Abstract: In this paper we consider non-smooth convex optimization problems with (possibly) infinite intersection of constraints. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets, which are easy to project onto, in this paper we consider that each constraint set is given as the level set of a convex but not necessarily differentiable function. For these settings we propose subgradient iterative algorithms with random minibatch feasibility updates. At each iteration, our algorithms take a subgradient step aimed at only minimizing the objective function and then a subsequent subgradient step minimizing the feasibility violation of the observed minibatch of constraints. The feasibility updates are performed based on either parallel or sequential random observations of several constraint components. We analyze the convergence behavior of the proposed algorithms for the case when the objective function is strongly convex and with bounded subgradients, while the functional constraints are endowed with a bounded first-order black-box oracle. For a diminishing stepsize, we prove sublinear convergence rates for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected suboptimality of the function values along the weighted averages. Our convergence rates are known to be optimal for subgradient methods on this class of problems. Moreover, the rates depend explicitly on the minibatch size and show when minibatching helps a subgradient scheme with random feasibility updates. PubDate: 2019-12-01

Abstract: The value function associated with an optimal control problem subject to the Navier–Stokes equations in dimension two is analyzed. Its smoothness is established around a steady state, moreover, its derivatives are shown to satisfy a Riccati equation at the order two and generalized Lyapunov equations at the higher orders. An approximation of the optimal feedback law is then derived from the Taylor expansion of the value function. A convergence rate for the resulting controls and closed-loop systems is demonstrated. PubDate: 2019-12-01

Abstract: We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in Dipierro et al. (arXiv:1808.07696, 2018), and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term. PubDate: 2019-12-01

Abstract: The equality between dissipation and energy drop is a structural property of gradient-flow dynamics. The classical implicit Euler scheme fails to reproduce this equality at the discrete level. We discuss two modifications of the Euler scheme satisfying an exact energy equality at the discrete level. Existence of discrete solutions and their convergence as the fineness of the partition goes to zero are discussed. Eventually, we address extensions to generalized gradient flows, GENERIC flows, and curves of maximal slope in metric spaces. PubDate: 2019-12-01

Abstract: In a Hilbert space \({{\mathcal {H}}}\) , we study the convergence properties of a class of relaxed inertial forward–backward algorithms. They aim to solve structured monotone inclusions of the form \(Ax + Bx \ni 0\) where \(A:{{\mathcal {H}}}\rightarrow 2^{{\mathcal {H}}}\) is a maximally monotone operator and \(B:{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}\) is a cocoercive operator. We extend to this class of problems the acceleration techniques initially introduced by Nesterov, then developed by Beck and Teboulle in the case of structured convex minimization (FISTA). As an important element of our approach, we develop an inertial and parametric version of the Krasnoselskii–Mann theorem, where joint adjustment of the inertia and relaxation parameters plays a central role. This study comes as a natural extension of the techniques introduced by the authors for the study of relaxed inertial proximal algorithms. An illustration is given to the inertial Nash equilibration of a game combining non-cooperative and cooperative aspects. PubDate: 2019-12-01

Abstract: We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the \(\sigma \) -gradient ( \(0<\sigma <1\) ). We establish continuous dependence results with respect to the data, including the threshold of the fractional \(\sigma \) -gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Using the approximation of the solutions with a family of quasilinear penalisation problems we show the existence of generalised Lagrange multipliers for the \(\sigma \) -gradient constrained problem, extending previous results for the classical gradient case, i.e., with \(\sigma =1\) . PubDate: 2019-12-01

Abstract: This paper associates a dual problem to the minimization of an arbitrary linear perturbation of the robust sum function introduced in Dinh et al. (Set Valued Var Anal, 2019). It provides an existence theorem for primal optimal solutions and, under suitable duality assumptions, characterizations of the primal–dual optimal set, the primal optimal set, and the dual optimal set, as well as a formula for the subdifferential of the robust sum function. The mentioned results are applied to get simple formulas for the robust sums of subaffine functions (a class of functions which contains the affine ones) and to obtain conditions guaranteeing the existence of best approximate solutions to inconsistent convex inequality systems. PubDate: 2019-12-01

Abstract: We consider a two player, zero sum differential game with a cost of Bolza type, subject to a state constraint. It is shown that, under a suitable hypothesis concerning existence of inward pointing velocity vectors for the minimizing player at the boundary of the constraint set, the lower value of the game is Lipschitz continuous and is the unique viscosity solution (appropriately defined) of the lower Hamilton-Jacobi-Isaacs equation. If the inward pointing hypothesis is satisfied by the maximizing player’s velocity set, then the upper game is Lipschitz continuous and is the unique solution of the upper Hamilton-Jacobi-Isaacs equation. Under the classical Isaacs condition, the upper and lower Hamilton-Jacobi-Isaacs equation coincide. In this case, even if the inward pointing hypothesis is satisfied w.r.t. both players, the value of the game might fail to exist; however imposing stronger constraint qualifications (involving the existence of inward pointing vectors associated with saddle points for the Hamiltonian) the game value does exist and is the unique solution to this Hamilton-Jacobi-Isaacs equation. The novelty of our work resides in the fact that we permit the two players’ controls to be completely coupled within the dynamic constraint, state constraint and the cost functional, in contrast to earlier work, in which the players’ controls are decoupled w.r.t. the dynamics and state constraint, and interaction between them only occurs through the cost function. Furthermore, the inward pointing hypotheses that we impose are of a verifiable nature and less restrictive than those earlier employed. PubDate: 2019-12-01

Abstract: The aim of the present work is to provide an explicit decomposition formula for the resolvent operator \(\mathrm {J}_{A+B}\) of the sum of two set-valued maps A and B in a Hilbert space. For this purpose we introduce a new operator, called the A-resolvent operator of B and denoted by \(\mathrm {J}^A_B\) , which generalizes the usual notion. Then, our main result lies in the decomposition formula \(\mathrm {J}_{A+B}=\mathrm {J}_A\circ \mathrm {J}^A_B\) holding true when A is monotone. Several properties of \(\mathrm {J}^A_B\) are deeply investigated in this paper. In particular the relationship between \(\mathrm {J}^A_B\) and an extended version of the classical Douglas–Rachford operator is established, which allows us to propose a weakly convergent algorithm that computes numerically \(\mathrm {J}^A_B\) (and thus \(\mathrm {J}_{A+B}\) from the decomposition formula) when A and B are maximal monotone. In order to illustrate our theoretical results, we give an application in elliptic PDEs. Precisely the decomposition formula is used to point out the relationship between the classical obstacle problem and a new nonlinear PDE involving a partially blinded elliptic operator. Some numerical experiments, using the finite element method, are carried out in order to support our approach. PubDate: 2019-12-01

Abstract: In this work, we propose a new algorithm for finding a zero of the sum of two monotone operators where one is assumed to be single-valued and Lipschitz continuous. This algorithm naturally arises from a non-standard discretization of a continuous dynamical system associated with the Douglas–Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretization with respect to one of the operators involved. Each iteration of the proposed algorithm requires the evaluation of one forward and one backward operator. PubDate: 2019-12-01

Abstract: In this paper, we first establish the existence of a trajectory attractor for the Navier–Stokes–Voight (NSV) equation and then prove upper semicontinuity of trajectory attractors of 3D incompressible Navier–Stokes equation when 3D NSV equation is considered as a perturbative equation of 3D incompressible Navier–Stokes equation. PubDate: 2019-11-13

Abstract: The superiorization methodology is intended to work with input data of constrained minimization problems, i.e., a target function and a constraints set. However, it is based on an antipodal way of thinking to the thinking that leads constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to reduce (not necessarily minimize) target function values. This is done while retaining the feasibility-seeking nature of the algorithm and without paying a high computational price. A guarantee that the local target function reduction steps properly accumulate to a global target function value reduction is still missing in spite of an ever-growing body of publications that supply evidence of the success of the superiorization method in various problems. We propose an analysis based on the principle of concentration of measure that attempts to alleviate this guarantee question of the superiorization method. PubDate: 2019-11-11

Abstract: This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier–Stokes equations with a non-homogeneous boundary condition on Lipschitz-like domains. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimates on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is perturbed from a stationary force by a time-dependent perturbation, we also prove the upper semi-continuity of the attractors as the non-autonomous perturbation vanishes. Lastly, we investigate the regularity of these attractors when smoother initial data are given. Our results are new even for smooth domains. PubDate: 2019-11-06

Abstract: The Baillon–Haddad theorem establishes that the gradient of a convex and continuously differentiable function defined in a Hilbert space is \(\beta \) -Lipschitz if and only if it is \(1/\beta \) -cocoercive. In this paper, we extend this theorem to Gâteaux differentiable and lower semicontinuous convex functions defined on an open convex set of a Hilbert space. Finally, we give a characterization of \(C^{1,+}\) convex functions in terms of local cocoercivity. PubDate: 2019-11-01

Abstract: We consider a local minimizer, in the sense of the \(W^{1,m}\) norm ( \(m\ge 1\) ), of the classical problem of the calculus of variations P $$\begin{aligned} {\left\{ \begin{array}{ll} {\mathrm{Minimize}}\quad &{}\displaystyle I(x):=\int _a^b\varLambda (t,x(t), x'(t))\,dt+\varPsi (x(a), x(b))\\ \text {subject to:} &{}x\in W^{1,m}([a,b];\mathbb {R}^n),\\ &{}x'(t)\in C\,\text { a.e., } \,x(t)\in \varSigma \quad \forall t\in [a,b].\\ \end{array}\right. } \end{aligned}$$ where \(\varLambda :[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}\cup \{+\infty \}\) is just Borel measurable, C is a cone, \(\varSigma \) is a nonempty subset of \(\mathbb {R}^n\) and \(\varPsi \) is an arbitrary possibly extended valued function. When \(\varLambda \) is real valued, we merely assume a local Lipschitz condition on \(\varLambda \) with respect to t, allowing \(\varLambda (t,x,\xi )\) to be discontinuous and nonconvex in x or \(\xi \) . In the case of an extended valued Lagrangian, we impose the lower semicontinuity of \(\varLambda (\cdot ,x,\cdot )\) , and a condition on the effective domain of \(\varLambda (t,x,\cdot )\) . We use a recent variational Weierstrass type inequality to show that the minimizers satisfy a relaxation result and an Erdmann – Du Bois-Reymond convex inclusion which, remarkably, holds whenever \(\varLambda (x,\xi )\) is autonomous and just Borel. Under a growth condition, weaker than superlinearity, we infer the Lipschitz continuity of minimizers. PubDate: 2019-11-01

Abstract: Within the framework of the cumulative prospective theory of Kahneman and Tversky, this paper considers a continuous-time behavioral portfolio selection problem whose model includes both running and terminal terms in the objective functional. Despite the existence of S-shaped utility functions and probability distortions, a necessary condition for the optimality is derived. The results are applied to a few examples. PubDate: 2019-10-30

Abstract: The main goal of this paper is to study the existence of solutions for optimal control problems governed by mixed quasi-equilibrium problems under monotonicity type conditions. More precisely, the state control system takes the general form of a mixed quasi-equilibrium problem described by the sum of a maximal monotone bifunction and a pseudomonotone bifunction in the sense of Brézis. As applications, we study the existence of solutions for optimal control problems governed by quasi-variational inequalities. In particularly, we consider the optimal control of an evolutionary quasi-variational inequality described by a p-Laplacian type operator. The results obtained in this paper are new and can be applied to the study of optimal control of a variety of systems whose formulations can be presented as a mixed equilibrium problem. PubDate: 2019-10-24

Abstract: In this paper, we deal with the existence of global mild solutions and asymptotic behavior to the viscous Camassa–Holm equation in the locally uniform spaces. First we establish the global well-posedness for the Cauchy problem of viscous Camassa–Holm equation in \({\mathbb {R}}^1\) for any initial data \(u_0\in {\dot{H}}^1_U({\mathbb {R}}^1).\) Then we study the long time dynamical behavior of non-autonomous viscous Camassa–Holm equation on \({\mathbb {R}}^1\) with a new class of external forces and show the existence of \((H^1_U({\mathbb {R}}^1),H^1_\phi ({\mathbb {R}}^1))\) -uniform(w.r.t. \(g\in \mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)\) ) attractor \(\mathcal {A}_{\mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)}\) with locally uniform external forces being translation uniform bounded but not translation compact in \(L_b^2({\mathbb {R}};L^2_U({\mathbb {R}}^1))\) . PubDate: 2019-10-24

Abstract: Within the framework of the cumulative prospective theory of Kahneman and Tversky, this paper considers a continuous-time behavioral portfolio selection problem whose model includes both running and terminal terms in the objective functional. Despite the existence of S-shaped utility functions and probability distortions, a necessary condition for the optimality is derived. The results are applied to a few examples. PubDate: 2019-10-23

Abstract: A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables. PubDate: 2019-10-21