In this paper, we establish a new version of Ekeland’s variational principle in a new setting of cone pseudo-quasimetric spaces. In constrast to metric spaces, we do not require that each forward Cauchy sequence is forward convergent and that each forward convergent sequence has the unique forward limit. The motivation of this paper comes from … Read more
This paper studies the iteration-complexity of new regularized hybrid proximal extragradient (HPE)-type methods for solving monotone inclusion problems (MIPs). The new (regularized HPE-type) methods essentially consist of instances of the standard HPE method applied to regularizations of the original MIP. It is shown that its pointwise iteration-complexity considerably improves the one of the HPE method … Read more
The paper introduces and characterizes new notions of Lipschitzian and H\”olderian full stability of solutions to general parametric variational systems described via partial subdifferential of prox-regular functions acting in finite-dimensional and Hilbert spaces. These notions, postulated certain quantitative properties of single-valued localizations of solution maps, are closely related to local strong maximal monotonicity of associated … Read more
This article extends to the vector setting the results of our previous work Kruger et al. (2015) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi, J. Math. Anal. Appl. 246(1), 308–319 (2000). We introduce and characterize two seemingly new natural concepts of epsilon-minimality, one … Read more
In this paper, we present a generic framework to extend existing uniformly optimal convex programming algorithms to solve more general nonlinear, possibly nonconvex, optimization problems. The basic idea is to incorporate a local search step (gradient descent or Quasi-Newton iteration) into these uniformly optimal convex programming methods, and then enforce a monotone decreasing property of … Read more
We present two algorithms for large-scale low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive semidefinite cone, yielding a combinatorial procedure that is provably robust and parallelizable. Our second algorithm is a first order method … Read more
This paper first proposes another proof of the \textit{necessary and sufficient conditions of solution uniqueness in 1-norm minimization} given recently by H. Zhang, W. Yin, and L. Cheng. The analysis avoids the need of the surjectivity assumption made by these authors and should be mainly appealing by its short length (it can therefore be proposed … Read more
In this paper we establish conditions under which uniqueness of market equilibrium is obtained in a setup where prior to trading of electricity, transmission capacities between different market regions are fixed. In our setup, firms facing fluctuating demand decide on the size and location of production facilities. They make production decisions constrained by the invested … Read more
The \emph{alternating direction method of multipliers} (ADMM) is a popular and efficient first-order method that has recently found numerous applications, and the proximal ADMM is an important variant of it. The main contributions of this paper are the proposition and the analysis of a class of inertial proximal ADMMs, which unify the basic ideas of … Read more
In this paper, we first propose a general inertial \emph{proximal point algorithm} (PPA) for the mixed \emph{variational inequality} (VI) problem. Based on our knowledge, without stronger assumptions, convergence rate result is not known in the literature for inertial type PPAs. Under certain conditions, we are able to establish the global convergence and nonasymptotic $O(1/k)$ convergence … Read more