Abstract: Abstract This paper deals with the study of weak sharp solutions for nonsmooth variational inequalities and finite convergence property of the proximal point method. We present several characterizations for weak sharpness of the solutions set of nonsmooth variational inequalities without using the gap functions. We show that under weak sharpness of the solutions set, the sequence generated by proximal point methods terminates after a finite number of iterations. We also give an upper bound for the number of iterations for which the sequence generated by the exact proximal point methods terminates. PubDate: 2020-02-18

Abstract: Abstract SIR models, also with age structure, can be used to describe the evolution of an infectious disease. A vaccination campaign influences this dynamics immunizing part of the susceptible individuals, essentially turning them into recovered individuals. We assume that vaccinations are dosed at prescribed times or ages which introduce discontinuities in the evolution of the S and R populations. It is then natural to seek the “best” vaccination strategies in terms of costs and/or effectiveness. This paper provides the basic well posedness and stability results on the SIR model with vaccination campaigns, thus ensuring the existence of optimal dosing strategies. PubDate: 2020-02-04

Abstract: We study the existence and uniqueness of mild and strict solutions for abstract neutral differential equations with state-dependent delay. Some examples related to partial neutral differential equations are presented. PubDate: 2020-02-01

Abstract: Abstract We consider a parametric Dirichlet problem driven by the sum of a p-Laplacian (\(p>2\)) and a Laplacian (a (p, 2)-equation). The reaction consists of an asymmetric \((p-1)\)-linear term which is resonant as \(x \rightarrow - \infty \), plus a concave term. However, in this case the concave term enters with a negative sign. Using variational tools together with suitable truncation techniques and Morse theory (critical groups), we show that when the parameter is small the problem has at least three nontrivial smooth solutions. PubDate: 2020-02-01

Abstract: Abstract Zero-sum games with risk-sensitive cost criterion are considered with underlying dynamics being given by controlled stochastic differential equations. Under the assumption of geometric stability on the dynamics, we completely characterize all possible saddle point strategies in the class of stationary Markov controls. In addition, we also establish existence-uniqueness result for the value function of the Hamilton–Jacobi–Isaacs equation. PubDate: 2020-02-01

Abstract: Abstract A parametric constrained convex optimal control problem, where the initial state is perturbed and the linear state equation contains a noise, is considered in this paper. Formulas for computing the subdifferential and the singular subdifferential of the optimal value function at a given parameter are obtained by means of some recent results on differential stability in mathematical programming. The computation procedures and illustrative examples are presented. PubDate: 2020-02-01

Abstract: Abstract A two-dimensional time dependent model of an interaction between a thin elastic plate and a Newtonian viscous fluid described by the non-steady Stokes equations is considered. It depends on a small parameter \(\varepsilon \) that is the ratio of the thicknesses of the plate and the fluid layer. The Young’s modulus of the plate and its density may be great or small parameters equal to some powers (positive or negative) of \(\varepsilon \) while the density and the viscosity of the fluid are supposed to be of order one. An asymptotic expansion is constructed and justified for various magnitudes of the rigidity and density of the plate. The limit problems are studied in all these cases. They are Stokes equations with some special coupled or uncoupled boundary conditions modeling the interaction with the plate. The estimates of the difference between the exact solution and a truncated asymptotic expansion are established. These estimates justify the asymptotic approximations. PubDate: 2020-02-01

Abstract: Abstract This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem. PubDate: 2020-02-01

Abstract: Abstract This paper stems from the idea of adopting a new approach to solve some classical optimal packing problems for balls. In fact, we attack this kind of problems (which are of discrete nature) by means of shape optimization techniques, applied to suitable \(\Gamma \)-converging sequences of energies associated to Cheeger type problems. More precisely, in a first step we prove that different optimal packing problems are limits of sequences of optimal clusters associated to the minimization of energies involving suitable (generalized) Cheeger constants. In a second step, we propose an efficient phase field approach based on a multiphase \(\Gamma \)-convergence result of Modica–Mortola type, in order to compute those generalized Cheeger constants, their optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions. Our continuous shape optimization approach to solve discrete packing problems circumvents the NP-hard character of these ones, and efficiently leads to configurations close to the global minima. PubDate: 2020-02-01

Abstract: Abstract We construct Cournot games with limited demand, resulting into capped sales volumes according to the respective production shares of the players. We show that such games admit three distinct equilibrium regimes, including an intermediate regime that allows for a range of possible equilibria. When information on demand is modeled by a delayed diffusion process, we also show that this intermediate regime collapses to a single equilibrium while the other regimes approximate the deterministic setting as the delay tends to zero. Moreover, as the delay approaches zero, the unique equilibrium achieved in the stochastic case provides a way to select a natural equilibrium within the range observed in the no lag setting. Numerical illustrations are presented when demand is modeled by an Ornstein–Uhlenbeck process and price is an affine function of output. PubDate: 2020-02-01

Abstract: Abstract In this paper, we investigate uniform decay estimate of the solution to the Petrovsky equation with memory $$\begin{aligned} u_{tt}+\Delta ^2u-\int _ 0^t g(t-s)\Delta ^2u(s)ds=0 \end{aligned}$$with initial conditions and boundary conditions, where g is a memory kernel function. The related energy has been shown to decay exponentially or polynomially as \(t\rightarrow +\infty \) by the theorem established under the assumption \(g'(t)\leqslant -k g^{1+\frac{1}{p}}(t)\) with \(p\in (2,\infty )\) and \(k>0\) in the reference(J Funct Anal 254(5):1342–1372, 2008). Using the ideas introduced by by Lasiecka and Wang (Springer INdAM Series 10, Chapter, vol 14, pp 271–303, 2014), we prove the optimized uniform general decay result under the assumption \(g'(t)+H(g(t))\le 0\), where the function \(H(\cdot )\in C^1({\mathbb {R}}^1)\) is positive, increasing and convex with \(H(0)=0\), which is introduced for the first time by Alabau-Boussouira and Cannarsa (C R Acad Sci Paris Ser I 347:867–872, 2009) and studied systematically by Lasiecka and Wang (Springer INdAM Series 10, Chapter, vol 14, pp 271–303, 2014). The exponential decay result and polynomial decay result in the reference (J Funct Anal 254(5):1342–1372, 2008) are the special cases of this paper by choosing special \(H(\cdot )\). PubDate: 2020-01-29

Abstract: Abstract In this paper, we establish the existence of normalized solutions to the following Kirchhoff-type equation $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{{\mathbb {R}}^3} \nabla u ^2{\mathrm {d}}x\right) \Delta u-\lambda u=K(x)f(u), &{} x\in {\mathbb {R}}^3; \\ u\in H^1({\mathbb {R}}^3), \end{array} \right. \end{aligned}$$where \(a, b> 0\), \(\lambda \) is unknown and appears as a Lagrange multiplier, \(K\in {\mathcal {C}}({\mathbb {R}}^3, {\mathbb {R}}^+)\) with \(0<\lim _{ y \rightarrow \infty }K(y)\le \inf _{{\mathbb {R}}^3} K\), and \(f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\) satisfies general \(L^2\)-supercritical or \(L^2\)-subcritical conditions. We introduce some new analytical techniques in order to exclude the vanishing and the dichotomy cases of minimizing sequences due to the presence of the potential K and the lack of the homogeneity of the nonlinearity f. This paper extends to the nonautonomous case previous results on prescribed \(L^2\)-norm solutions of Kirchhoff problems. PubDate: 2020-01-29

Abstract: Abstract In this paper, we investigate open-loop and weak closed-loop solvabilities of stochastic linear quadratic (LQ, for short) optimal control problem of Markovian regime switching system. Interestingly, these two solvabilities are equivalent on [0, T). We first provide an alternative characterization of the open-loop solvability of LQ problem using a perturbation approach. Then, we study the weak closed-loop solvability of LQ problem of Markovian regime switching system, and establish the equivalent relationship between open-loop and weak closed-loop solvabilities. Finally, we present an example to shed on light on finding weak closed-loop optimal strategies within the framework of Markovian regime switching system. PubDate: 2020-01-28

Abstract: Abstract In Delong [8] we investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. We assume that the insurer’s risk aversion coefficient consists of a constant risk aversion and a small amount of wealth-dependent risk aversion. We apply perturbation theory and expand the equilibrium value function of the optimization problem on the parameter \(\epsilon \) controlling the degree of the insurer’s risk aversion depending on wealth. We derive a candidate for the first-order approximation to the equilibrium investment strategy. In this paper we formally show that the zeroth-order investment strategy \(\pi _0^*\) postulated by Delong (Math Methods Oper Res 89:73–113, 2019) performs better than any strategy \(\pi _0\) when we compare the asymptotic expansions of the objective functions up to order \({\mathcal {O}}(1)\) as \(\epsilon \rightarrow 0\), and the first-order investment strategy \(\pi _0^*+\pi _1^*\epsilon \) postulated by Delong (Math Methods Oper Res 89:73–113, 2019) is the equilibrium strategy in the class of strategies \(\pi ^*_0+\pi _1\epsilon \) when we compare the asymptotic expansions of the objective functions up to order \({\mathcal {O}}(\epsilon ^2)\) as \(\epsilon \rightarrow 0\), where \(\epsilon \) denotes the parameter controlling the degree of the insurer’s risk aversion depending on wealth. PubDate: 2020-01-25

Abstract: Abstract Motivated by empirical evidence for rough volatility models, this paper investigates continuous-time mean–variance (MV) portfolio selection under the Volterra Heston model. Due to the non-Markovian and non-semimartingale nature of the model, classic stochastic optimal control frameworks are not directly applicable to the associated optimization problem. By constructing an auxiliary stochastic process, we obtain the optimal investment strategy, which depends on the solution to a Riccati–Volterra equation. The MV efficient frontier is shown to maintain a quadratic curve. Numerical studies show that both roughness and volatility of volatility materially affect the optimal strategy. PubDate: 2020-01-24

Abstract: Abstract Gradient descent is a simple and widely used optimization method for machine learning. For homogeneous linear classifiers applied to separable data, gradient descent has been shown to converge to the maximal-margin (or equivalently, the minimal-norm) solution for various smooth loss functions. The previous theory does not, however, apply to the non-smooth hinge loss which is widely used in practice. Here, we study the convergence of a homotopic variant of gradient descent applied to the hinge loss and provide explicit convergence rates to the maximal-margin solution for linearly separable data. PubDate: 2020-01-23

Abstract: Abstract Motivated from time-inconsistent stochastic control problems, we introduce a new type of coupled forward–backward stochastic systems, namely, flows of forward–backward stochastic differential equations. They are coupled systems consisting of a single forward SDE and a continuum of BSDEs, which are defined on different time-intervals. We formulate a notion of equilibrium solutions in a general framework and prove small-time well-posedness of the equations. We also consider discretized flows and show that their equilibrium solutions approximate the original one, together with an estimate of the convergence rate. PubDate: 2020-01-20

Abstract: Abstract We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the \(L_2\)-norm difference of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform \(L_{\infty }\) bound, and \(W_2^{1,1}\)-energy estimate fo the discrete nonlinear PDE problem with discontinuous coefficient. PubDate: 2020-01-20

Abstract: Abstract Well-posedness is proved for the stochastic viscous Cahn–Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have any growth at infinity (in particular, also super-polynomial) provided that it is everywhere defined on the real line. A vanishing viscosity argument is carried out and the convergence of the solutions to the ones of the pure Cahn–Hilliard equation is shown. Some refined regularity results are also deduced for both the viscous and the non-viscous case. PubDate: 2020-01-14

Abstract: Abstract In this paper, we develop a Mean Field Games approach to Cluster Analysis. We consider a finite mixture model, given by a convex combination of probability density functions, to describe the given data set. We interpret a data point as an agent of one of the populations represented by the components of the mixture model, and we introduce a corresponding optimal control problem. In this way, we obtain a multi-population Mean Field Games system which characterizes the parameters of the finite mixture model. Our method can be interpreted as a continuous version of the classical Expectation–Maximization algorithm. PubDate: 2020-01-10