Recently, a strictly contractive Peaceman- Rachford splitting method (SC-PRSM) was proposed to solve a convex minimization model with linear constraints and a separable objective function which is the sum of two functions without coupled variables. We show by an example that the SC-PRSM cannot be directly extended to the case where the objective function is … Read more
In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a Pareto-Clarke critical point. Our method may be seen as an extension, for … Read more
The augmented Lagrangian method (ALM) is a benchmark for solving convex minimization problems with linear constraints. When the objective function of the model under consideration is representable as the sum of some functions without coupled variables, a Jacobian or Gauss-Seidel decomposition is often implemented to decompose the ALM subproblems so that the functions’ properties could … Read more
In this paper we propose an inexact proximal point method to solve constrained minimization problems with locally Lipschitz quasiconvex objective functions. Assuming that the function is also bounded from below, lower semicontinuous and using proximal distances, we show that the sequence generated for the method converges to a stationary point of the problem. Citation July … Read more
We propose a generalized proximal point algorithm (PPA), in the generic setting of finding a zero point of a maximal monotone operator. In addition to the classical PPA, a number of benchmark operator splitting methods in PDE and optimization literatures such as the Douglas-Rachford splitting method, Peaceman-Rachford splitting method, alternating direction method of multipliers, generalized … Read more
This paper presents a proximal point approach for computing the Riemannian or intrinsic Karcher mean of symmetric positive definite matrices. Our method derives from proximal point algorithm with Schur decomposition developed to compute minimum points of convex functions on symmetric positive definite matrices set when it is seen as a Hadamard manifold. The main idea … Read more
We present a new barrier-based method of constructing smoothing approximations for the Euclidean projector onto closed convex cones. These smoothing approximations are used in a smoothing proximal point algorithm to solve monotone nonlinear complementarity problems (NCPs) over a convex cones via the normal map equation. The smoothing approximations allow for the solution of the smoothed … Read more
In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex locally Lipschitz objective functions on Hadamard manifolds. To reach this goal, we use the concept of Clarke subdifferential on Hadamard manifolds and assuming that the function is bounded from below, we prove the global convergence of the … Read more
This paper is mainly devoted to the study and applications of H\”older metric subregularity (or metric $q$-subregularity of order $q\in(0,1]$) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and pointbased sufficient conditions as well as necessary conditions for $q$-metric subregularity with evaluating the exact … Read more
We present an extension of the proximal point method with Bregman distances to solve Variational Inequality Problems (VIP) on Hadamard manifolds (simply connected finite dimensional Riemannian manifold with nonpositive sectional curvature). Under some natural assumption, as for example, the existence of solutions of the (VIP) and the monotonicity of the multivalued vector field, we prove … Read more