Approximate Stationary Solutions Research Articles

Code and Data Repository for Penalty Decomposition Methods for Second-Best Congestion Pricing Problems on Large-Scale Networks

he second-best congestion pricing (SBCP) problem is one of the most challenging problems in transportation due to its two-level hierarchical structure. In spite of various intriguing attempts for solving SBCP, existing solution methods are either heuristic without convergence guarantee or suitable for solving SBCP on small networks only. In this paper, we first reveal some convexity-based structural properties of the marginal value function reformation of SBCP and then, by effectively exploiting these structural properties, we propose two dedicated decomposition methods for solving SBCP on large-scale networks which are different from existing methods in that they avoid linearizing nonconvex functions. We establish the convergence of the two decomposition methods under commonly used conditions and provide the maximum number of iterations for deriving an approximate stationary solution. The computational experiments based on a collection of real road networks show that in comparison with three existing popular methods, the two proposed methods are capable of solving SBCP on larger-scale networks; and for instances that can be solved by existing methods, the two proposed methods are substantially faster.

Multitask-constrained reentry trajectory planning for hypersonic gliding vehicle

This paper studies the reentry trajectory planning problems for hypersonic gliding vehicle under multiple tasks. Different from the former achievements, this paper takes into account the practical tasks involved in the reentry phase, including penetration of interceptors, evasion of the no-fly zones, and the arrival of the waypoints. Firstly, the constraints during the reentry phase are analyzed in detail, and the original trajectory planning problem is formulated. Secondly, the hp-adaptive pseudospectral discretization method is proposed to effectively reduce the discretization error. Relevant variables are introduced to relax and transform severe intractable constraints of multiple nonconvex forms, thus enhancing the robustness of the planning process. Thirdly, the improved sequential convex programming with decision variables algorithm is proposed to ensure the converged trajectory satisfies complicated tasks. The theoretical analysis is also presented to demonstrate that the converged trajectory is the approximate stationary solution of the discrete form of the original problem. Finally, the effectiveness of the proposed algorithms is validated through numerical simulation.

Alternating and Parallel Proximal Gradient Methods for Nonsmooth, Nonconvex Minimax: A Unified Convergence Analysis

There is growing interest in nonconvex minimax problems that is driven by an abundance of applications. Our focus is on nonsmooth, nonconvex-strongly concave minimax, thus departing from the more common weakly convex and smooth models assumed in the recent literature. We present proximal gradient schemes with either parallel or alternating steps. We show that both methods can be analyzed through a single scheme within a unified analysis that relies on expanding a general convergence mechanism used for analyzing nonconvex, nonsmooth optimization problems. In contrast to the current literature, which focuses on the complexity of obtaining nearly approximate stationary solutions, we prove subsequence convergence to a critical point of the primal objective and global convergence when the latter is semialgebraic. Furthermore, the complexity results we provide are with respect to approximate stationary solutions. Lastly, we expand the scope of problems that can be addressed by generalizing one of the steps with a Bregman proximal gradient update, and together with a few adjustments to the analysis, this allows us to extend the convergence and complexity results to this broader setting. Funding: The research of E. Cohen was partially supported by a doctoral fellowship from the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240]. The research of M. Teboulle was partially supported by the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240].

Bifurcation and stability analysis of a hybrid energy harvester with fractional-order proportional–integral–derivative controller and Gaussian white noise excitations

Hybrid energy harvester (HEH) has received much attention in the field of energy harvesting. The HEH inevitably suffers from external stochastic disturbances in the working environment such as winds, waves, and ocean currents. Only few works deal with the stochastic dynamics of a hybrid energy harvester with fractional-order proportional–integral–derivative (PID) controller. This paper aims to investigate bifurcation control and stability analysis of a HEH with fractional-order PID controller subjected to Gaussian white noise excitations. An approximately equivalent dimensionally reduced system is formulated via the variables transformation method, and its approximate stationary solutions are derived through the stochastic averaging method. One example is worked out in detail to verify the effectiveness of the proposed procedure. It is shown that the analytical results provide a good approximation to the numerical simulation results. The influences of system parameters on the stochastic bifurcation and the asymptotic Lyapunov stability are also discussed.

Iteration Complexity of an Inner Accelerated Inexact Proximal Augmented Lagrangian Method Based on the Classical Lagrangian Function

This paper establishes the iteration complexity of an inner accelerated inexact proximal augmented Lagrangian (IAIPAL) method for solving linearly constrained smooth nonconvex composite optimization problems that is based on the classical augmented Lagrangian (AL) function. More specifically, each IAIPAL iteration consists of inexactly solving a proximal AL subproblem by an accelerated composite gradient (ACG) method followed by a classical Lagrange multiplier update. Under the assumption that the domain of the composite function is bounded and the problem has a Slater point, it is shown that IAIPAL generates an approximate stationary solution in ACG iterations where is a tolerance for both stationarity and feasibility. Moreover, the above bound is derived without assuming that the initial point is feasible. Finally, numerical results are presented to demonstrate the strong practical performance of IAIPAL.

Iteration Complexity of a Proximal Augmented Lagrangian Method for Solving Nonconvex Composite Optimization Problems with Nonlinear Convex Constraints

This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving constrained nonconvex composite optimization problems, where the constraints are smooth and convex with respect to the order given by a closed convex cone. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient method followed by a Lagrange multiplier update. Under some mild assumptions, a complexity bound for NL-IAPIAL to obtain an approximate stationary solution of the problem is also derived. Numerical experiments are also given to illustrate the computational efficiency of the proposed method. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grant PGSD3-516700-2018]; Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant 312559/2019-4]; Fundação de Amparo à Pesquisa do Estado de Goiás; Office of Naval Research [Grant N00014-18-1-2077]; Exascale Computing Project [Grant 17-SC-20-SC]; Air Force Office of Scientific Research [Grant FA9550-22-1-0088]; UT-Battelle, LLC [Grant DE-AC05-00OR22725].

On Approximate Stationary Radial Solutions for a Class of Boundary Value Problems Arising in Epitaxial Growth Theory

In this paper, we consider a non-self-adjoint, singular, nonlinear fourth order boundary value problem which arises in the theory of epitaxial growth. It is possible to reduce the fourth order equation to a singular boundary value problem of second order given by w''-1/r w'=w^2/(2r^2 )+1/2 λ r^2. The problem depends on the parameter λ and admits multiple solutions. Therefore, it is difficult to pick multiple solutions together by any discrete method like finite difference method, finite element method etc. Here, we propose a new technique based on homotopy perturbation method and variational iteration method. We compare numerically the approximate solutions computed by Adomian decomposition method. We study the convergence analysis of both iterative schemes in C^2 ([0,1]). For small values of λ, solutions exist whereas for large values of λ solutions do not exist.

Vortex Stationary Karman Structures in Flows of a Rotating Incompressible Polymer Fluid

The paper considers stationary solutions for the motion problem of a finite layer of an incompressible polymer fluid over an infinite rotating disk. An approximate stationary solution representation is used, similar to the self-similar Karman solution for a viscous fluid. Examples of stationary numerical solutions are given for various values of the problem’s parameters.

Stability of finite perturbations of the phase transition interface for one problem of water evaporation in a porous medium

We study global dynamics of phase transition evaporation interfaces in horizontally extended domains of porous layers where a water located over a vapor. The derivation of the model equation describing the secondary structures, which bifurcate from the ground state in a small neighborhood of the instability threshold in the case of a quasi-stationary approach to the description of the diffusion process, is presented. The resulting equation is reduced to the equation in the form of Kolmogorov-Petrovsky-Piskounov. The obtained approximate equation predicts the existence of stationary solutions in the full problem. To verify the obtained results, the numerical solution of the problem of the motion of the phase transition interface is performed using the original program code developed by the authors. The results of numerical simulation are used to verify the possibility of using stationary solutions obtained in the weakly nonlinear approximation to determine the scenario for the development of the initial localized finite amplitude perturbation. It is shown that the obtained approximate stationary solutions accurately predict the behavior of the perturbation in the vicinity of the turning point of the bifurcation diagram. A modification of the formulas describing an approximate stationary soliton-like solution is proposed in the case when the perturbation amplitude is comparable with the height of a low-permeable layer of a porous medium in which the phase transition interface is located. By numerical simulation it is shown that this modified approximate solution is in good agreement with the results of numerical calculation for the full problem.

Stochastic Averaging for the Piezoelectric Energy Harvesting System With Fractional Derivative Element

Combining the stochastic averaging method and generalized harmonic function, we propose a procedure to investigate the fractional-order energy harvesting system. First, we can obtain the approximately equivalent system by means of the generalized harmonic transformation; Second, the stochastic averaging method is utilized to get the approximate stationary solutions of the simplified system. Finally, In order to illustrate the accuracy of the developed scheme, two examples are carried out. The results show that the developed procedure has a satisfactory reliability. The influences of the intensity of noise excitation, the fractional order, the fractional coefficient and the nonlinear restoring force on the mean-square voltage (MSV) and mean output power (MOP) are also investigated.

An approximate global solution of Einstein’s equations for a differentially rotating compact body

We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be thought as a simple star model: a self-gravitating perfect fluid ball with a differential rotation motion pattern. Using the post--Minkowskian formalism (weak field approximation) and considering rotation as a perturbation (slow rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic coordinates. Interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to get a global solution. The resulting metric depends on four arbitrary constants: mass density, rotational velocity at $r=0$, a parameter which takes into account of the change in the rotational velocity through the star and the star radius in the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined in terms of these four parameters.

Stationary Response Probability Distribution of SDOF Nonlinear Stochastic Systems

There has been no significant progress in developing new techniques for obtaining exact stationary probability density functions (PDFs) of nonlinear stochastic systems since the development of the method of generalized probability potential in 1990s. In this paper, a general technique is proposed for constructing approximate stationary PDF solutions of single degree of freedom (SDOF) nonlinear systems under external and parametric Gaussian white noise excitations. This technique consists of two novel components. The first one is the introduction of new trial solutions for the reduced Fokker–Planck–Kolmogorov (FPK) equation. The second one is the iterative method of weighted residuals to determine the unknown parameters in the trial solution. Numerical results of four challenging examples show that the proposed technique will converge to the exact solutions if they exist, or a highly accurate solution with a relatively low computational effort. Furthermore, the proposed technique can be extended to multi degree of freedom (MDOF) systems.

An Efficient Global Algorithm for Single-Group Multicast Beamforming

Consider the single-group multicast beamforming problem, where multiple users receive the same data stream simultaneously from a single transmitter. The problem is NP-hard and all existing algorithms for the problem either find suboptimal approximate or local stationary solutions. In this paper, we propose an efficient branch-and-bound algorithm for the problem that is guaranteed to find its global solution. To the best of our knowledge, our proposed algorithm is the first tailored global algorithm for the single-group multicast beamforming problem. Simulation results show that our proposed algorithm is computationally efficient (albeit its theoretical worst-case iteration complexity is exponential with respect to the number of receivers) and it significantly outperforms a state-of-the-art general-purpose global optimization solver called Baron. Our proposed algorithm provides an important benchmark for performance evaluation of existing algorithms for the same problem. By using it as the benchmark, we show that two state-of-the-art algorithms, semidefinite relaxation algorithm and successive linear approximation algorithm, work well when the problem dimension (i.e., the number of antennas at the transmitter and the number of receivers) is small but their performance deteriorates quickly as the problem dimension increases.

Rate matrix estimation from site frequency data

A procedure is described for estimating evolutionary rate matrices from observed site frequency data. The procedure assumes (1) that the data are obtained from a constant size population evolving according to a stationary Wright–Fisher or decoupled Moran model; (2) that the data consist of a multiple alignment of a moderate number of sequenced genomes drawn randomly from the population; and (3) that within the genome a large number of independent, neutral sites evolving with a common mutation rate matrix can be identified. No restrictions are imposed on the scaled rate matrix other than that the off-diagonal elements are positive, their sum is ≪1, and that the rows of the matrix sum to zero. In particular the rate matrix is not assumed to be reversible. The key to the method is an approximate stationary solution to the diffusion limit, forward Kolmogorov equation for neutral evolution in the limit of low mutation rates.

Approximate explicit stationary solutions to a Vlasov equation for planetary rings

In this paper we consider a Vlasov or collisionless Boltzmann equation describing the dynamics of planetary rings. We propose a simple physical model, where the particles of the rings move under the gravitational Newtonian potential of two primary bodies. We neglect the gravitational forces between the particles. We use a perturbative technique, which allows to find explicit solutions at the first order and approximate solutions at the second order, by solving a set of two linear ordinary differential equations.

Hermite–Gaussian stationary solutions in strongly nonlocal nonlinear optical media

Approximate analytical stationary solutions (SSs) of a cluster of Hermite–Gaussian (HG) shape is obtained in strongly nonlocal nonlinear media by the variational approach. The evolution of the HG SSs shows that when the order n⩽3, they propagate stably and form solitons; otherwise, when n⩾4, they always propagate unstably and evolve into self-trapped speckle-like beams. However, all these SSs maintain nearly invariant statistic beam-width during their propagation. Furthermore, when the input power deviates from the so-called critical power, the unstable HG beam will adjust its beam-width to form a new self-trapped beam, unlike the soliton which will turn to be a breather. But the average beam-widths are independent of the stability of the propagation of the HG SSs.

An approximate stationary solution for multi-allele neutral diffusion with low mutation rates

We address the problem of determining the stationary distribution of the multi-allelic, neutral-evolution Wright–Fisher model in the diffusion limit. A full solution to this problem for an arbitrary K×K mutation rate matrix involves solving for the stationary solution of a forward Kolmogorov equation over a (K−1)-dimensional simplex, and remains intractable. In most practical situations mutations rates are slow on the scale of the diffusion limit and the solution is heavily concentrated on the corners and edges of the simplex. In this paper we present a practical approximate solution for slow mutation rates in the form of a set of line densities along the edges of the simplex. The method of solution relies on parameterising the general non-reversible rate matrix as the sum of a reversible part and a set of (K−1)(K−2)/2 independent terms corresponding to fluxes of probability along closed paths around faces of the simplex. The solution is potentially a first step in estimating non-reversible evolutionary rate matrices from observed allele frequency spectra.

Response analysis of a class of quasi-linear systems with fractional derivative excited by Poisson white noise

The Poisson white noise, as a typical non-Gaussian excitation, has attracted much attention recently. However, little work was referred to the study of stochastic systems with fractional derivative under Poisson white noise excitation. This paper investigates the stationary response of a class of quasi-linear systems with fractional derivative excited by Poisson white noise. The equivalent stochastic system of the original stochastic system is obtained. Then, approximate stationary solutions are obtained with the help of the perturbation method. Finally, two typical examples are discussed in detail to demonstrate the effectiveness of the proposed method. The analysis also shows that the fractional order and the fractional coefficient significantly affect the responses of the stochastic systems with fractional derivative.

Response of strongly nonlinear vibro-impact system with fractional derivative damping under Gaussian white noise excitation

This paper aims to investigate the strongly nonlinear vibro-impact dynamic system with fractional derivative damping under Gaussian white noise excitation, and this system considers both nonlinear factors and non-smooth factors. With the help of non-smooth transformation, the original system is rewritten as a smooth system, which is an easier handled form. For the altered function, we obtain the approximate stationary solutions analytically by generalized stochastic averaging method. To prove the validity of the approximate analytical methods, an efficient scheme with high accuracy is adopt to simulate the fractional derivative, then the fourth-order Runge–Kutta approach is used to obtain the numerically response statistics. Meanwhile the analytical solutions are verified by the numerical simulation solutions. At last, we research the responses of this system. In this context, we can consider the influences to this system caused by the fractional order, the restitution coefficient and the noise intensity through changing the values of corresponding parameters. The observation and investigation state that fractional derivative term, impact conditions and stochastic excitation can influence the responses of the fractional vibro-impact dynamic system.

Liquid toroidal drop in compressional Stokes flow

The deformation of an immiscible toroidal drop embedded in axisymmetric compressional Stokes flow is analysed via the boundary integral formulation in the case of equal viscosity. Numerical simulations are performed for the drop having initially the shape of a torus with circular cross-section. The quasi-stationary dynamic simulations reveal that, when the viscous forces, proportional to the intensity of the flow, are relatively weak compared with the surface tension (the ratio of these forces is characterized by the capillary number,$Ca$), three different scenarios of drop evolution are possible: indefinite expansion of the liquid torus, contraction to the centre and a stationary toroidal shape. When the intensity of the flow is low, the stationary shapes are shown to be close to circular tori. Once the outer flow strengthens, the cross-section of the stationary torus assumes first an elliptic and then an egg-like shape. For the capillary number greater than a critical value,$Ca_{cr}$, toroidal stationary shapes were not found. Remarkably,$Ca_{cr}$is close to the critical capillary number found previously for a simply connected drop flattened in compressional flow. Thus, a new example of non-uniqueness of stationary drop shape in viscous flow is obtained. Approximate stationary solutions in the form of tori with circular and elliptic cross-sections are obtained by minimizing the normal velocity over the drop interface. They are shown to be in good agreement with the stationary shapes from quasi-dynamic simulations for the corresponding intervals of the capillary number.