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Convex Analysisの二,三の進展について

A critical hurdle of a nonlinear vibration system in a fractal space is the inefficiency in modelling the system. Specifically, the differential equation models cannot elucidate the effect of porosity size and distribution of the periodic property. This paper establishes a fractal-differential model for this purpose, and a fractal Duffing-Van der Pol oscillator (DVdP) with two-scale fractal derivatives and a forced term is considered as an example to reveal the basic properties of the fractal oscillator. Utilizing the two-scale transforms and He-Laplace method, an analytic approximate solution may be attained. Unfortunately, this solution is not physically preferred. It has to be modified along with the nonlinear frequency analysis, and the stability criterion for the equation under consideration is obtained. On the other hand, the linearized stability theory is employed in the autonomous arrangement. Consequently, the phase portraits around the equilibrium points are sketched. For the non-autonomous organization, the stability criteria are analyzed via the multiple time scales technique. Numerical estimations are designed to confirm graphically the analytical approximate solutions as well as the stability configuration. It is revealed that the exciting external force parameter plays a destabilizing role. Furthermore, both of the frequency of the excited force and the stiffness parameter, execute a dual role in the stability picture.

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Forced nonlinear oscillator in a fractal space

We investigate the dynamics of a Van der Pol–Duffing forced oscillator, which is modelled by a five-parameter second order nonautonomous nonlinear ordinary differential equation. Firstly we fix three of these parameters, and investigate the dynamics of this system by varying the other two, namely the amplitude and the angular frequency of the external forcing. We also investigate the existence of different attractors, periodic, quasiperiodic, and chaotic. Finally, we investigate the occurrence of multistability in the considered Van der Pol–Duffing forced oscillator, for some fixed sets of parameters.

Multistability and organization of periodicity in a Van der Pol–Duffing oscillator

In this paper, a Van der Pol–Duffing (VdPD) jerk oscillator is designed. The proposed VdPD jerk oscillator is built by converting the autonomous two-dimensional VdPD oscillator to a jerk oscillator. Dynamical behaviours of the proposed VdPD jerk oscillator are investigated analytically, numerically and analogically. The numerical results indicate that the proposed VdPD jerk oscillator displays chaotic oscillations, symmetrical bifurcations and coexisting attractors. The physical existence of the chaotic behaviour found in the proposed VdPD jerk oscillator is verified by using Orcad-PSpice software. A good qualitative agreement is shown between the numerical simulations and the PSpice results. Moreover, the fractional-order form of the proposed VdPD jerk oscillator is studied using stability theorem of fractional-order systems and numerical simulations. It is found that chaos, periodic oscillations and coexistence of attractors exist in the fractional-order form of the proposed jerk oscillator with order less than three. The effect of fractional-order derivative on controlling chaos is illustrated. It is shown that chaos control is achieved in fractional-order form of the proposed VdPD jerk oscillator only for the values of linear controller used. Finally, the problem of drive–response synchronisation of the fractional-order form of the chaotic proposed VdPD jerk oscillators is considered using active control technique.

Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis, chaos control and synchronisation in its fractional-order form

This paper proposes an inertial Bregman proximal gradient method for minimizing the sum of two possibly nonconvex functions This method includes two different inertial steps and adopts the Bregman regularization in solving the subproblem Under some general parameter constraints, we prove the subsequential convergence that each generated sequence converges to the stationary point of the considered problem To overcome the parameter constraints, we further propose a nonmonotone line search strategy to make the parameter selections more flexible The subsequential convergence of the proposed method with line search is established When the line search is monotone, we prove the stronger global convergence and linear convergence rate under Kurdyka–Łojasiewicz framework Moreover, numerical results on SCAD and MCP nonconvex penalty problems are reported to demonstrate the effectiveness and superiority of the proposed methods and line search strategy

Inertial proximal gradient methods with Bregman regularization for a class of nonconvex optimization problems

Based on the homotopy analysis method (HAM), the high accuracy frequency response curve and the stable/unstable periodic solutions of the Van der Pol-Duffing forced oscillator with the variation of the forced frequency are obtained and studied. The stability of the periodic solutions obtained is analyzed by use of Floquet theory. Furthermore, the results are validated in the light of spectral analysis and bifurcation theory.

https://iopscience.iop.org/article/10.1088/0031-8949/91/1/015201/pdf

Stability analysis for periodic solutions of the Van der Pol–Duffing forced oscillator

This paper describes and establishes the iteration-complexity of a doubly accelerated inexact proximal point (D-AIPP) method for solving the nonconvex composite minimization problem whose objective function is of the form $f+h$ where $f$ is a (possibly nonconvex) differentiable function whose gradient is Lipschitz continuous and $h$ is a closed convex function with bounded domain. D-AIPP performs two types of iterations, namely, inner and outer ones. Its outer iterations correspond to the ones of the accelerated inexact proximal point scheme. Its inner iterations are the ones performed by an accelerated composite gradient method for inexactly solving the convex proximal subproblems generated during the outer iterations. Thus, D-AIPP employs both inner and outer accelerations.

A Doubly Accelerated Inexact Proximal Point Method for Nonconvex Composite Optimization Problems

This paper presents a proximal bundle variant, namely, the relaxed proximal bundle (RPB) method, for solving convex nonsmooth composite optimization problems. Like other proximal bundle variants, RPB solves a sequence of prox bundle subproblems whose objective functions are regularized composite cutting-plane models. Moreover, RPB uses a novel condition to decide whether to perform a serious or null iteration which does not necessarily yield a function value decrease. Optimal iteration-complexity bounds for RPB are established for a large range of prox stepsizes, both in the convex and strongly convex settings. To the best of our knowledge, this is the first time that a proximal bundle variant is shown to be optimal for a large range of prox stepsizes. Finally, iteration-complexity results for RPB to obtain iterates satisfying practical termination criteria, rather than near optimal solutions, are also derived.

A proximal bundle variant with optimal iteration-complexity for a large range of prox stepsizes

In this paper, we describe and establish iteration-complexity of two accelerated composite gradient (ACG) variants to solve a smooth nonconvex composite optimization problem whose objective function is the sum of a nonconvex differentiable function $ f $ with a Lipschitz continuous gradient and a simple nonsmooth closed convex function $ h $. When $f$ is convex, the first ACG variant reduces to the well-known FISTA for a specific choice of the input, and hence the first one can be viewed as a natural extension of the latter one to the nonconvex setting. The first variant requires an input pair $(M,m)$ such that $f$ is $m$-weakly convex, $\nabla f$ is $M$-Lipschitz continuous, and $m \le M$ (possibly $m<M$), which is usually hard to obtain or poorly estimated. The second variant on the other hand can start from an arbitrary input pair $(M,m)$ of positive scalars and its complexity is shown to be not worse, and better in some cases, than that of the first variant for a large range of the input pairs. Finally, numerical results are provided to illustrate the efficiency of the two ACG variants.

A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems

The branch-and-bound optimization algorithm for mixed-integer model predictive control (MI-MPC) solves several convex quadratic program relaxations, but often the solutions are discarded based on already known integer feasible solutions. This letter presents a projection and early termination strategy for infeasible interior point methods to reduce the computational effort of finding a globally optimal solution for MI-MPC. The method is shown to be also effective for infeasibility detection of the convex relaxations. We present numerical simulation results with a reduction of the total number of solver iterations by 42% for an MI-MPC example of decision making for automated driving with obstacle avoidance constraints.

Jiaming Liang

Papers

Stability analysis for periodic solutions of the Van der Pol–Duffing forced oscillator

A Doubly Accelerated Inexact Proximal Point Method for Nonconvex Composite Optimization Problems

A proximal bundle variant with optimal iteration-complexity for a large range of prox stepsizes

A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems

Early Termination of Convex QP Solvers in Mixed-Integer Programming for Real-Time Decision Making