We show that the the error bound property, postulating that the step lengths of the proximal gradient method linearly bound the distance to the solution set, is equivalent to a standard quadratic growth condition. We exploit this equivalence in an analysis of asymptotic linear convergence of the proximal gradient algorithm for structured problems, which lack … Read more
The Euclidean distance degree of a real variety is an important invariant arising in distance minimization problems. We show that the Euclidean distance degree of an orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances. Article Download … Read more
This paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar’s proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar’s PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm … Read more
We present a new method for minimizing the sum of a differentiable convex function and an $\ell_1$-norm regularizer. The main features of the new method include: $(i)$ an evolving set of indices corresponding to variables that are predicted to be nonzero at a solution (i.e., the support); $(ii)$ a reduced-space subproblem defined in terms of … Read more
An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimates of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool … Read more
Local graph clustering methods aim to identify well-connected clusters around a given “seed set” of reference nodes. The main focus of prior theoretical work has been on worst-case running time properties or on implicit statistical regularization; and the focus of prior empirical work has been to identify structure in large social and information networks. Here, … Read more
The primal-dual hybrid gradient method (PDHG) originates from the Arrow-Hurwicz method, and it has been widely used to solve saddle point problems, particularly in image processing areas. With the introduction of a combination parameter, Chambolle and Pock proposed a generalized PDHG scheme with both theoretical and numerical advantages. It has been analyzed that except for … Read more
This paper is devoted to the study of unconstrained planar multiobjective location problems, where distances between points are defined by means of the Manhattan norm. By identifying all nonessential objectives, we develop an effective algorithm for generating the whole set of efficient solutions. We prove the correctness of this algorithm and present some computational results, … Read more
We first study the fast minimization properties of the trajectories of the second-order evolution equation \begin{equation*} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \beta \nabla^2 \Phi (x(t))\dot{x} (t) + \nabla \Phi (x(t)) = 0, \end{equation*} where $\Phi : \mathcal H \to \mathbb R$ is a smooth convex function acting on a real Hilbert space $\mathcal H$, and … Read more
We consider a model of a pay-as-bid electricity market based on a multi-leader-common-follower approach where the producers as leaders are at the upper level and the regulator as a common follower is at the lower level. We fully characterise Nash equilibria for this model by describing necessary and sufficient conditions for their existence as well … Read more