combines the spectral conjugate gradient method for smooth problems with the spectral subgradient method for nonsmooth problems. We study the effect of two different choices of line search, as well as three formulas for determining the conjugate directions. In addition to numerical experiments with standard nonsmooth test problems, we also apply the method to several image reconstruction problems in computed tomography, using total variation regularization. Performance profiles are used to compare the performance of the algorithm using different line search strategies and conjugate directions to that of the original spectral subgradient method. Our results show that the spectral conjugate subgradient algorithm outperforms the original spectral subgradient method, and that the use of the Polak–Ribière formula for conjugate directions provides the best and most robust performance.

An effective subgradient algorithm via Mifflin’s line search for nonsmooth nonconvex multiobjective optimization

We propose a descent subgradient algorithm for unconstrained nonsmooth nonconvex multiobjective optimization problems. To find a descent direction, we present an iterative process that efficiently approximates the ɛ-subdifferential of each objective function. To this end, we develop a new variant of Mifflin’s line search in which the subgradients are arbitrary and its finite convergence is proved under a semismooth assumption. To reduce the number of subgradient evaluations, we employ a backtracking line search that identifies the objectives requiring an improvement in the current approximation of the ɛ-subdifferential. Meanwhile, for the remaining objectives, new subgradients are not computed. Unlike bundle-type methods, the proposed approach can handle nonconvexity without the need for algorithmic adjustments. Moreover, the quadratic subproblems have a simple structure, and hence the method is easy to implement. We analyze the global convergence of the proposed method and prove that any accumulation point of the generated sequence satisfies a necessary Pareto optimality condition. Furthermore, our convergence analysis addresses a theoretical challenge in a recently developed subgradient method. Through numerical experiments, we observe the practical capability of the proposed method and evaluate its efficiency when applied to a diverse range of nonsmooth test problems.

Spatial–Temporal Joint Design and Optimization of Phase-Coded Waveform for MIMO Radar

By simultaneously transmitting multiple different waveform signals, a multiple-input multiple-output (MIMO) radar possesses higher degrees of freedom and potential in many aspects compared to a traditional phased-array radar. The spatial–temporal characteristics of waveforms are the key to determining their performance. In this paper, a transmitting waveform design method based on spatial–temporal joint (STJ) optimization for a MIMO radar is proposed, where waveforms are designed not only for beam-pattern matching (BPM) but also for minimizing the autocorrelation sidelobes (ACSLs) of the spatial synthesis signals (SSSs) in the directions of interest. Firstly, the STJ model is established, where the two-step strategy and least squares method are utilized for BPM, and the L2p-Norm of the ACSL is constructed as the criterion for temporal characteristics optimization. Secondly, by transforming it into an unconstrained optimization problem about the waveform phase and using the gradient descent (GD) algorithm, the hard, non-convex, high-dimensional, nonlinear optimization problem is solved efficiently. Finally, the method’s effectiveness is verified through numerical simulation. The results show that our method is suitable for both orthogonal and partial-correlation MIMO waveform designs and efficiently achieves better spatial–temporal characteristic performances simultaneously in comparison with existing methods.

Adaptive Dynamic Programming for Robust Event-Driven Tracking Control of Nonlinear Systems With Asymmetric Input Constraints.

This article considers the robust dynamic event-driven tracking control problem of nonlinear systems having mismatched disturbances and asymmetric input constraints. Initially, to tackle the asymmetric constraints, a novel nonquadratic value function is constructed for the original system. This makes the asymmetrically constrained tracking control problem transformed into an unconstrained optimal regulation problem. Then, a dynamic event-driven mechanism is proposed. Meanwhile, the event-driven Hamilton-Jacobi-Bellman equation (ED-HJBE) is developed for the optimal regulation problem in order to acquire the optimal control with distinctly decreased computational burden. To solve the ED-HJBE, a single critic neural network (CNN) is designed in the adaptive dynamic programming framework. Meanwhile, the gradient descent method is employed to update the CNN's weights. After that, both the weight estimation error and the tracking error are proved to be uniformly ultimately bounded via Lyapunov's direct method. Finally, simulations of the spring-mass-damper system and the pendulum plant are separately utilized to validate the established theoretical claims.

An improved descent Perry‐type algorithm for large‐scale unconstrained nonconvex problems and applications to image restoration problems

AbstractConjugate gradient methods are much effective for large‐scale unconstrained optimization problems by their simple computations and low memory requirements. The Perry conjugate gradient method has been considered to be one of the most efficient methods in the context of unconstrained minimization. However, a globally convergent result for general functions has not been established yet. In this paper, an improved three‐term Perry‐type algorithm is proposed which automatically satisfies the sufficient descent property independent of the accuracy of line search strategy. Under the standard Wolfe line search technique and a modified secant condition, the proposed algorithm is globally convergent for general nonlinear functions without convexity assumption. Numerical results compared with the Perry method for stability, two modified Perry‐type conjugate gradient methods and two effective three‐term conjugate gradient methods for large‐scale problems up to 300,000 dimensions indicate that the proposed algorithm is more efficient and reliable than the other methods for the testing problems. Additionally, we also apply it to some image restoration problems.