A Variable Metric Extension of the Forward–Backward–Forward Algorithm for Monotone Operators

We propose a variable metric extension of the forward–backward-forward algorithm for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. In turn, this framework provides a variable metric splitting algorithm for solving monotone inclusions involving sums of composite operators. Monotone operator splitting methods recently proposed in the literature are recovered as special cases.     

Four-Operator Splitting via a Forward–Backward–Half-Forward Algorithm with Line Search

In this article, we provide a splitting method for solving monotone inclusions in a real Hilbert space involving four operators: a maximally monotone, a monotone-Lipschitzian, a cocoercive, and a monotone-continuous operator. The proposed method takes advantage of the intrinsic properties of each operator, generalizing the forward–backward–half-forward splitting and the Tseng’s algorithm with line search. At each iteration, our algorithm defines the step size by using a line search in which the monotone-Lipschitzian and the cocoercive operators need only one activation. We also derive a method for solving nonlinearly constrained composite convex optimization problems in real Hilbert spaces. Finally, we implement our algorithm in a nonlinearly constrained least-square problem and we compare its performance with available methods in the literature.

     

Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function

We consider the minimization of a function G defined on \({ \mathbb{R} } ^{N}\) , which is the sum of a (not necessarily convex) differentiable function and a (not necessarily differentiable) convex function. Moreover, we assume that G satisfies the Kurdyka–?ojasiewicz property. Such a problem can be solved with the Forward–Backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose an acceleration strategy based on the use of variable metrics and of the Majorize–Minimize principle. We give conditions under which the sequence generated by the resulting Variable Metric Forward–Backward algorithm converges to a critical point of G. Numerical results illustrate the performance of the proposed algorithm in an image reconstruction application.     

On the Stability of a Forward–Backward Heat Equation

In this paper we examine spectral properties of a family of periodic singular Sturm?CLiouville problems which are highly non-self-adjoint but have purely real spectrum. The problem originated from the study of the lubrication approximation of a viscous fluid film in the inner surface of a rotating cylinder and has received a substantial amount of attention in recent years. Our main focus will be the determination of Schatten class inclusions for the resolvent operator and regularity properties of the associated evolution equation.     

An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations

In this paper, we will study an indefinite stochastic linear quadratic optimal control problem, where the controlled system is described by a stochastic differential equation with delay. By introducing the relaxed compensator as a novel method, we obtain the well-posedness of this linear quadratic problem for indefinite case. And then, we discuss the uniqueness and existence of the solutions for a kind of anticipated forward–backward stochastic differential delayed equations. Based on this, we derive the solvability of the corresponding stochastic Hamiltonian systems, and give the explicit representation of the optimal control for the linear quadratic problem with delay in an open-loop form. The theoretical results are validated as well on the control problems of engineering and economics under indefinite condition.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

Lagrangian Penalization Scheme with Parallel Forward–Backward Splitting

We propose a new iterative algorithm for the numerical approximation of the solutions to convex optimization problems and constrained variational inequalities, especially when the functions and operators involved have a separable structure on a product space, and exhibit some dissymmetry in terms of their component-wise regularity. Our method combines Lagrangian techniques and a penalization scheme with bounded parameters, with parallel forward–backward iterations. Conveniently combined, these techniques allow us to take advantage of the particular structure of the problem. We prove the weak convergence of the sequence generated by this scheme, along with worst-case convergence rates in the convex optimization setting, and for the strongly non-degenerate monotone operator case. Implementation issues related to the penalization of the constraint set are discussed, as well as applications in image recovery and non-Newtonian fluids modeling. A numerical illustration is also given, in order to prove the performance of the algorithm.     

Fast Kalman Filtering and Forward–Backward Smoothing via a Low-Rank Perturbative Approach

Kalman filtering-smoothing is a fundamental tool in statistical time-series analysis. However, standard implementations of the Kalman filter-smoother require O(d³) time and O(d²) space per time step, where d is the dimension of the state variable, and are therefore impractical in high-dimensional problems. In this article we note that if a relatively small number of observations are available per time step, the Kalman equations may be approximated in terms of a low-rank perturbation of the prior state covariance matrix in the absence of any observations. In many cases this approximation may be computed and updated very efficiently (often in just O(k²d) or O(k²d + kdlog?d) time and space per time step, where k is the rank of the perturbation and in general k ? d), using fast methods from numerical linear algebra. We justify our approach and give bounds on the rank of the perturbation as a function of the desired accuracy. For the case of smoothing, we also quantify the error of our algorithm because of the low-rank approximation and show that it can be made arbitrarily low at the expense of a moderate computational cost. We describe applications involving smoothing of spatiotemporal neuroscience data. This article has online supplementary material.     

Forward–Backward–Half Forward Dynamical Systems for Monotone Inclusion Problems with Application to v-GNE

Journal of Optimization Theory and Applications - In this paper, the first-order forward–backward–half forward dynamical systems associated with the inclusion problem consisting of...     

Discrete-time approximation of decoupled Forward–Backward SDE with jumps

We study a discrete-time approximation for solutions of systems of decoupled Forward–Backward Stochastic Differential Equations (FBSDEs) with jumps. Assuming that the coefficients are Lipschitz-continuous, we prove the convergence of the scheme when the number of time steps n$n$ goes to infinity. The rate of convergence is at least n^−1/2+ε$n^{-1/2+\epsilon }$, for any ε>0$\epsilon >0$. When the jump coefficient of the first variation process of the forward component satisfies a non-degeneracy condition which ensures its inversibility, we achieve the optimal convergence rate n^−1/2$n^{-1/2}$. The proof is based on a generalization of a remarkable result on the path-regularity of the solution of the backward equation derived by Zhang [J. Zhang, A numerical scheme for BSDEs, Annals of Applied Probability 14 (1) (2004) 459–488] in the no-jump case.     

The Thomas–Fermi Problem and Solutions of the Emden–Fowler Equation

Computational Mathematics and Mathematical Physics - A two-point boundary value problem is considered for the Emden–Fowler equation, which is a singular nonlinear ordinary differential...     

Linear Forward—Backward Stochastic Differential Equations

The problem of finding adapted solutions to systems of coupled linear forward—backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sufficient condition for the solvability of a class of linear FBSDEs. Then a Riccati-type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati-type equation is studied which leads to a representation of adapted solutions to linear FBSDEs. Accepted 29 April 1997     

Forward–Backward Penalty Scheme for Constrained Convex Minimization Without Inf-Compactness

In order to solve constrained minimization problems, Attouch et al. propose a forward–backward algorithm that involves an exterior penalization scheme in the forward step. They prove that every sequence generated by the algorithm converges weakly to a solution of the minimization problem if either the objective function or the penalization function corresponding to the feasible set is inf-compact. Unfortunately, this assumption leaves out problems that are not coercive, as well as several interesting applications in infinite-dimensional spaces. The purpose of this short article is to show this convergence result without the inf-compactness assumption.     

The Ambrosetti–Prodi Problem for the p-Laplace Operator

In this work we study the existence of a solution for the problem ? Δ _p u = f(u) + tΦ(x) + h(x), with homogeneous Dirichlet boundary conditions. Here the nonlinear term f(u) is a so-called jumping nonlinearity. In the proofs we use topological arguments and the sub-supersolutions method, together with comparison principles for the p-Laplacian.     

Maximum Principles of Markov Regime-Switching Forward–Backward Stochastic Differential Equations with Jumps and Partial Information

This paper presents three versions of maximum principle for a stochastic optimal control problem of Markov regime-switching forward–backward stochastic differential equations with jumps. First, a general sufficient maximum principle for optimal control for a system, driven by a Markov regime-switching forward–backward jump–diffusion model, is developed. In the regime-switching case, it might happen that the associated Hamiltonian is not concave and hence the classical maximum principle cannot be applied. Hence, an equivalent type maximum principle is introduced and proved. In view of solving an optimal control problem when the Hamiltonian is not concave, we use a third approach based on Malliavin calculus to derive a general stochastic maximum principle. This approach also enables us to derive an explicit solution of a control problem when the concavity assumption is not satisfied. In addition, the framework we propose allows us to apply our results to solve a recursive utility maximization problem.     

Convergence rate of inertial Forward–Backward algorithm beyond Nesterov’s rule

Mathematical Programming - In this paper we study the convergence of an Inertial Forward–Backward algorithm, with a particular choice of an over-relaxation term. In particular we show that...