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Register a new userStrong Convergence of Forward-Backward-Forward Methods for Pseudo-monotone Variational Inequalities with Applications to Dynamic User Equilibrium in Traffic Networks
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In infinite-dimensional Hilbert spaces we device a class of strongly convergent primal-dual schemes for solving variational inequalities defined by a Lipschitz continuous and pseudomonote map. Our novel numerical scheme is based on Tsengs forward-backward-forward scheme, which is known to display weak convergence, unless very strong global monotonicity assumptions are made on the involved operators. We provide a simple augmentation of this algorithm which is computationally cheap and still guarantees strong convergence to a minimal norm solution of the underlying problem. We provide an adaptive extension of the algorithm, freeing us from requiring knowledge of the global Lipschitz constant. We test the performance of the algorithm in the computationally challenging task to find dynamic user equilibria in traffic networks and verify that our scheme is at least competitive to state-of-the-art solvers, and in some case even improve upon them.
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We introduce a relaxed inertial forward-backward-forward (RIFBF) splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitz continuous operator. This work aims to extend Tsengs forward-backward-forward method by both using inertial effects as well as relaxation parameters. We formulate first a second order dynamical system which approaches the solution set of the monotone inclusion problem to be solved and provide an asymptotic analysis for its trajectories. We provide for RIFBF, which follows by explicit time discretization, a convergence analysis in the general monotone case as well as when applied to the solving of pseudo-monotone variational inequalities. We illustrate the proposed method by applications to a bilinear saddle point problem, in the context of which we also emphasize the interplay between the inertial and the relaxation parameters, and to the training of Generative Adversarial Networks (GANs).
We consider monotone inclusions defined on a Hilbert space where the operator is given by the sum of a maximal monotone operator $T$ and a single-valued monotone, Lipschitz continuous, and expectation-valued operator $V$. We draw motivation from the seminal work by Attouch and Cabot on relaxed inertial methods for monotone inclusions and present a stochastic extension of the relaxed inertial forward-backward-forward (RISFBF) method. Facilitated by an online variance reduction strategy via a mini-batch approach, we show that (RISFBF) produces a sequence that weakly converges to the solution set. Moreover, it is possible to estimate the rate at which the discrete velocity of the stochastic process vanishes. Under strong monotonicity, we demonstrate strong convergence, and give a detailed assessment of the iteration and oracle complexity of the scheme. When the mini-batch is raised at a geometric (polynomial) rate, the rate statement can be strengthened to a linear (suitable polynomial) rate while the oracle complexity of computing an $epsilon$-solution improves to $O(1/epsilon)$. Importantly, the latter claim allows for possibly biased oracles, a key theoretical advancement allowing for far broader applicability. By defining a restricted gap function based on the Fitzpatrick function, we prove that the expected gap of an averaged sequence diminishes at a sublinear rate of $O(1/k)$ while the oracle complexity of computing a suitably defined $epsilon$-solution is $O(1/epsilon^{1+a})$ where $a>1$. Numerical results on two-stage games and an overlapping group Lasso problem illustrate the advantages of our method compared to stochastic forward-backward-forward (SFBF) and SA schemes.
Monotone inclusions play an important role in studying various convex minimization problems. In this paper, we propose a forward-partial inverse-half-forward splitting (FPIHFS) algorithm for finding a zero of the sum of a maximally monotone operator, a monotone Lipschitzian operator, a cocoercive operator, and a normal cone of a closed vector subspace. The FPIHFS algorithm is derived from a combination of the partial inverse method with the forward-backward-half-forward splitting algorithm. As applications, we employ the proposed algorithm to solve several composite monotone inclusion problems, which include a finite sum of maximally monotone operators and parallel-sum of operators. In particular, we obtain a primal-dual splitting algorithm for solving a composite convex minimization problem, which has wide applications in many real problems. To verify the efficiency of the proposed algorithm, we apply it to solve the Projection on Minkowski sums of convex sets problem and the generalized Heron problem. Numerical results demonstrate the effectiveness of the proposed algorithm.
In this paper, we consider a generalized forward-backward splitting (G-FBS) operator for solving the monotone inclusions, and analyze its nonexpansive properties in a context of arbitrary variable metric. Then, for the associated fixed-point iterations (i.e. the G-FBS algorithms), the global ergodic and pointwise convergence rates of metric distance are obtained from the nonexpansiveness. The convergence in terms of objective function value is also investigated, when the G-FBS operator is applied to a minimization problem. A main contribution of this paper is to show that the G-FBS operator provides a simplifying and unifying framework to model and analyze a great variety of operator splitting algorithms, where the convergence behaviours can be easily described by the fixed-point construction of this simple operator.
In this paper, we study a class of generalized monotone variational inequality (GMVI) problems whose operators are not necessarily monotone (e.g., pseudo-monotone). We present non-Euclidean extragradient (N-EG) methods for computing approximate strong solutions of these problems, and demonstrate how their iteration complexities depend on the global Lipschitz or H{o}lder continuity properties for their operators and the smoothness properties for the distance generating function used in the N-EG algorithms. We also introduce a variant of this algorithm by incorporating a simple line-search procedure to deal with problems with more general continuous operators. Numerical studies are conducted to illustrate the significant advantages of the developed algorithms over the existing ones for solving large-scale GMVI problems.
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