Strong Monotonicity Assumption Research Articles

Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback

Feasible Online Learning with Gradient Feedback Online gradient descent (OGD) is well-known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of [Formula: see text] for strongly convex cost functions, and (2) in the multiagent setting of strongly monotone games with each agent employing OGD we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of [Formula: see text]. Whereas these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In “Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback,” M. Jordan, T. Lin, and Z. Zhou design a fully adaptive OGD algorithm, AdaOGD, that does not require a priori knowledge of these parameters. In the single-agent setting, the algorithm achieves [Formula: see text] regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs AdaOGD in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of [Formula: see text], again optimal up to log factors. The algorithms are illustrated in a learning version of the classic newsvendor problem, in which, because of lost sales, only (noisy) gradient feedback can be observed. The results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multiretailer settings.

Distributed Nash Equilibrium Seeking Over Random Graphs

Dear Editor, This letter is concerned with the distributed Nash equilibrium (NE) seeking in an N-player game over random graphs. We develop a distributed stochastic forward-backward (DSFB) algorithm based on local information exchange between agents. We prove that the DSFB algorithm can converge to an NE almost surely, and analyze the convergence rate of the proposed algorithm. Compared with the existing works on distributed NE seeking, the communication graph in this letter is supposed to be time-varying and stochastic, which makes the NE seeking algorithm more suitable for practical scenarios, but brings a great challenge in both the design and convergence analysis of the algorithm. Besides, by establishing a variational inequality on NE, we relax the co-coercivity or strong monotonicity assumption on the extended pseudo-gradient.

EXISTENCE AND UNIQUENESS RESULTS FOR A CLASS OF PARAMETRIC MIXED VARIATIONAL-HEMIVARIATIONAL PROBLEMS

In this paper, we introduce a general class of parametric mixed variational-hemivariational problems involving the Clarke's generalized derivatives and equilibrium functions (for brevity, PMVHP). Then, based on the technique involving the Tarafdar’s fixed point theorem and some arguments in the nonsmooth analysis, the existence of solutions for the problem PMVHP is studied. Furthermore, we establish the uniqueness of the solution to the problem PMVHP under some strong monotonicity assumptions. O ur main results in this paper extend the corresponding results in Matei (2019, 2022). In this paper, we introduce a general class of parametric mixed variational-hemivariational problems involving the Clarke's generalized derivatives and equilibrium functions (for brevity, PMVHP). Then, based on the technique involving the Tarafdar’s fixed point theorem and some arguments in the nonsmooth analysis, the existence of solutions for the problem PMVHP is studied. Furthermore, we establish the uniqueness of the solution to the problem PMVHP under some strong monotonicity assumptions. O ur main results in this paper extend the corresponding results in Matei (2019, 2022). Keywords: Clarke's generalized derivative; existence and uniqueness; fixed points for set-valued mappings ; parametric mixed variational-hemivariational problem

Convergence of a subgradient extragradient algorithm for solving monotone variational inequalities

In this paper, we introduce a new iterative algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a step size; an advantage of the algorithm is the computation of only one value of the mapping and one projection onto the admissible set per one iteration. The convergence of the algorithm is established without the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Also, we generalize the method with Bregman projection. Finally, some numerical experiments are presented to show the efficiency of the proposed algorithm.

A forward–backward penalty scheme with inertial effects for monotone inclusions. Applications to convex bilevel programming

ABSTRACTWe investigate a forward–backward splitting algorithm of penalty type with inertial effects for finding the zeros of the sum of a maximally monotone operator and a cocoercive one and the convex normal cone to the set of zeroes of an another cocoercive operator. Weak ergodic convergence is obtained for the iterates, provided that a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions, strong convergence for the sequence of generated iterates is proved. As a particular instance we consider a convex bilevel minimization problem including the sum of a non-smooth and a smooth function in the upper level and another smooth function in the lower level. We show that in this context weak non-ergodic and strong convergence can be also achieved under inf-compactness assumptions for the involved functions.

A Modified Projected Gradient Method for Monotone Variational Inequalities

In this paper, we investigate and analyze classical variational inequalities with Lipschitz continuous and monotone mapping in real Hilbert space. The projected reflected gradient method, with varying step size, requires at most two projections onto the feasible set and one value of the mapping per iteration. We modify the method with a simple structure; a weak convergence theorem for our algorithm is proved without any requirement of additional projections and the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Preliminary results from numerical experiments are performed.

The choice of effect measure for binary outcomes: Introducing counterfactual outcome state transition parameters.

Standard measures of effect, including the risk ratio, the odds ratio, and the risk difference, are associated with a number of well-described shortcomings, and no consensus exists about the conditions under which investigators should choose one effect measure over another. In this paper, we introduce a new framework for reasoning about choice of effect measure by linking two separate versions of the risk ratio to a counterfactual causal model. In our approach, effects are defined in terms of "counterfactual outcome state transition parameters", that is, the proportion of those individuals who would not have been a case by the end of follow-up if untreated, who would have responded to treatment by becoming a case; and the proportion of those individuals who would have become a case by the end of follow-up if untreated who would have responded to treatment by not becoming a case. Although counterfactual outcome state transition parameters are generally not identified from the data without strong monotonicity assumptions, we show that when they stay constant between populations, there are important implications for model specification, meta-analysis, and research generalization.

Tight global linear convergence rate bounds for Douglas\u2013Rachford splitting

Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.

Penalty schemes with inertial effects for monotone inclusion problems

We introduce a penalty term-based splitting algorithm with inertial effects designed for solving monotone inclusion problems involving the sum of maximally monotone operators and the convex normal cone to the (nonempty) set of zeros of a monotone and Lipschitz continuous operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the monotone inclusion problem, provided a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions we can even show strong nonergodic convergence of the iterates. This approach constitutes the starting point for investigating from a similar perspective monotone inclusion problems involving linear compositions of parallel-sum operators and, further, for the minimization of a complexly structured convex objective function subject to the set of minima of another convex and differentiable function.

Lagrange multipliers in elastic–plastic torsion problem for nonlinear monotone operators

The existence of Lagrange multipliers as a Radon measure is ensured for an elastic–plastic torsion problem associated to a nonlinear strictly monotone operator. A regularization of this result, namely the existence of Lp Lagrange multipliers, is obtained under strong monotonicity assumption on the operator. Moreover, the relationships between elastic–plastic torsion problem and the obstacle problem are investigated. Finally, an example of the so-called “Von Mises functions” is provided, namely of solutions of the elastic–plastic torsion problem, associated to nonlinear monotone operators, which are not obtained by means of the obstacle problem in the case f=constant.

Projected Reflected Gradient Methods for Monotone Variational Inequalities

This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. First, we propose the projected reflected gradient algorithm with a constant stepsize. It is similar to the projected gradient method, namely, the method requires only one projection onto the feasible set and only one value of the mapping per iteration. This distinguishes our method from most other projection-type methods for variational inequalities with monotone mapping. Also we prove that it has R-linear rate of convergence under the strong monotonicity assumption. The usual drawback of algorithms with constant stepsize is the requirement to know the Lipschitz constant of the mapping. To avoid this, we modify our first algorithm so that the algorithm needs at most two projections per iteration. In fact, our computational experience shows that such cases with two projections are very rare. This scheme, at least theoretically, seems to be very effective. All methods are shown to be globally convergent to a solution of the variational inequality. Preliminary results from numerical experiments are quite promising.

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

We present two modified versions of the primal-dual splitting algorithm relying on forward---backward splitting proposed in V $$\tilde{\mathrm{u}}$$ u ~ (Adv Comput Math 38(3):667---681, 2013) for solving monotone inclusion problems. Under strong monotonicity assumptions for some of the operators involved we obtain for the sequences of iterates that approach the solution orders of convergence of $$\mathcal{{O}}(\frac{1}{n})$$ O ( 1 n ) and $$\mathcal{{O}}(\omega ^n)$$ O ( ? n ) , for $$\omega \in (0,1)$$ ? ? ( 0 , 1 ) , respectively. The investigated primal-dual algorithms are fully decomposable, in the sense that the operators are processed individually at each iteration. We also discuss the modified algorithms in the context of convex optimization problems and present numerical experiments in image processing and pattern recognition in cluster analysis.

Convergence Analysis for a Primal-Dual Monotone + Skew Splitting Algorithm with Applications to Total Variation Minimization

In this paper we investigate the convergence behavior of a primal-dual splitting method for solving monotone inclusions involving mixtures of composite, Lipschitzian and parallel sum type operators proposed by Combettes and Pesquet (in Set-Valued Var. Anal. 20(2):307---330, 2012). Firstly, in the particular case of convex minimization problems, we derive convergence rates for the partial primal-dual gap function associated to a primal-dual pair of optimization problems by making use of conjugate duality techniques. Secondly, we propose for the general monotone inclusion problem two new schemes which accelerate the sequences of primal and/or dual iterates, provided strong monotonicity assumptions for some of the involved operators are fulfilled. Finally, we apply the theoretical achievements in the context of different types of image restoration problems solved via total variation regularization.

Identifiability of Causal Effects for Binary Variables with Baseline Data Missing Due to Death

We discuss identifiability and estimation of causal effects of a treatment in subgroups defined by a covariate that is sometimes missing due to death, which is different from a problem with outcomes censored by death. Frangakis et al. (2007, Biometrics 63, 641-662) proposed an approach for estimating the causal effects under a strong monotonicity (SM) assumption. In this article, we focus on identifiability of the joint distribution of the covariate, treatment and potential outcomes, show sufficient conditions for identifiability, and relax the SM assumption to monotonicity (M) and no-interaction (NI) assumptions. We derive expectation-maximization algorithms for finding the maximum likelihood estimates of parameters of the joint distribution under different assumptions. Further we remove the M and NI assumptions, and prove that signs of the causal effects of a treatment in the subgroups are identifiable, which means that their bounds do not cover zero. We perform simulations and a sensitivity analysis to evaluate our approaches. Finally, we apply the approaches to the National Study on the Costs and Outcomes of Trauma Centers data, which are also analyzed by Frangakis et al. (2007) and Xie and Murphy (2007, Biometrics 63, 655-658).

Regularization and iterative method for general variational inequality problem in hilbert spaces

Without the strong monotonicity assumption of the mapping, we provide a regularization method for general variational inequality problem, when its solution set is related to a solution set of an inverse strongly monotone mapping. Consequently, an iterative algorithm for finding such a solution is constructed, and convergent theorem of the such algorithm is proved. It is worth pointing out that, since we do not assume strong monotonicity of general variational inequality problem, our results improve and extend some well-known results in the literature.

A quasi-variational approach to a competitive economic equilibrium problem without strong monotonicity assumption

This paper is focused on the investigation of the Walrasian economic equilibrium problem involving utility functions. The equilibrium problem is here reformulated by means of a quasi-variational inequality problem. Our goal is to give an existence result without assuming strong monotonicity conditions. To this end, we make use of a perturbation procedure. In particular, we will consider suitable perturbed utility functions whose gradient satisfies a strong monotonicity condition and whose associated equilibrium problem admits a solution. Then, we will prove that the limit solution solves the unperturbed problem. We stress out that our result allows us to consider a wide class of utility functions in which the Walrasian equilibrium problem may be solved.

Asymptotic Behavior of Spherically Symmetric Marginally Trapped Tubes

We give conditions on a general stress-energy tensor T_{\alpha \beta} in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay per se is not required for this asymptotic result, and in this general setting, such decay only implies that the marginally trapped tube has finite length with respect to the induced metric. We do, however, impose a smallness condition (B1) which one may obtain in practice by imposing decay on the T_{vv} component of the stress-energy tensor. We give two applications of the theorem to self-gravitating Higgs field spacetimes, one using weak Price law decay, the other certain strong smallness and monotonicity assumptions.

Error Bounds of Regularized Gap Functions for Nonsmooth Variational Inequality Problems

We study the Clarke–Rockafellar directional derivatives of the regularized gap functions (and of some modified ones) for the variational inequality problem (VIP) defined by a locally Lipschitz but not necessarily differentiable function on a closed convex set in an Euclidean space. As applications we show that, under the strong monotonicity assumption, the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of Armijo type converge to the solution of the (VIP).

A Combined Relaxation Method for Nonlinear Variational Inequalities

When applied to variational inequalities, combined relaxation (CR) methods are convergent under mild assumptions. Namely, the underlying mapping need not be strictly monotone. In this paper, we describe a class of CR methods for nonlinear variational inequality problems (NVI), which involve two, rather than one, nonlinear mappings and a nonsmooth convex function. We establish a convergence result for the CR method in the monotone case and also show that it attains a linear rate of convergence under the additional strong monotonicity assumption. Implementation issues are also discussed.

Equivalent Unconstrained Minimization and Global Error Bounds for Variational Inequality Problems

New merit functions for variational inequality problems are constructed through the Moreau--Yosida regularization of some gap functions. The proposed merit functions constitute unconstrained optimization problems equivalent to the original variational inequality problem under suitable assumptions. Conditions are studied for those merit functions to be differentiable and for any stationary point of those functions to be a solution of the original variational inequality problem. Moreover, those functions are shown to provide global error bounds for general variational inequality problems under the strong monotonicity assumption only.