The Proximal Point Method for Nonmonotone Variational Inequalities

We present a unified convergence rate analysis of iterative methods for solving the variational inequality problem. Our results are based on certain error bounds; they subsume and extend the linear and sublinear rates of convergence established in several previous studies. We also derive a new error bound for $\gamma$-strictly monotone variational inequalities. The class of algorithms covered by our analysis in fairly broad. It includes some classical methods for variational inequalities, e.g., the extragradient, matrix splitting, and proximal point methods. For these methods, our analysis gives estimates not only for linear convergence (which had been studied extensively), but also sublinear, depending on the properties of the solution. In addition, our framework includes a number of algorithms to which previous studies are not applicable, such as the infeasible projection methods, a separation-projection method, (inexact) hybrid proximal point methods, and some splitting techniques. Finally, our analysis covers certain feasible descent methods of optimization, for which similar convergence rate estimates have been recently obtained by Luo [14].

The proximal point method (PPM) for solving maximal monotone operator inclusion problem is a highly powerful tool for algorithm design, analysis and interpretation. To accelerate convergence of the PPM, inertial PPM (iPPM) was proposed in the literature. In this note, we point out that some of the attractive properties of the PPM, e.g., the generated sequence is contractive with the set of solutions, do not hold anymore for iPPM. To partially inherit the advantages of the PPM and meanwhile incorporate inertial extrapolation steps, we propose an iPPM with alternating inertial steps. Our analyses show that the even subsequence generated by the proposed iPPM is contractive with the set of solutions. Moreover, we establish global convergence result under much relaxed conditions on the inertial extrapolation stepsizes, e.g., monotonicity is no longer needed and the stepsizes are significantly enlarged compared to existing methods. Furthermore, we establish certain $o(1/k)$ convergence rate results, where $k$ denotes the iteration counter. These features are new to inertial type PPMs.

We study the relationships between three concepts which arise in connection with variational inequality problems: central paths defined by arbitrary barriers, generalized proximal point methods (where a Bregman distance substitutes for the Euclidean one), and Cauchy trajectory in Riemannian manifolds. First we prove that under rather general hypotheses the central path defined by a general barrier for a monotone variational inequality problem is well defined, bounded, and continuous and converges to the analytic center of the solution set (with respect to the given barrier), thus generalizing results which deal only with complementarity problems and with the logarithmic barrier. Next we prove that a sequence generated by the proximal point method with the Bregman distance naturally induced by the barrier function converges precisely to the same point. Furthermore, for a certain class of problems (including linear programming), such a sequence is contained in the central path, making the concepts of central path and generalized proximal point sequence virtually equivalent. Finally we prove that for this class of problems the central path also coincides with the Cauchy trajectory in the Riemannian manifold defined on the positive orthant by a metric given by the Hessian of the barrier (i.e., a curve whose direction at each point is the negative gradient of the objective function at that point in the Riemannian metric).

This paper studies the nonlinear inverse problem of identifying certain material parameters in the fourth-order boundary value problem representing the beam model. The inverse problem is solved by posing a convex optimization problem whose solution is an approximation of the sought parameters. The optimization problem is solved by the gradient based approaches, and in this setting, the most challenging aspect is the computation of the gradient of the objective functional. We present a detailed treatment of the adjoint stiffness matrix based approach for the gradient computation. We employ recently proposed self-adaptive inexact proximal point methods by Hager and Zhang [6] to solve the inverse problem. It is known that the regularization features of the proximal point methods are quite different from that of the Tikhonov regularization. We present a comparative analysis of the numerical efficiency of the used proximal point methods without using the Tikhonov regularization.

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to obtain a suitable ODE from a given DTA, and (ii) it is unclear the connection between the convergence of a DTA and its corresponding ODEs. In this paper, we propose a new machinery—an $$O(s^r)$$ -resolution ODE framework—for analyzing the behavior of a generic DTA, which (partially) answers the above two questions. The framework contains three steps: 1. To obtain a suitable ODE from a given DTA, we define a hierarchy of $$O(s^r)$$ -resolution ODEs of a DTA parameterized by the degree r, where s is the step-size of the DTA. We present a principal approach to construct the unique $$O(s^r)$$ -resolution ODEs from a DTA; 2. To analyze the resulting ODE, we propose the $$O(s^r)$$ -linear-convergence condition of a DTA with respect to an energy function, under which the $$O(s^r)$$ -resolution ODE converges linearly to an optimal solution; 3. To bridge the convergence properties of a DTA and its corresponding ODEs, we define the properness of an energy function and show that the linear convergence of the $$O(s^r)$$ -resolution ODE with respect to a proper energy function can automatically guarantee the linear convergence of the DTA. To better illustrate this machinery, we utilize it to study three classic algorithms—gradient descent ascent (GDA), proximal point method (PPM) and extra-gradient method (EGM)—for solving the unconstrained minimax problem $$\min _{x\in \mathbb {R}^n} \max _{y\in \mathbb {R}^m} L(x,y)$$ . Their O(s)-resolution ODEs explain the puzzling convergent/divergent behaviors of GDA, PPM and EGM when L(x, y) is a bilinear function, and showcase that the interaction terms help the convergence of PPM/EGM but hurts the convergence of GDA. Furthermore, their O(s)-linear-convergence conditions not only unify the known scenarios when PPM and EGM have linear convergence, but also showcase that these two algorithms exhibit linear convergence in much broader contexts, including when solving a class of nonconvex-nonconcave minimax problems. Finally, we show how this ODE framework can help design new optimization algorithms for minimax problems, by studying the difference between the O(s)-resolution ODE of GDA and that of PPM/EGM.

The EM algorithm is a popular method for maximum likelihood estimation from incomplete data. This method may be viewed as a proximal point method for maximizing the log-likelihood function using an integral form of the Kullback-Leibler distance function. Motivated by this interpretation, we consider a proximal point method using an integral form of entropy-like distance function. We give a convergence analysis of the resulting proximal point method in the case where the cluster points lie in the interior of the objective function domain. This result is applied to a normal/independent example and a Gaussian mixture example to establish convergence of the EM algorithm on these examples. Further convergence analysis of the method for maximization over an orthant is given in low dimensions. Sublinear convergence and schemes for accelerating convergence are also discussed.

Splitting methods have been extensively studied in the context of convex programming and variational inequalities with separable structures. Recently, a parallel splitting method based on the augmented Lagrangian method (abbreviated as PSALM) was proposed in He (Comput. Optim. Appl. 42:195---212, 2009) for solving variational inequalities with separable structures. In this paper, we propose the inexact version of the PSALM approach, which solves the resulting subproblems of PSALM approximately by an inexact proximal point method. For the inexact PSALM, the resulting proximal subproblems have closed-form solutions when the proximal parameters and inexact terms are chosen appropriately. We show the efficiency of the inexact PSALM numerically by some preliminary numerical experiments.

This paper deals with the study of weak sharp solutions for nonsmooth variational inequalities and finite convergence property of the proximal point method. We present several characterizations for weak sharpness of the solutions set of nonsmooth variational inequalities without using the gap functions. We show that under weak sharpness of the solutions set, the sequence generated by proximal point methods terminates after a finite number of iterations. We also give an upper bound for the number of iterations for which the sequence generated by the exact proximal point methods terminates.

We analyse a proximal point method for equilibrium problems in Hilbert spaces, improving upon previously known convergence results. We prove global weak convergence of the generated sequence to a solution of the problem, assuming existence of solutions and rather mild monotonicity properties of the bifunction which defines the equilibrium problem, and we establish existence of solutions of the proximal subproblems. We also present a new reformulation of equilibrium problems as variational inequalities ones.

The primary objective of this work is to present a rigorous treatment of various iterative methods for solving the elastography inverse problem of identifying cancerous tumors. From a mathematical standpoint, this inverse problem requires the identification of a variable parameter in a system of partial differential equations. We pose the nonlinear inverse problem as an optimization problem by using an output least-squares (OLS) and a modified output least-squares (MOLS) formulation. The optimality conditions then lead to a variational inequality problem which is solved using various gradient, extragradient, and proximal-point methods. Previously, only a few of these methods have been implemented, and there is currently no understanding of their relative efficiency and effectiveness. We present a thorough numerical comparison of the 15 iterative solvers which emerge from a variational inequality formulation.

We study the convergence of exact and inexact versions of the proximal point method with a generalized regularization function in Hadamard manifolds for solving scalar and vectorial optimization problems involving Lipschitz functions. We consider a local dominance property of the directional derivative of the objective function over the regularization term in order to obtain that cluster points of the sequence are stationary points. Under an additional assumption, we prove that every cluster point of the sequence is a minimizer in the scalar case and a weak efficient point in the vectorial case. Our results extend some of the existing ones in the literature about optimization on manifolds.

Metric subregularity and the proximal point method

We introduce a new iteration method for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of strict pseudocontractions in a real Hilbert space. The weak convergence of the iterative sequences generated by the method is obtained thanks to improve and extend some recent results under the assumptions that the cost mapping associated with the variational inequality problem only is pseudomonotone and not necessarily inverse strongly monotone. Finally, we present some numerical examples to illustrate the behaviour of the proposed algorithm.

We consider a general equilibrium problem defined on a convex set, whose cost bifunction may not be monotone. We show that this problem can be solved by the inexact proximal point method if there exists a solution to the dual problem. An application of this approach to nonlinearly constrained problems is also suggested.

This paper is mainly devoted to the study and applications of Hölder metric subregularity (or metric $q$-subregularity of order $q\in(0,1]$) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and point-based sufficient conditions as well as necessary conditions for $q$-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (first-order) metric subregularity in both finite and infinite dimensions. In this way we also obtain new fractional error bound results for composite polynomial systems with explicit calculating fractional exponents. Finally, metric $q$-subregularity is applied to conduct a quantitative convergence analysis of the classical proximal point method (PPM) for finding zeros of maximal monotone operators on Hilbert spaces.