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Proximal bundle methods are a class of optimisation algorithms that leverage the proximal operator to address nonsmoothness in the objective function efficiently. This study focuses on a derivative-free (DFO) proximal bundle method and one of its applications called the DFO VU -algorithm. These algorithms incorporate approximate proximal points as subprocedures in order to optimise convex nonsmooth functions based on approximated subdifferential information. Interestingly, the classical VU -algorithm, which operates on true subgradient values, achieves superlinear convergence. At each iteration, the algorithm divides the whole space into two: the smooth U -space and the nonsmooth V -space. It takes a Newton-like step on the U -space and a proximal-point step on the V -space, enabling it to handle both smooth and nonsmooth parts effectively and converge faster. In this work, we reveal the worst possible convergence rate for the DFO VU -method by showing the linear convergence of the DFO proximal bundle method. This will be done by presenting a suitable framework and using the subdifferential-based error bound on the distance to critical points. [ABSTRACT FROM AUTHOR]

An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator T on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation x ˙ (t) + T λ (t) (x (t) − α (t) T λ (t) (x (t))) = 0 , t ≥ t 0 ≥ 0 , to the solution set T − 1 (0) of a generalized equation 0 ∈ T (x). With smart choices of parameters λ (t) and α (t) , we prove the weak convergence of the trajectory to some point of T − 1 (0) with ‖ x ˙ (t) ‖ ≤ O (1 / t) as t → + ∞. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory x h (t) → x (t) on interval [ 0 , + ∞) is demonstrated when the step size h → 0. [ABSTRACT FROM AUTHOR]

A class of proximal gradient-type algorithm for bilevel nonlinear nondifferentiable programming problems with smooth substructure is developed in this paper. The original problem is approximately reformulated by explicit slow control technique to a parameterized family function which makes full use of the information of smoothness. At each iteration, we only need to calculate one proximal point analytically or with low computational cost. We prove that the accumulation iterations generated by the algorithms are solutions of the original problem. Moreover, some results of complexity of the algorithms are presented in convergence analysis. Numerical experiments are implemented to verify the efficiency of the proximal gradient algorithms for solving this kind of bilevel programming problems. [ABSTRACT FROM AUTHOR]

We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We argue that the strict saddle property may be a realistic assumption in applications, since it provably holds for generic semi-algebraic optimization problems. [ABSTRACT FROM AUTHOR]

Supported by the recent contributions in multiple domains, the first-order splitting became algorithms of choice for structured nonsmooth optimization. The large-scale noisy contexts make available stochastic information on the objective function and thus, the extension of proximal gradient schemes to stochastic oracles is heavily based on the tractability of the proximal operator corresponding to nonsmooth component, which has been highly exploited in the literature. However, some questions remained about the complexity of the composite models with proximal untractable terms. In this paper we tackle composite optimization problems, assuming only the access to stochastic information on both smooth and nonsmooth components, with a stochastic proximal first-order scheme with stochastic proximal updates. We provide sublinear O 1 k convergence rates (in expectation of squared distance to the optimal set) under the strong convexity assumption on the objective function. Also, linear convergence is achieved for convex feasibility problems. The empirical behavior is illustrated by numerical tests on parametric sparse representation models. [ABSTRACT FROM AUTHOR]

In this work, we give a tight estimate of the rate of convergence for the Halpern-iteration for approximating a fixed point of a nonexpansive mapping in a Hilbert space. Specifically, using semidefinite programming and duality we prove that the norm of the residuals is upper bounded by the distance of the initial iterate to the closest fixed point divided by the number of iterations plus one. [ABSTRACT FROM AUTHOR]

We propose an infeasible interior-point algorithm for constrained linear least-squares problems based on the primal-dual regularization of convex programs of Friedlander and Orban. Regularization allows us to dispense with the assumption that the active gradients are linearly independent. At each iteration, a linear system with a symmetric quasi-definite (SQD) matrix is solved. This matrix is shown to be uniformly bounded and nonsingular. While the linear system may be solved using sparse LDL T factorization, we observe that other approaches may be used. In particular, we build on the connection between SQD linear systems and unconstrained linear least-squares problems to solve the linear system with sparse QR factorization. We establish conditions under which a solution of the original, constrained least-squares problem is recovered. We report computational experience with the sparse QR factorization and illustrate the potential advantages of our approach. [ABSTRACT FROM AUTHOR]

We consider the problems of minimizing the sum of a continuously differentiable convex function and a nonsmooth convex function in this paper. These problems arise in many applications of practical interest. A class of alternating linearization methods is presented for solving these problems. The global convergence rate is also obtained under certain mild conditions. Numerical experiments validate the theoretical convergence analysis and verify the implementation of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Locating proximal points is a component of numerous minimization algorithms. This work focuses on developing a method to find the proximal point of a convex function at a point, given an inexact oracle. Our method assumes that exact function values are at hand, but exact subgradients are either not available or not useful. We use approximate subgradients to build a model of the objective function, and prove that the method converges to the true prox-point within acceptable tolerance. The subgradient gk used at each step k is such that the distance from gk to the true subdifferential of the objective function at the current iteration point is bounded by some fixed ε > 0. The algorithm includes a novel tilt-correct step applied to the approximate subgradient. [ABSTRACT FROM AUTHOR]

For a class of nonconvex nonsmooth functions, we consider the problem of computing an approximate critical point, in the case when only inexact information about the function and subgradient values is available. We assume that the errors in function and subgradient evaluations are merely bounded, and in principle need not vanish in the limit. We examine the redistributed proximal bundle approach in this setting, and show that reasonable convergence properties are obtained. We further consider a battery of difficult nonsmooth nonconvex problems, made even more difficult by introducing inexactness in the available information. We verify that very satisfactory outcomes are obtained in our computational implementation of the inexact algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the problems of minimizing the sum of two nonsmooth functions which are possibly nonconvex. These problems arise in many applications of practical interests. We present a proximal alternating linearization algorithm which alternately generates two approximate proximal points of the original objective function. It is proved that the accumulation points of iterations converge to a stationary point of the problem. Numerical experiments validate the theoretical convergence analysis and verify the implementation of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

An alternating linearization method with inexact data, for the bilevel problem of minimizing a nonsmooth convex function over the optimal solution set of another nonsmooth convex problem, is presented in this paper. The problem is first approximately transformed into an unconstrained optimization with the help of a penalty function and we prove that the penalty function admits exact penalization under some mild conditions. The objective function of this unconstrained problem is the sum of two nonsmooth convex functions and in the algorithm each iteration involves solving two easily solved subproblems. It is shown that the iterative sequence converges to a solution of the original problem by parametric minimization theorem. Numerical experiments validate the theoretical convergence analysis and illustrate the implementation of the alternating linearization algorithm for this bilevel program. [ABSTRACT FROM AUTHOR]

Given a point on the standard simplex, we calculate a proximal point on the regular grid which is closest with respect to any norm in a large class, including all $$\ell ^p$$ -norms for $$p\ge 1$$ . We show that the minimal $$\ell ^p$$ -distance to the regular grid on the standard simplex can exceed one, even for very fine mesh sizes in high dimensions. Furthermore, for $$p=1$$ , the maximum minimal distance approaches the $$\ell ^1$$ -diameter of the standard simplex. We also put our results into perspective with respect to the literature on approximating global optimization problems over the standard simplex by means of the regular grid. [ABSTRACT FROM AUTHOR]

Abstract: The backward–backward algorithm is a tool for finding minima of a regularization of the sum of two convex functions in Hilbert spaces. We generalize this setting to Hadamard spaces and prove the convergence of an error-tolerant version of the backward–backward method. [Copyright &y& Elsevier]

This paper presents an accelerated variant of the hybrid proximal extragradient (HPE) method for convex optimization, referred to as the accelerated HPE (A-HPE) framework. Iteration-complexity results are established for the A-HPE framework, as well as a special version of it, where a large stepsize condition is imposed. Two specific implementations of the A-HPE framework are described in the context of a structured convex optimization problem whose objective function consists of the sum of a smooth convex function and an extended real-valued nonsmooth convex function. In the first implementation, a generalization of a variant of Nesterov's method is obtained for the case where the smooth component of the objective function has Lipschitz continuous gradient. In the second implementation, an accelerated Newton proximal extragradient (A-NPE) method is obtained for the case where the smooth component of the objective function has Lipschitz continuous Hessian. It is shown that the A-NPE method has a O(1/k7/2) convergence rate, which improves upon the O(1/k³) convergence rate bound for another accelerated Newton-type method presented by Nesterov. Finally, while Nesterov's method is based on exact solutions of subproblems with cubic regularization terms, the A-NPE method is based on inexact solutions of subproblems with quadratic regularization terms and hence is potentially more tractable from a computational point of view. [ABSTRACT FROM AUTHOR]

In this article we study thehybrid extragradient methodcoupled with approximation and penalty schemes for convex minimization problems. Under certain hypotheses, which include, for example, the case of Tikhonov regularization, we prove asymptotic convergence of the method to the solution set of our minimization problem. When we use schemes of penalization or barrier, we can show asymptotic convergence using the well-known fast/slow parameterization techniques and exploiting the existence and finite length of an optimal path. [ABSTRACT FROM AUTHOR]

We consider the development of single-timescale schemes for the distributed computation of equilibria associated with Nash games in which each player solves a convex program. Equilibria associated with such games are wholly captured by the solution set of a variational inequality. Our focus is on a class of games, termed monotone Nash games, that lead to monotone variational inequalities. Distributed extensions of standard approaches for solving such variational problems are characterized by two challenges: (1) Unless suitable assumptions (such as strong monotonicity) are imposed on the mapping arising in the specification of the variational inequality, iterative methods often require the solution of a sequence of regularized problems, a naturally two-timescale process that is harder to implement in practice. (2) Additionally, algorithm parameters for all players (such as steplengths and regularization parameters) have to be chosen centrally and communicated to all players; importantly, these parameters cannot be independently chosen by a player. Motivated by these shortcomings, we present two practically implementable distributed regularization schemes that work on a single timescale; specifically, each scheme requires precisely one gradient or projection step at every iteration. Of these, the first is an iterative Tikhonov regularization (ITR) scheme, while the second is an analogously constructed iterative proximal-point (IPP) method. Both schemes are characterized by the property that the regularization/centering parameter are updated after every iteration, rather than when one has approximately solved the regularized problem. To aid in distributed settings requiring limited coordination across players, the schemes allow players to select their parameters independently and do not insist on central prescription of such parameters. We conclude with an application of these schemes on a networked Cournot game with nonlinear prices. [ABSTRACT FROM AUTHOR]

In a recent paper by Monteiro and Svaiter, a hybrid proximal extragradient (HPE) framework was used to study the iteration-complexity of a first-order (or, in the context of optimization, second-order) method for solving monotone nonlinear equations. The purpose of this paper is to extend this analysis to study a prox-type first-order method for monotone smooth variational inequalities and inclusion problems consisting of the sum of a smooth monotone map and a maximal monotone point-to-set operator. Each iteration of the method computes an approximate solution of a proximal subproblem obtained by linearizing the smooth part of the operator in the corresponding proximal equation for the original problem, which is then used to perform an extragradient step as prescribed by the HPE framework. Both pointwise and ergodic iteration-complexity results are derived for the aforementioned first-order method using corresponding results obtained here for a subfamily of the HPE framework. [ABSTRACT FROM AUTHOR]

In this note, an inertial and relaxed version of a diagonal hybrid projection-proximal point algorithm is considered, in order to find the minimum of a function f approximated by a sequence of functions (in general, smoother than f or taking into account some constraints of the problem). Two convergence theorems are proved under different kind of assumptions, which allows to apply the method in various cases. [ABSTRACT FROM AUTHOR]

Abstract: We examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators. We begin by showing how metric subregularity is sufficient for local linear convergence to a zero of a maximal monotone operator. This result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods. [Copyright &y& Elsevier]

We consider the problem of minimizing nonsmooth convex functions, defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite Hessian on the set of interest. This is a particular case of functions with primal-dual gradient structure, a notion closely related to the so-called VU space decomposition: At a given point, nonsmoothness is locally restricted to the directions of the subspace V, while along the subspace U the behavior of the function is twice differentiable. Constructive identification of the two subspaces is important, because it opens the way to devising fast algorithms for nonsmooth optimization (by following iteratively the manifold of smoothness on which superlinear U-Newton steps can be computed). In this work we show that, for the class of functions in consideration, the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points, provided a minimizer satisfies the nondegeneracy and strong transversality conditions. [ABSTRACT FROM AUTHOR]

The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the original function. This suggests that the many uses of proximal points, and their corresponding proximal envelopes (Moreau envelopes), will have a natural extension from convex optimization to nonconvex optimization. For example, the inexact proximal point methods for convex optimization might be redesigned to work for nonconvex functions. In order to begin the practical implementation of proximal points in a nonconvex setting, a first crucial step would be to design efficient methods of approximating nonconvex proximal points. This would provide a solid foundation on which future design and analysis for nonconvex proximal point methods could flourish. In this paper we present a methodology based on the computation of proximal points of piecewise affine models of the nonconvex function. These models can be built with only the knowledge obtained from a black box providing, for each point, the function value and one subgradient. Convergence of the method is proved for the class of nonconvex functions that are prox-bounded and lower- $${\mathcal{C}}^2$$ and encouraging preliminary numerical testing is reported. [ABSTRACT FROM AUTHOR]

We study a family of penalty functions for augmented Lagrangian methods, and concentrate on a penalty based on the modified logarithmic barrier function. The convex conjugate of this penalty induces a Bregman distance, and the dual iterates associated with the augmented Lagrangian algorithm correspond to the iterates produced by a proximal point algorithm based on this distance. The global convergence of the dual iterates is then proved. Moreover, the level curves of the quadratic approximation of the dual kernels associated with these penalty functions are the Dikin ellipsoids. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We propose a class of self-adaptive proximal point methods suitable for degenerate optimization problems where multiple minimizers may exist, or where the Hessian may be singular at a local minimizer. If the proximal regularization parameter has the form µ(x) = β||∇ f (x)||η where η ∈ [0, 2) and β > 0 is a constant, we obtain convergence to the set of minimizers that is linear for ∇ = 0 and β sufficiently small, superlinear for ∇ ∈ (0, 1), and at least quadratic for ∇ ∈ [1, 2). Two different acceptance criteria for an approximate solution to the proximal problem are analyzed. These criteria are expressed in terms of the gradient of the proximal function, the gradient of the original function, and the iteration difference. With either acceptance criterion, the convergence results are analogous to those of the exact iterates. Preliminary numerical results are presented using some ill-conditioned CUTE test problems. [ABSTRACT FROM AUTHOR]

Finite dimensional local convergence results for self-adaptive proximal point methods and nonlinear functions with multiple minimizers are generalized and extended to a Hilbert space setting. The principle assumption is a local error bound condition which relates the growth in the function to the distance to the set of minimizers. A local convergence result is established for almost exact iterates. Less restrictive acceptance criteria for the proximal iterates are also analyzed. These criteria are expressed in terms of a subdifferential of the proximal function and either a subdifferential of the original function or an iteration difference. If the proximal regularization parameter μ(x) is sufficiently small and bounded away from zero and f is sufficiently smooth, then there is local linear convergence to the set of minimizers. For a locally convex function, a convergence result similar to that for almost exact iterates is established. For a locally convex solution set and smooth functions, it is shown that if the proximal regularization parameter has the form μ(x) = β∥f′[x]∥η, where η ϵ (0, 2), then the convergence is at least superlinear if η ϵ (0, 1) and at least quadratic if η [1, 2). [ABSTRACT FROM AUTHOR]

In order to minimize a closed convex function that is approximated by a sequence of better behaved functions, we investigate the global convergence of a general hybrid iterative algorithm, which consists of an inexact relaxed proximal point step followed by a suitable orthogonal projection onto a hyperplane. The latter permits to consider a fixed relative error criterion for the proximal step. We provide various sets of conditions ensuring the global convergence of this algorithm. The analysis is valid for nonsmooth data in infinite-dimensional Hilbert spaces. Some examples are presented, focusing on penalty/barrier methods in convex programming. We also show that some results can be adapted to the zero-finding problem for a maximal monotone operator. [ABSTRACT FROM AUTHOR]

We consider the vector optimization problem of finding weakly efficient points for maps from a Hilbert space X to a Banach space Y with respect to the partial order induced by a closed, convex, and pointed cone $C\subset Y$ with a nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued convex optimization. In this extension, the subproblems consist of finding weakly efficient points for suitable regularizations of the original map. We present both an exact and an inexact version, in which the subproblems are solved only approximately, within a constant relative tolerance. In both cases, we prove weak convergence of the generated sequence to a weakly efficient point, assuming convexity of the map with respect to C and C-completeness of the initial section. In cases where this last assumption fails, we still establish that the generating sequence is, in a suitable sense, a minimizing one. We also exhibit a particular instance of the algorithm for which, under a mild hypothesis on C, the weak limit of the generated sequence is an efficient, rather than a weakly efficient, point. [ABSTRACT FROM AUTHOR]

In this work, we consider the convex quadratic programming problem arising in support vector machine (SVM), which is a technique designed to solve a variety of learning and pattern recognition problems. Since the Hessian matrix is dense and real applications lead to large-scale problems, several decomposition methods have been proposed, which split the original problem into a sequence of smaller subproblems. SVM light algorithm is a commonly used decomposition method for SVM, and its convergence has been proved only recently under a suitable block-wise convexity assumption on the objective function. In SVM light algorithm, the size q of the working set, i.e. the dimension of the subproblem, can be any even number. In the present paper, we propose a decomposition method on the basis of a proximal point modification of the subproblem and the basis of a working set selection rule that includes, as a particular case, the one used by the SVM light algorithm. We establish the asymptotic convergence of the method, for any size q ≥ 2 of the working set, and without requiring any further block-wise convexity assumption on the objective function. Furthermore, we show that the algorithm satisfies in a finite number of iterations a stopping criterion based on the violation of the optimality conditions. [ABSTRACT FROM AUTHOR]

This paper introduces a general implicit iterative method for finding zeros of a maximal monotone operator in a Hilbert space which unifies three previously studied strategies: relaxation, inertial type extrapolation and projection step. The first two strategies are intended to speed up the convergence of the standard proximal point algorithm, while the third permits one to perform inexact proximal iterations with fixed relative error tolerance. The paper establishes the global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters. [ABSTRACT FROM AUTHOR]

Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be represented using firmly nonexpansive operators, even when the original observation process is discontinuous. The proposed framework thus captures a large body of classical and contemporary best approximation problems arising in areas such as harmonic analysis, statistics, interpolation theory, and signal processing. The resulting problem is recast in terms of a common fixed point problem and solved with a new block-iterative algorithm that features approximate projections onto the individual sets as well as an extrapolated relaxation scheme that exploits the possible presence of affine constraints. A numerical application to signal recovery is demonstrated. [ABSTRACT FROM AUTHOR]

We propose a proximal Newton method for solving nondifferentiable convex optimization. This method combines the generalized Newton method with Rockafellar’s proximal point algorithm. At each step, the proximal point is found approximately and the regularization matrix is preconditioned to overcome inexactness of this approximation. We show that such a preconditioning is possible within some accuracy and the second-order differentiability properties of the Moreau-Yosida regularization are invariant with respect to this preconditioning. Based upon these, superlinear convergence is established under a semismoothness condition. [ABSTRACT FROM AUTHOR]

To minimize a convex function, we combine Moreau-Yosida regularizations, quasi-Newton matrices and bundling mechanisms. First we develop conceptual forms using “reversal” quasi-Newton formulae and we state their global and local convergence. Then, to produce implementable versions, we incorporate a bundle strategy together with a “curve-search”. No convergence results are given for the implementable versions; however some numerical illustrations show their good behaviour even for large-scale problems. [ABSTRACT FROM AUTHOR]

This paper introduces an algorithm for convex minimization which includes quasi-Newton updates within a proximal point algorithm that depends on a preconditioned bundle subalgorithm. The method uses the Hessian of a certain outer function which depends on the Jacobian of a proximal point mapping which, in turn, depends on the preconditioner matrix and on a Lagrangian Hessian relative to a certain tangent space. Convergence is proved under boundedness assumptions on the preconditioner sequence. [ABSTRACT FROM AUTHOR]