A non-smooth model for signalized road network design problems

A signalized road network is considered where the set of link capacity expansions and signal setting variables are simultaneously determined. This paper addresses a new optimization scheme for a signalized road network design problem (SRNDP). A SRNDP can be formulated as a mathematical program with equilibrium constraints (MPEC) where user equilibrium is expressed as a variational inequality problem. Due to non-differentiability of the perturbed solutions in equilibrium constraints, a non-smooth model is established. A bundle subgradient projection (BSP) method is presented with global convergence. Numerical calculations are conducted on a real data city road network and large-scale grid networks where promising results are obtained.     

Stability of the Minimizers of Least Squares with a Non-Convex Regularization. Part II: Global Behavior

We address estimation problems where the sought-after solution is defined as the minimizer of an objective function composed of a quadratic data-fidelity term and a regularization term. We especially focus on non-convex and possibly non-smooth regularization terms because of their ability to yield good estimates. This work is dedicated to the stability of the minimizers of such piecewise C^m, with m ≥ 2, non-convex objective functions. It is composed of two parts. In the previous part of this work we considered general local minimizers. In this part we derive results on global minimizers. We show that the data domain contains an open, dense subset such that for every data point therein, the objective function has a finite number of local minimizers, and a unique global minimizer. It gives rise to a global minimizer function which is C^m-1 everywhere on an open and dense subset of the data domain.     

Nonlinear travelling waves in a hyperelastic rod composed of a compressible Mooney-Rivlin material

In this paper, we study strongly nonlinear axisymmetric waves in a circular cylindrical rod composed of a compressible Mooney-Rivlin material. To consider the travelling wave solutions for the governing partial differential system, we first reduce it to a nonlinear ordinary differential equation. By using the bifurcation theory of planar dynamical systems, we show that the reduced system has seven periodic annuluses with different boundaries which depend on four parameters. We further consider the bifurcation behavior of the phase portraits for the reduced one-parameter vector fields when other three parameters are fixed. Corresponding to seven different periodic annuluses, we obtain seven types of travelling wave solutions, including solitary waves of radial contraction, solitary waves of radial expansion, solitary shock waves of radial contraction, solitary shock waves of radial expansion, periodic waves and two types of periodic shock waves. These are physically acceptable solutions by the governing partial differential system. The rigorous parameter conditions for the existence of these waves are given.     

Non-smooth modelling of electrical systems using the flux approach

The non-smooth modelling of electrical systems, which allows for idealised switching components, is described using the flux approach. The formulations and assumptions used for non-smooth mechanical systems are adopted for electrical systems using the position–flux analogy. For the most important non-smooth electrical elements, like diodes and switches, set-valued branch relations are formulated and related to analogous mechanical elements. With the set-valued branch relations, the dynamics of electrical circuits are described as measure differential inclusions. For the numerical solution, the measure differential inclusions are formulated as a measure complementarity system and discretised with a difference scheme, known in mechanics as time-stepping. For every time-step a linear complementarity problem is obtained. Using the example of the DC–DC buck converter, the formulation of the measure differential inclusions, state reduction and their numerical solution using the time-stepping method is shown for the flux approach.     

A subgradient optimization model for continuous road network design problem

This paper addresses a subgradient optimization model for a well-known continuous network design problem (CNDP). A continuous network design problem can be formulated as a mathematical program with equilibrium constraints (MPEC) where the user equilibrium flows are considered. By contrast to previous studies, in this paper, a conjugate subgradient projection method is presented to efficiently solve the continuous network design problem with global convergence. Numerical calculations are conducted on a real data of road network and various grid-size networks where encouraging results are reported when compared to earlier studies.     

Nash-type equilibria on Riemannian manifolds: A variational approach

Motivated by Nash equilibrium problems on ‘curved’ strategy sets, the concept of Nash–Stampacchia equilibrium points is introduced via variational inequalities on Riemannian manifolds. Characterizations, existence, and stability of Nash–Stampacchia equilibria are studied when the strategy sets are compact/noncompact geodesic convex subsets of Hadamard manifolds, exploiting two well-known geometrical features of these spaces both involving the metric projection map. These properties actually characterize the non-positivity of the sectional curvature of complete and simply connected Riemannian spaces, delimiting the Hadamard manifolds as the optimal geometrical framework of Nash–Stampacchia equilibrium problems. Our analytical approach exploits various elements from set-valued and variational analysis, dynamical systems, and non-smooth calculus on Riemannian manifolds. Examples are presented on the Poincaré upper-plane model and on the open convex cone of symmetric positive definite matrices endowed with the trace-type Killing form.     

On an inverse problem: Recovery of non-smooth solutions to backward heat equation

We have recently developed two quasi-reversibility techniques in combination with Euler and Crank–Nicolson schemes and applied successfully to solve for smooth solutions of backward heat equation. In this paper, we test the viability of using these techniques to recover non-smooth solutions of backward heat equation. In particular, we numerically integrate the backward heat equation with smooth initial data up to a time of singularity (corners and discontinuities) formation. Using three examples, it is shown that the numerical solutions are very good smooth approximations to these singular exact solutions. The errors are shown using pseudo-L- and U-curves and compared where available with existing works. The limitations of these methods in terms of time of simulation and accuracy with emphasis on the precise set of numerical parameters suitable for producing smooth approximations are discussed. This paper also provides an opportunity to gain some insight into developing more sophisticated filtering techniques that can produce the fine-scale features (singularities) of the final solutions. Techniques are general and can be applied to many problems of scientific and technological interests.     

Periodic impact behavior of a class of Hamiltonian oscillators with obstacles

In this paper, we study the existence of harmonic and subharmonic solutions of a class of non-smooth Hamiltonian systems, then apply its results to the vibration problems

     

Model reference adaptive control of a nonsmooth dynamical system

In this paper a modified model reference adaptive control (MRAC) technique is presented which can be used to control systems with nonsmooth characteristics. Using unmodified MRAC on (noisy) nonsmooth systems leads to destabilization of the controller. A localized analysis is presented which shows that the mechanism behind this behavior is the presence of a time invariant zero eigenvalue in the system. The modified algorithm is designed to eliminate this zero eigenvalue, making all the system eigenvalues stable. Both the modified and unmodified strategies are applied to an experimental system with a nonsmooth deadzone characteristic. As expected the unmodified algorithm cannot control the system, whereas the modified algorithm gives stable robust control, which has significantly improved performance over linear fixed gain control.     

Smoothing Technique and its Applications in Semidefinite Optimization

In this paper we extend the smoothing technique (Nesterov in Math Program 103(1): 127–152, 2005; Nesterov in Unconstrained convex mimimization in relative scale, 2003) onto the problems of semidefinite optimization. For that, we develop a simple framework for estimating a Lipschitz constant for the gradient of some symmetric functions of eigenvalues of symmetric matrices. Using this technique, we can justify the Lipschitz constants for some natural approximations of maximal eigenvalue and the spectral radius of symmetric matrices. We analyze the efficiency of the special gradient-type schemes on the problems of minimizing the maximal eigenvalue or the spectral radius of the matrix, which depends linearly on the design variables. We show that in the first case the number of iterations of the method is bounded by \(O({1}/{\epsilon})\), where \(\epsilon\) is the required absolute accuracy of the problem. In the second case, the number of iterations is bounded by \({({4}/{\delta})} \sqrt{(1 + \delta) r\, \ln r }\), where δ is the required relative accuracy and r is the maximal rank of corresponding linear matrix inequality. Thus, the latter method is a fully polynomial approximation scheme.