First-order optimization algorithms via inertial systems with Hessian driven damping

For controlled discrete-time stochastic processes we introduce a new class of dynamic risk measures, which we call process-based. Their main feature is that they measure risk of processes that are functions of the history of a base process. We introduce a new concept of conditional stochastic time consistency and we derive the structure of process-based risk measures enjoying this property. We show that they can be equivalently represented by a collection of static law-invariant risk measures on the space of functions of the state of the base process. We apply this result to controlled Markov processes and we derive dynamic programming equations. We also derive dynamic programming equations for multistage stochastic programming with decision-dependent distributions.

     

Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities

In this paper we present the relaxed inertial proximal algorithm for Ky Fan minimax inequalities. Based on Opial lemma, we propose a weak convergence result to a solution of the problem by eliminating in the algorithm (RIPAFAN) the Browder–Halpern’s factor of contraction. We present after, a first result of strong convergence by adding a strong monotonicity condition. Secondly, we eliminate the strong monotonicity and add a Browder–Halpern’s contraction factor in the algorithm (RIPAFAN) and then ensure the strong convergence to a selected solution with respect to the contraction factor. Some examples are proposed. The first one concerns the convex minimization where the objective function is only controlled with a provided well conditioning. In the second one, we propose monotone set-valued variational inequalities. The last example deals with the problem of fixed point for a nonexpansive set-valued operator.     

Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

Mathematical Programming - In a Hilbert space setting $${{\mathcal {H}}}$$ , we study the fast convergence properties as $$t \rightarrow + \infty $$ of the trajectories of the second-order...     

Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.     

KKM-Fan’s lemma for solving nonlinear vector evolution equations with nonmonotonic perturbations

In this paper we are interested in the existence of solutions of the following initial value problem: on (0,T) with u(0)=u₀ where A:V→V^′ is a monotone operator, G:V→V^′ is a nonlinear nonmonotone operator and f:(0,T)→V^′ is a measurable function, by means of a recent generalization of the famous KKM-Fan’s lemma.     

Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics

Journal of Optimization Theory and Applications - In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve...