Nonsmooth Method Research Articles

Sparse optimal control of Timoshenko's beam using a locking‐free finite element approximation

AbstractThis paper addresses the optimal control problem with sparse controls of a Timoshenko beam, its numerical approximation using the finite element method, and the numerical solution via nonsmooth methods. Incorporating sparsity‐promoting terms in the cost function is practically useful for beam vibration models and results in the localization of the control action that facilitates the placement of actuators or control devices. We consider two types of sparsity‐inducing penalizers: the ‐norm and the ‐penalizer, which measures function support. We analyze discretized problems utilizing linear finite elements with a locking‐free scheme to approximate the states and adjoint states. We confirm that this approximation has the looking‐free property required to achieve a linear convergence linear order of approximation for control case and depending on the set of switching points in the controls. This is similar to the purely ‐norm penalized optimal control, where the order of approximation is independent of the thickness of the beam.

Fast optimization methods for high-dimensional row-sparse multivariate quantile linear regression

Sparse quantile regression is a useful tool for variable selection, robust estimation and heteroscedasticity treatment in high-dimensional data analysis. Due to the non-smooth of quantile loss, the computation is heavier than the least square models. In the literature, there are various numerical methods for linear quantile regression, such as ADMM developed in Gu et al. [Technometrics. 2017;60(3):319–331]. However, the computation of multivariate quantile regression has not yet been fully resolved, especially when the dimension is high. Motivated by this, we will focus on the design of fast numerical algorithms for row-sparse multivariate quantile regression model. By virtue of proximal operator and Majorize–Minimization, four smoothed algorithms are designed. For all the obtained algorithms, we analyse their convergence and the parameter selection. We conduct plenty of simulations and four real data sets analysis. Finally, we conclude that the smoothed method is faster than the non-smooth method, especially when the number of predictors is large.

A nonsmooth method for spatial frictional contact dynamics of flexible multibody systems with large deformation

AbstractA nonsmooth contact dynamics computation methodology for spatial frictional contact dynamics of flexible multibody systems with large deformation is presented. The contact and friction conditions between contact bodies are formulated as a cone complementary problem (CCP). Based on the mortar discretization method, nonsmooth frictional contact formulations for surface‐to‐surface and beam‐to‐beam contact problems are derived. These formulations can be used to study a wide range of contact problems between three‐dimensional bodies, thin plates and beams. The nonsmooth multibody dynamics equations are solved by using the generalized‐ method. The CCP is solved by using the accelerated projected gradient descend (APGD) algorithm in each Newton's iteration. Finally, six numerical examples are provided to validate the proposed computation methodology.

A Stochastic Nesterov’s Smoothing Accelerated Method for General Nonsmooth Constrained Stochastic Composite Convex Optimization

A Stochastic Nesterov’s Smoothing Accelerated Method for General Nonsmooth Constrained Stochastic Composite Convex Optimization

Efficient projection onto the intersection of a half-space and a box-like set and its generalized Jacobian

This paper focuses on efficient projection onto the intersection of a half-space and a box-like set and its generalized Jacobian. Based on the Lagrangian duality theory, we deal with the projection problem via a semismooth Newton algorithm with line search safeguard, which admits global and locally quadratic convergence, to solve a univariate semismooth equation. Numerical experiments show that our proposed algorithm outperforms favourably the existing state-of-the-art standard solvers and is able to reliably solve very large-scale projection problems. Besides, we derive an explicit expression of a generalized Jacobian of the studied projection, which is an essential component of second-order nonsmooth methods.

A novel nonsmooth approach for flexible multibody systems with contact and friction in 3D space

For computational multibody system dynamics, contact and friction problems are very complex and important problems. Therefore, this paper proposes a novel nonsmooth method for flexible multibody systems with contact and friction in 3D space. Considering the nonsmooth effect of contact and friction on the state variable of the multibody systems, the proposed method is divided into two parts: (i) the change of state variables under the action of smooth force and (ii) the change of state variables caused by contact and friction. Furthermore, for the contact part, the Newton’s impact law and the classical Coulomb friction model are employed to deal with normal impact contact impulse and tangential friction contact impulse, respectively. In addition, Fischer–Burmeister function and Karush–Kuhn–Tucker conditions are also used to solve the normal impact contact impulse and tangential friction contact impulse. What’s more, since the symplectic discrete format performs well robustness to numerical results, the discrete format of the proposed method is based on the symplectic discreteness, and it is expected to obtain the robust numerical results. Finally, several numerical examples are tested by the proposed nonsmooth method, and the numerical simulation results verify the effectiveness of the proposed approach.

Nonsmooth Decentralized Stabilization for Interconnected Systems Subject to Strongly Coupled Uncertain Interactions

This paper revisits the decentralized stabilization problem for a class of interconnected nonlinear systems subject to strongly coupled uncertain interactions. As a main contribution, we show that, without any priori required restrictive nonlinear growth conditions, a ${\mathcal {C}}^{1}$ condition of the nonlinear interactions is sufficient to derive a finite-time convergent decentralized stabilizing control law under a semi-global control objective. Another new feature is the adoption of a novel one step nonsmooth synthesis method which leads to a simple form of stabilizing control law with easily implementable gain tuning mechanisms. A numerical example and its simulation verification are provided to illustrate the simplicity and effectiveness of the proposed method.

Index-2 hybrid DAE: a case study with well-posedness and numerical analysis

Abstract In this work, we study differential algebraic equations with constraints defined in a piecewise manner using a conditional statement. Such models classically appear in systems where constraints can evolve in a very small time frame compared to the observed time scale. The use of conditional statements or hybrid automata is a powerful way to describe such systems and are, in general, well suited to simulation with event driven numerical schemes. However, such methods are often subject to chattering at mode switch in presence of sliding modes, or can result in Zeno behaviours. In contrast, the representation of such systems using differential inclusions and method from non-smooth dynamics are often closer to the physical theory but may be harder to interpret. Associated time-stepping numerical methods have been extensively used in mechanical modelling with success and then extended to other fields such as electronics and system biology. In a similar manner to the previous application of non-smooth methods to the simulation of piecewise linear ODEs, non-smooth event-capturing numerical schemes are applied to piecewise linear DAEs. In particular, the detailed study of a 2-D dynamical system of index-2 with a switching constraint using set-valued operators, is presented.

A parallel-GPU code for asteroid aggregation problems with angular particles

ABSTRACTThe paper presents a numerical implementation of the gravitational N-body problem with contact interactions between non-spherically shaped bodies. The work builds up on a previous implementation of the code and extends its capabilities. The number of bodies handled is significantly increased through the use of a CUDA/GPU-parallel octree structure. The implementation of the code is discussed and its performance is compared against direct N2 integration. The code features both smooth (force-based) and non-smooth (impulse-based) methods, as well as a visco-elastic non-smooth method, to handle contact interaction between bodies. The numerical problem of simulating ‘rubble-pile’ asteroid gravitational aggregation processes is addressed. We discuss the features of the problem and derive criteria to set up the numerical simulation from the dynamical constraints of the combined gravitational–collisional problem. Examples of asteroid aggregation scenarios that could benefit from such implementation are finally presented.

Data-driven method of damage detection using sparse sensors installation by SEREPa

This paper presents a model-based method of damage detection and severity identification in structural elements. The model-based method is performed with a finite element model. One of the important challenges in damage identification problems is lack of measured degrees of freedom and limitations of installed sensors on the structures. The new approach of this study is the use of expanded mode shapes data to train artificial neural network (ANN). In this study, the measured mode shapes are expanded by SEREPa. SEREPa expansion is developed based on the System-Equivalent Reduction and Expansion Process (SEREP), which is a non-smooth method and protects the measured data. ANN was then trained through the expanded data as inputs, location and severity of damage as outputs. The algorithm used to train ANN is scaled conjugate gradient. The advantage of this algorithm is that less data storage space is used and lower computation costs are needed. To show SEREPa’s efficiency in estimating unmeasured mode shapes, an experimental example containing a truss tower was presented. Two numerical examples including a plane truss and a space truss were presented to illustrate efficiency of damage detection method. Finally, the proposed method was verified by an experimental example. Damage prediction results for both numerical and experimental examples indicated an acceptable accuracy of the proposed method.

The Strength of Nesterov's Extrapolation in the Individual Convergence of Nonsmooth Optimization.

The extrapolation strategy raised by Nesterov, which can accelerate the convergence rate of gradient descent methods by orders of magnitude when dealing with smooth convex objective, has led to tremendous success in training machine learning tasks. In this article, the convergence of individual iterates of projected subgradient (PSG) methods for nonsmooth convex optimization problems is theoretically studied based on Nesterov's extrapolation, which we name individual convergence. We prove that Nesterov's extrapolation has the strength to make the individual convergence of PSG optimal for nonsmooth problems. In light of this consideration, a direct modification of the subgradient evaluation suffices to achieve optimal individual convergence for strongly convex problems, which can be regarded as making an interesting step toward the open question about stochastic gradient descent (SGD) posed by Shamir. Furthermore, we give an extension of the derived algorithms to solve regularized learning tasks with nonsmooth losses in stochastic settings. Compared with other state-of-the-art nonsmooth methods, the derived algorithms can serve as an alternative to the basic SGD especially in coping with machine learning problems, where an individual output is needed to guarantee the regularization structure while keeping an optimal rate of convergence. Typically, our method is applicable as an efficient tool for solving large-scale l1 -regularized hinge-loss learning problems. Several comparison experiments demonstrate that our individual output not only achieves an optimal convergence rate but also guarantees better sparsity than the averaged solution.

Minimization of marginal functions in mathematical programming based on continuous outer subdifferentials

Typically, exact information of the whole subdifferential is not available for intrinsically nonsmooth objective functions such as for marginal functions. Therefore, the semismoothness of the objective function cannot be proved or is even violated. In particular, in these cases standard nonsmooth methods cannot be used. In this paper, we propose a new approach to develop a converging descent method for this class of nonsmooth functions. This approach is based on continuous outer subdifferentials introduced by us. Further, we introduce on this basis a conceptual optimization algorithm and prove its global convergence. This leads to a constructive approach enabling us to create a converging descent method. Within the algorithmic framework, neither semismoothness nor calculation of exact subgradients are required. This is in contrast to other approaches which are usually based on the assumption of semismoothness of the objective function.

A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm

The Interior Epigraph Directions (IED) method for solving constrained nonsmooth and nonconvex optimization problem via Generalized Augmented Lagrangian Duality considers the dual problem induced by a Generalized Augmented Lagrangian Duality scheme and obtains the primal solution by generating a sequence of iterates in the interior of the epigraph of the dual function. In this approach, the value of the dual function at some point in the dual space is given by minimizing the Lagrangian. The first version of the IED method uses the Matlab routine fminsearch for this minimization. The second version uses NFDNA, a tailored algorithm for unconstrained, nonsmooth and nonconvex problems. However, the results obtained with fminsearch and NFDNA were not satisfactory. The current version of the IED method, presented in this work, employs a Genetic Algorithm, which is free of any strategy to handle the constraints, a difficult task when a metaheuristic, such as GA, is applied alone to solve constrained optimization problems. Two sets of constrained optimization problems from mathematics and mechanical engineering were solved and compared with literature. It is shown that the proposed hybrid algorithm is able to solve problems where fminsearch and NFDNA fail.

RBF Nonsmooth Control Method for Vibration of Building Structure with Actuator Failure

In order to accommodate the actuator failure, the finite-time stable nonsmooth control method with RBF neural network is used to suppress the structural vibration. The traditional designed control methods neglect influence of actuator failure in structural vibration. By Lyapunov stable theory, the designed control method is demonstrated to suppress the building structural vibration with actuator failure. Finally, there are some examples to numerically simulate the three-layer building structure which is affected by El Centro seismic wave. Control effect of nonsmooth control is compared with no control and LQR control. The simulation results demonstrate that the designed control method is great for vibration of building structure with actuator failure and great antiseismic effect.

A nonconvex nonsmooth regularization method for compressed sensing and low rank matrix completion

In this paper, a nonconvex and nonsmooth method for compressed sensing and low-rank matrix completion is studied. The proposed model is formulated as nonconvex regularized least square optimization problem. At first, an alternating minimization scheme is developed in which the problem can be decomposed into three subproblems, two of them are convex and the remaining one is smooth. Then, the convergence of the sequence which is generated by the alternating minimization algorithm is proved. In addition, some recovery guarantees are also analyzed. Finally, various numerical simulations are performed to test the efficiency of the method.

Feasible methods for nonconvex nonsmooth problems with applications in green communications

We propose a general feasible method for nonsmooth, nonconvex constrained optimization problems. The algorithm is based on the (inexact) solution of a sequence of strongly convex optimization subproblems, followed by a step-size procedure. Key features of the scheme are: (i) it preserves feasibility of the iterates for nonconvex problems with nonconvex constraints, (ii) it can handle nonsmooth problems, and (iii) it naturally leads to parallel/distributed implementations. We illustrate the application of the method to an open problem in green communications whereby the energy consumption in MIMO multiuser interference networks is minimized, subject to nonconvex Quality-of-Service constraints.

A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles

We propose an algorithm for solving nonsmooth, nonconvex, constrained optimization problems as well as a new set of visualization tools for comparing the performance of optimization algorithms. Our algorithm is a sequential quadratic optimization method that employs Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton Hessian approximations and an exact penalty function whose parameter is controlled using a steering strategy. While our method has no convergence guarantees, we have found it to perform very well in practice on challenging test problems in controller design involving both locally Lipschitz and non-locally-Lipschitz objective and constraint functions with constraints that are typically active at local minimizers. In order to empirically validate and compare our method with available alternatives—on a new test set of 200 problems of varying sizes—we employ new visualization tools which we call relative minimization profiles. Such profiles are designed to simultaneously assess the relative performance of several algorithms with respect to objective quality, feasibility, and speed of progress, highlighting the trade-offs between these measures when comparing algorithm performance.

Response analysis of Rayleigh–Van der Pol vibroimpact system with inelastic impact under two parametric white-noise excitations

In the present paper, the random vibration problem of Rayleigh–Van der Pol oscillator is considered by means of a single-degree-of-freedom system with a unilateral rigid barrier at its equilibrium position. The random excitations are two parametric Gaussian white noises. The nonsmooth transformation method is applied to convert the vibroimpact system to an equivalent system without velocity jump. The modified stochastic averaging technique for energy envelope is used to deal with the transformed system, and the averaging Ito stochastic differential equation is obtained. The analytical solution of the response of the original vibroimpact system is derived by solving the corresponding Fokker–Planck equation and using the inverse transformation of nonsmooth transformation. The validity of the analytical results is verified by those from Monte Carlo simulations based on original system. Effects of different system parameters and parametric white noises on the response of the system are examined. In addition, the critical condition of stochastic bifurcations is also explored.

Diagonal Bundle Method for Nonsmooth Sparse Optimization

We propose an efficient diagonal bundle method for sparse nonsmooth, possibly nonconvex optimization. The convergence of the proposed method is proved for locally Lipschitz continuous functions, which are not necessary differentiable or convex. The numerical experiments have been made using problems with up to million variables. The results to be presented confirm the usability of the diagonal bundle method especially for extremely large-scale problems.

Levenberg-Marquardt method for the eigenvalue complementarity problem.

The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method.