TY - JOUR AB - Weak convergence of inertial iterative method for solving variational inequalities is the focus of this paper. The cost function is assumed to be non-Lipschitz and monotone. We propose a projection-type method with inertial terms and give weak convergence analysis under appropriate conditions. Some test results are performed and compared with relevant methods in the literature to show the efficiency and advantages given by our proposed methods. AU - Shehu, Yekini AU - Iyiola, Olaniyi S. ID - 7577 IS - 1 JF - Applicable Analysis SN - 0003-6811 TI - Weak convergence for variational inequalities with inertial-type method VL - 101 ER - TY - CONF AB - The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs. AU - Harris, David G. AU - Iliopoulos, Fotis AU - Kolmogorov, Vladimir ID - 10072 SN - 1868-8969 T2 - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques TI - A new notion of commutativity for the algorithmic Lovász Local Lemma VL - 207 ER - TY - CONF AB - We study a class of convex-concave saddle-point problems of the form minxmaxy⟨Kx,y⟩+fP(x)−h∗(y) where K is a linear operator, fP is the sum of a convex function f with a Lipschitz-continuous gradient and the indicator function of a bounded convex polytope P, and h∗ is a convex (possibly nonsmooth) function. Such problem arises, for example, as a Lagrangian relaxation of various discrete optimization problems. Our main assumptions are the existence of an efficient linear minimization oracle (lmo) for fP and an efficient proximal map for h∗ which motivate the solution via a blend of proximal primal-dual algorithms and Frank-Wolfe algorithms. In case h∗ is the indicator function of a linear constraint and function f is quadratic, we show a O(1/n2) convergence rate on the dual objective, requiring O(nlogn) calls of lmo. If the problem comes from the constrained optimization problem minx∈Rd{fP(x)|Ax−b=0} then we additionally get bound O(1/n2) both on the primal gap and on the infeasibility gap. In the most general case, we show a O(1/n) convergence rate of the primal-dual gap again requiring O(nlogn) calls of lmo. To the best of our knowledge, this improves on the known convergence rates for the considered class of saddle-point problems. We show applications to labeling problems frequently appearing in machine learning and computer vision. AU - Kolmogorov, Vladimir AU - Pock, Thomas ID - 10552 T2 - 38th International Conference on Machine Learning TI - One-sided Frank-Wolfe algorithms for saddle problems ER - TY - CONF AB - The convex grabbing game is a game where two players, Alice and Bob, alternate taking extremal points from the convex hull of a point set on the plane. Rational weights are given to the points. The goal of each player is to maximize the total weight over all points that they obtain. We restrict the setting to the case of binary weights. We show a construction of an arbitrarily large odd-sized point set that allows Bob to obtain almost 3/4 of the total weight. This construction answers a question asked by Matsumoto, Nakamigawa, and Sakuma in [Graphs and Combinatorics, 36/1 (2020)]. We also present an arbitrarily large even-sized point set where Bob can obtain the entirety of the total weight. Finally, we discuss conjectures about optimum moves in the convex grabbing game for both players in general. AU - Dvorak, Martin AU - Nicholson, Sara ID - 9592 KW - convex grabbing game KW - graph grabbing game KW - combinatorial game KW - convex geometry T2 - Proceedings of the 33rd Canadian Conference on Computational Geometry TI - Massively winning configurations in the convex grabbing game on the plane ER - TY - JOUR AB - In this paper, we consider reflected three-operator splitting methods for monotone inclusion problems in real Hilbert spaces. To do this, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the reflected Krasnosel'skiĭ-Mann iteration for finding a fixed point of nonexpansive mapping in real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. We then apply our results to three-operator splitting for the monotone inclusion problem and consequently obtain the corresponding convergence analysis. Furthermore, we derive reflected primal-dual algorithms for highly structured monotone inclusion problems. Some numerical implementations are drawn from splitting methods to support the theoretical analysis. AU - Iyiola, Olaniyi S. AU - Enyi, Cyril D. AU - Shehu, Yekini ID - 9469 JF - Optimization Methods and Software SN - 1055-6788 TI - Reflected three-operator splitting method for monotone inclusion problem ER -