Abstract: This paper is concerned with the integrodifferential Benjamin-Bona-Mahony equation $$\begin{aligned} u_t - u_{txx} + \alpha u - \int _0^\infty g(s) u_{xx}(t-s) \mathrm{d}s + (f(u))_x = h \end{aligned}$$ complemented with Dirichlet boundary conditions, in the presence of a possibly large external force h. The nonlinearity f is allowed to exhibit a superquadratic growth, and the dissipation is due to the simultaneous interaction between the nonlocal memory term and the Rayleigh friction. The existence of regular global and exponential attractors of finite fractal dimension is shown. PubDate: 2019-04-16

Abstract: In this paper, we investigate the homotopy path related to \(\ell _{1}\) -norm minimization problems with \(\ell _{\infty }\) -norm constraints. We establish an enhanced upper bound on the number of linear segments in the path and provide an example showing that the number of segments is exponential in the number of variables in the worst case. We also use the homotopy framework to develop grid independent (cross-)validation schemes for sparse linear discriminant analysis and classification that make use of the entire path. Several numerical and statistical examples illustrate the applicability of the framework. PubDate: 2019-04-13

Abstract: The principal component analysis (PCA) is a powerful standard tool for reducing the dimensionality of data. Unfortunately, it is sensitive to outliers so that various robust PCA variants were proposed in the literature. This paper addresses the robust PCA by successively determining the directions of lines having minimal Euclidean distances from the data points. The corresponding energy functional is non-differentiable at a finite number of directions which we call anchor directions. We derive a Weiszfeld-like algorithm for minimizing the energy functional which has several advantages over existing algorithms. Special attention is paid to carefully handling the anchor directions, where the relation between local minima and one-sided derivatives of Lipschitz continuous functions on submanifolds of \(\mathbb {R}^d\) is taken into account. Using ideas for stabilizing the classical Weiszfeld algorithm at anchor points and the Kurdyka–Łojasiewicz property of the energy functional, we prove global convergence of the whole sequence of iterates generated by the algorithm to a critical point of the energy functional. Numerical examples demonstrate the very good performance of our algorithm. PubDate: 2019-04-05

Abstract: A stochastic control representation for solution of the Schrödinger equation is obtained, utilizing complex-valued diffusion processes. The Maslov dequantization is employed, where the domain is complex-valued in the space variable. The notion of stationarity is utilized to relate the Hamilton–Jacobi form of the dequantized Schrödinger equation to its stochastic control representation. Convexity is not required, and consequently, there is no restriction on the duration of the problem. Additionally, existence is reduced to a real-valued domain case. PubDate: 2019-04-01

Abstract: This paper investigates the optimal contraception control for a nonlinear size-structured population model with three kinds of mortality rates: intrinsic, intra-competition and female sterilant. First, we transform the model to a system of two subsystems, and establish the existence of a unique non-negative solution by means of frozen coefficients and fixed point theory, and show the continuous dependence of the population density on control variable. Then, the existence of an optimal control strategy is proved via compactness and extremal sequence. Next, necessary optimality conditions of first order are established in the form of an Euler–Lagrange system by the use of tangent-normal cone technique and adjoint system. Moreover, a numerical result for the optimal control strategy is presented. Our conclusions would be useful for managing the vermin. PubDate: 2019-04-01

Abstract: In this paper we analyze an homogeneous and isotropic mixture of viscoelastic solids. We propose conditions to guarantee the coercivity of the internal energy and also of the dissipation, first in dimension two and later in dimension three. We obtain an uniqueness result for the solutions when the dissipation is positive and without any hypothesis over the internal energy. When the internal energy and the dissipation are both positive, we prove the existence of solutions as well as their analyticity. Exponential stability and impossibility of localization of the solutions are immediate consequences. PubDate: 2019-04-01

Abstract: In this work we first present the existence, uniqueness and regularity of the strong solution of the tidal dynamics model perturbed by Lévy noise. Monotonicity arguments have been exploited in the proofs. We then formulate a martingale problem of Stroock and Varadhan associated to an initial value control problem and establish existence of optimal controls. PubDate: 2019-04-01

Abstract: In this paper, we study the existence of solutions for a time and state-dependent discontinuous nonconvex second order sweeping process with a multivalued perturbation. The moving set is assumed to be prox-regular, relatively ball-compact with a bounded variation. The perturbation of the normal cone is a scalarly upper semicontinuous convex valued multimapping satisfying a linear growth condition possibly time-dependent. As an application of the theoretical results, we investigate the theory of evolution quasi-variational inequalities. PubDate: 2019-04-01

Abstract: We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than q impulses (q can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality. PubDate: 2019-04-01

Abstract: In this paper, by introducing a new concept of the (f, g, h)-quasimonotonicity and applying the maximal monotonicity of bifunctions and KKM technique, we show the existence results of solutions for quasi mixed equilibrium problems when the constraint set is compact, bounded and unbounded, respectively, which extends and improves several well-known results in many respects. Next, we also obtain a result of optimal control to a minimization problem. Our main results can be applied to the problems of evolution equations, differential inclusions and hemivariational inequalities. PubDate: 2019-04-01

Abstract: We investigate gradient flows of some homogeneous functionals in \(\mathbb R^N\) , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong self-interaction (super critical case), the functional possesses a cone of negative energy. It is immediate to see that solutions with negative energy at some time become singular in finite time, meaning that a subset of particles concentrate at a single point. Here, we establish that all solutions become singular in finite time, in the super critical case, for the class of functionals under consideration. The paper is completed with numerical simulations illustrating the striking non linear dynamics when initial data have positive energy. PubDate: 2019-04-01

Abstract: We consider an optimal control problem for the 3D Navier–Stokes–Voigt equations in bounded domains, where the time needed to reach a desired state plays an essential role. We first prove the existence of optimal solutions, and then establish the first-order necessary optimality conditions and the second-order sufficient optimality conditions. PubDate: 2019-04-01

Abstract: In the present paper, we focus on the vector optimization problems with constraints, where objective functions and constrained functions are Fréchet differentiable, and whose gradient mapping is locally Lipschitz. By using the second-order symmetric subdifferential and the second-order tangent set, we introduce some new types of second-order regularity conditions in the sense of Abadie. Then we establish some second-order necessary optimality conditions Karush–Kuhn–Tucker-type for local efficient (weak efficient, Geoffrion properly efficient) solutions of the considered problem. In addition, we provide some sufficient conditions for a local efficient solution to such problem. PubDate: 2019-04-01

Abstract: The optimal allocation of resources for maximizing influence, spread of information or coverage, has gained attention in the past years, in particular in machine learning and data mining. But in applications, the parameters of the problem are rarely known exactly, and using wrong parameters can lead to undesirable outcomes. We hence revisit a continuous version of the Budget Allocation or Bipartite Influence Maximization problem introduced by Alon et al. (in: WWW’12 - Proceedings of the 21st Annual Conference on World Wide, ACM, New York, 2012) from a robust optimization perspective, where an adversary may choose the least favorable parameters within a confidence set. The resulting problem is a nonconvex–concave saddle point problem (or game). We show that this nonconvex problem can be solved exactly by leveraging connections to continuous submodular functions, and by solving a constrained submodular minimization problem. Although constrained submodular minimization is hard in general, here, we establish conditions under which such a problem can be solved to arbitrary precision \(\varepsilon \) . PubDate: 2019-03-29

Abstract: We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the Euclidean case to the Riemannian case. Thus, the variable lives on a known smooth manifold and is further constrained. In doing so, we exploit the growing literature on unconstrained Riemannian optimization. For the special case where the manifold is itself described by equality constraints, one could in principle treat the whole problem as a constrained problem in a Euclidean space. The main hypothesis we test here is whether it is sometimes better to exploit the geometry of the constraints, even if only for a subset of them. Specifically, this paper extends an augmented Lagrangian method and smoothed versions of an exact penalty method to the Riemannian case, together with some fundamental convergence results. Numerical experiments indicate some gains in computational efficiency and accuracy in some regimes for minimum balanced cut, non-negative PCA and k-means, especially in high dimensions. PubDate: 2019-03-28

Abstract: We investigate the long-time dynamics and optimal control problem of a thermodynamically consistent diffuse interface model that describes the growth of a tumor in presence of a nutrient and surrounded by host tissues. The state system consists of a Cahn–Hilliard type equation for the tumor cell fraction and a reaction–diffusion equation for the nutrient. The possible medication that serves to eliminate tumor cells is in terms of drugs and is introduced into the system through the nutrient. In this setting, the control variable acts as an external source in the nutrient equation. First, we consider the problem of “long-time treatment” under a suitable given mass source and prove the convergence of any global solution to a single equilibrium as \(t\rightarrow +\infty \) . Second, we consider the “finite-time treatment” that corresponds to an optimal control problem. Here we allow the objective cost functional to depend on a free time variable, which represents the unknown treatment time to be optimized. We prove the existence of an optimal control and obtain first order necessary optimality conditions for both the drug concentration and the treatment time. One of the main aim of the control problem is to realize in the best possible way a desired final distribution of the tumor cells, which is expressed by the target function \(\phi _\Omega \) . By establishing the Lyapunov stability of certain equilibria of the state system (without external source), we show that \(\phi _{\Omega }\) can be taken as a stable configuration, so that the tumor will not grow again once the finite-time treatment is completed. PubDate: 2019-03-15

Abstract: Recent results have proven the minimax optimality of LASSO and related algorithms for noisy linear regression. However, these results tend to rely on variance estimators that are inefficient or optimizations that are slower than LASSO itself. We propose an efficient estimator for the noise variance in high dimensional linear regression that is faster than LASSO, only requiring p matrix–vector multiplications. We prove this estimator is consistent with a good rate of convergence, under the condition that the design matrix satisfies the restricted isometry property (RIP). In practice, our estimator scales incredibly well into high dimensions, is highly parallelizable, and only incurs a modest bias. PubDate: 2019-03-14

Abstract: We consider an elliptic variational–hemivariational inequality with constraints in a reflexive Banach space, denoted \(\mathcal{P}\) , to which we associate a sequence of inequalities \(\{\mathcal{P}_n\}\) . For each \(n\in \mathbb {N}\) , \(\mathcal{P}_n\) is a variational–hemivariational inequality without constraints, governed by a penalty parameter \(\lambda _n\) and an operator \(P_n\) . Such inequalities are more general than the penalty inequalities usually considered in literature which are constructed by using a fixed penalty operator associated to the set of constraints of \(\mathcal{P}\) . We provide the unique solvability of inequality \(\mathcal{P}_n\) . Then, under appropriate conditions on operators \(P_n\) , we state and prove the convergence of the solution of \(\mathcal{P}_n\) to the solution of \(\mathcal{P}\) . This convergence result extends the results previously obtained in the literature. Its generality allows us to apply it in various situations which we present as examples and particular cases. Finally, we consider a variational–hemivariational inequality with unilateral constraints which arises in Contact Mechanics. We illustrate the applicability of our abstract convergence result in the study of this inequality and provide the corresponding mechanical interpretations. PubDate: 2019-03-14

Abstract: In this paper, we consider a linear one-dimensional thermoelastic Bresse system with second sound consisting of three hyperbolic equations and two parabolic equations coupled in a certain manner under mixed homogeneous Dirichlet–Neumann boundary conditions, where the heat conduction is given by Cattaneo’s law. Only the longitudinal displacement is damped via the dissipation from the two parabolic equations, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, non exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. Our method of proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach. PubDate: 2019-02-26

Authors:Tim Leung; Hongzhong Zhang Abstract: Trailing stop is a popular stop-loss trading strategy by which the investor will sell the asset once its price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing to buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we study an optimal double stopping problem with a random path-dependent maturity. Specifically, we first analytically solve the optimal liquidation problem with a trailing stop, and in turn derive the optimal timing to buy the asset. Our method of solution reduces the problem of determining the optimal trading regions to solving the associated differential equations. For illustration, we implement an example and conduct a sensitivity analysis under the exponential Ornstein–Uhlenbeck model. PubDate: 2019-02-22 DOI: 10.1007/s00245-019-09559-0