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In this paper, we introduce a generalized inexact scalarized proximal point algorithm to find Pareto-Clarke critical points and Pareto efficient solutions of quasiconvex multivalued functions defined on Hadamard manifolds considering vectorial and scalar errors to find a critical point of the regularized proximal function in each iteration. Under some assumptions on the problem, we obtain the global convergence of the sequence to a Pareto-Clarke critical point and assuming an extra condition on the proximal parameters we establish convergence to a Pareto efficient solution, approximately linear/superlinear rate of convergence and finite termination of the algorithm. In the convex case, we prove the convergence to a Pareto efficient solution point (more than a weak Pareto efficient solution point). The results of the paper are new even in the Euclidean space. [ABSTRACT FROM AUTHOR]

Inertial procedures attached to classical methods for solving monotone inclusion and optimization problems, which arise from an implicit discretization of second-order differential equations, have shown a remarkable acceleration effect with respect to these classical algorithms. Among these classical methods, one can mention steepest descent, alternate directions, and the proximal point methods. For the problem of finding zeroes of set-valued operators, the convergence analysis of all existing inertial-proximal methods requires the monotonicity of the operator. We present here a new inertial-proximal point algorithm for finding zeroes of set-valued operators, whose convergence is established for a relevant class of nonmonotone operators, namely the hypomonotone ones. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and analyse the strong convergence of the sequence generated by an inexact proximal point method with possible unbounded errors for finding zeros of monotone operators in Hadamard spaces. We show that the boundedness of the generated sequence is equivalent to the zero set of the operator to be nonempty. In this case, we prove the strong convergence of the generated sequence to a zero of the operator. We also provide some applications of our main results and give a numerical example to show the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of unconstrained multiobjective optimization problems (abbreviated as, MPQs), where the components of the objective function are locally Lipschitz and quasiconvex. To solve MPQs, we introduce an inexact proximal point method with Bregman distances (abbreviated as, IPPMB) via Mordukhovich limiting subdifferentials. We establish the well-definedness of the sequence generated by the IPPMB algorithm. Based on two versions of error criteria, we introduce two variants of IPPMB, namely, IPPMB1 and IPPMB2. Moreover, we establish that the sequences generated by the IPPMB1 and IPPMB2 algorithms converge to the Pareto–Mordukhovich critical point of the problem MPQ. In addition, we derive that if the components of the objective function of MPQ are convex, then the sequences converge to the weak Pareto efficient solution of MPQ. Furthermore, we discuss the linear and superlinear convergence of the sequence generated by the IPPMB2 algorithm. We furnish several non-trivial numerical examples to demonstrate the effectiveness of the proposed algorithms and solve them by employing MATLAB R2023b. To demonstrate the applicability and significance of the IPPMB algorithm, we solve a nonsmooth large-scale sparse quasiconvex multiobjective optimization by employing MATLAB R2023b. [ABSTRACT FROM AUTHOR]

In this paper we present an inexact scalarization proximal point algorithm to solve unconstrained multiobjective minimization problems where the objective functions are quasiconvex and locally Lipschitz. Under some natural assumptions on the problem, we prove that the sequence generated by the algorithm is well defined and converges. Then providing two error criteria we obtain two versions of the algorithm and it is proved that each sequence converges to a Pareto–Clarke critical point of the problem; furthermore, it is also proved that assuming an extra condition, the convergence rate of one of these versions is linear when the regularization parameters are bounded and superlinear when these parameters converge to zero. [ABSTRACT FROM AUTHOR]

The goal of this paper is to propose and investigate new iterative methods for examining an approximate solution of a fixed-point problem, an equilibrium problem, and a finite collection of variational inclusions in the Hadamard manifold's structure. Operating under some assumptions, we extend the proximal point algorithm to estimate the common solution of stated problems and obtain a strong convergence theorem for the common solution. We also present several consequences of the proposed iterative methods and their convergence results. [ABSTRACT FROM AUTHOR]

This paper focuses on efficient algorithms for finding the Dantzig selector which was first proposed by Candès and Tao as an effective variable selection technique in the linear regression. This paper first reformulates the Dantzig selector problem as an equivalent convex composite optimization problem and proposes a semismooth Newton augmented Lagrangian (Ssnal) algorithm to solve the equivalent form. This paper also applies a proximal point dual semismooth Newton (PpdSsn) algorithm to solve another equivalent form of the Dantzig selector problem. Comprehensive results on the global convergence and local asymptotic superlinear convergence of the Ssnal and PpdSsn algorithms are characterized under very mild conditions. The computational costs of a semismooth Newton algorithm for solving the subproblems involved in the Ssnal and PpdSsn algorithms can be cheap by fully exploiting the second order sparsity and employing efficient techniques. Numerical experiments on the Dantzig selector problem with synthetic and real data sets demonstrate that the Ssnal and PpdSsn algorithms substantially outperform the state-of-the-art first order algorithms even for the required low accuracy, and the proposed algorithms are able to solve the large-scale problems robustly and efficiently to a relatively high accuracy. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. First, we examine convergence properties of an inexact proximal point method with a quasi distance as a regularization term in order to find a critical point (in the sense of Toland) of a DC function (difference of two convex functions). Global convergence of the sequence and some convergence rates are obtained with additional assumptions. Second, as an application and its inspiration, we study in a dynamic setting, the very important and difficult problem of the limit of the firm and the time it takes to reach it (maturation time), when increasing returns matter in the short run. Both the formalization of the critical size of the firm in term of a recent variational rationality approach of human dynamics and the speed of convergence results are new in Behavioral Sciences. [ABSTRACT FROM AUTHOR]

This paper proposes an efficient adaptive variant of a quadratic penalty accelerated inexact proximal point (QP-AIPP) method proposed earlier by the authors. Both the QP-AIPP method and its variant solve linearly set constrained nonconvex composite optimization problems using a quadratic penalty approach where the generated penalized subproblems are solved by a variant of the underlying AIPP method. The variant, in turn, solves a given penalized subproblem by generating a sequence of proximal subproblems which are then solved by an accelerated composite gradient algorithm. The main difference between AIPP and its variant is that the proximal subproblems in the former are always convex while the ones in the latter are not necessarily convex due to the fact that their prox parameters are chosen as aggressively as possible so as to improve efficiency. The possibly nonconvex proximal subproblems generated by the AIPP variant are also tentatively solved by a novel adaptive accelerated composite gradient algorithm based on the validity of some key convergence inequalities. As a result, the variant generates a sequence of proximal subproblems where the stepsizes are adaptively changed according to the responses obtained from the calls to the accelerated composite gradient algorithm. Finally, numerical results are given to demonstrate the efficiency of the proposed AIPP and QP-AIPP variants. [ABSTRACT FROM AUTHOR]

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Weak sharp solutions of mixed variational inequalities are introduce and studied in Hilbert spaces. Several characterizations of weak sharpness of solutions of mixed variational inequalities without using gap functions are given. It is proved that sequences generated by an inexact proximal point algorithm terminate after a finite number of iterations for solutions of mixed variational inequalities provided that their solution sets are weakly sharp. Examples are also given to illustrate our results. [ABSTRACT FROM AUTHOR]

We consider the Nash-Cournot oligopolistic equilibrium models involving separable concave cost functions. In contrast to the models with linear and convex cost functions, a local equilibrium point may not be a global one in the models. We propose two algorithms for finding global and local equilibrium points of the models having separable concave cost functions by using a DC decomposition of a gap function. The first algorithm uses the convex envelope of a separable concave cost function to approximate a concave cost model with affine cost ones. The latter is equivalent to strongly convex quadratic programs that can be solved efficiently. To obtain better approximate solutions, the first algorithm uses an adaptive rectangular bisection which is performed only in the space of concave variables. The second algorithm is an extension of the proximal method to the models. [ABSTRACT FROM AUTHOR]

Results on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton-type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton's method, the nonsmooth Newton's method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton's method is discussed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a fast algorithm for MR image reconstruction based on the TV- /TGV- model. By utilizing the connection between the cut operator (called projection operator) and the shrink operator, the proposed algorithm adopts a semi-implicit scheme to get a compact iteration sequence so as to reduce computational cost. By combining the proximal point scheme, we can ensure that the system of linear equations to be solved at each iteration is well-conditioned. In addition, we also give convergence analysis of the proposed method. Numerical results with comparison to the split Bregman algorithm are supplied to demonstrate the efficiency of the proposed algorithm. [Copyright &y& Elsevier]

The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature. [ABSTRACT FROM AUTHOR]

This paper proposes an iterative method for solving strongly monotone equilibrium problems by using gap functions combined with double projection-type mappings. Global convergence of the proposed algorithm is proved and its complexity is estimated. This algorithm is then coupled with the proximal point method to generate a new algorithm for solving monotone equilibrium problems. A class of linear equilibrium problems is investigated and numerical examples are implemented to verify our algorithms. [ABSTRACT FROM AUTHOR]

Abstract: Let T : K → H be a continuous (η) –pseudomonotone and (η, L) –relaxed monotone mapping from a nonempty closed invex subset K of a finite-dimensional Hilbert space H into H. Let f : K → R be proper, invex and lower semicontinuous on K and let h : K → R be continuously Fréchet-differentiable on K with h′, the gradient of h, (η, α) –strongly monotone and (η, β) –Lipschitz continuous on K. Let x ∗ ∈ K be any fixed solution of the variational inequality problem (VIP): find an element x ∗ ∈ K such thatThen the sequence {x k } generated by the proximal point algorithm converges to x ∗. [Copyright &y& Elsevier]

We consider two different algorithms for solving the inclusion T ( x ) ∋ x , where T is a metrically regular set-valued mapping acting from a Banach space X to itself. The first one is given by the following Mann-type iteration: where λ n is a sequence of positive scalars in (0, 1) such that λ n ↑ 1. The second one is the classical proximal point algorithm adapted to metrically regular mappings: where μ n is a sequence of positive scalars such that μ n ↓ 0. We prove that if the modulus of regularity of T is sufficiently small then both algorithms (*) and (**) converge locally superlinearly to a fixed point of T . Similar convergence results are obtained for the cases when the mapping T is strongly subregular and strongly regular. [ABSTRACT FROM AUTHOR]

We present a primal-dual application of the proximal point algorithm to solve convex constrained minimization problems. Motivated by the work of Eckstein [Math. Oper. Res. 18 (1993) 203] about the generalized proximal point method, we propose here a mixed proximal multipliers method where we improve the result of Eckstein [Math. Oper. Res. 18 (1993) 203] Theorem 1. [Copyright &y& Elsevier]

Conditions are given for the viability and the weak convergence of an inexact, relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a Hilbert space. In turn, new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose a new approach to globalizing the Josephy-Newton algorithm for solving the monotone variational inequality problem. Known globalization strategies rely either on minimization of a suitable merit function, or on a projection-type approach. The technique proposed here is based on a linesearch in the regularized Josephy-Newton direction which finds a trial point and a proximal point subproblem (i.e., subproblem with suitable parameters), for which this trial point is an acceptable approximate solution. We emphasize that this requires only checking a certain approximation criterion, and in particular, does not entail actually solving any nonlinear proximal point subproblems. The method converges globally under very mild assumptions. Furthermore, an easy modification of the method secures the local superlinear rate of convergence under standard conditions. [ABSTRACT FROM AUTHOR]

We generalize a proximal-like method for finding zeroes of maximal monotone operators in Hilbert spaces with quadratic regularization due to Solodov and Svaiter, making it possible the use of other kind of regularizations and extending it to Banach spaces. In particular, we introduce an appropriate error criterium to obtain an inexact proximal iteration based on Bregman functions and construct a hyperplane which strictly separates the current iterate from the solution set. A Bregman projection onto this hyperplane is then used to obtain the next iterate. Boundedness of the sequence and optimality of the weak accumulation points are established under suitable assumptions on the regularizing function, which hold for any power greater than one of the norm of any uniformly smooth and uniformly convex Banach space, without any assumption on the operator other than existence of zeroes. These assumptions let us, also, obtain similar results in Banach spaces for the Hybrid Extragradient-Generalized Proximal Point method, proposed by Solodov and Svaiter for finite dimensional spaces. We then transpose such methods to generate augmented Lagrangian methods for LP-constrained convex optimization problems in Banach spaces, obtaining two alternative procedures which allow for sequences, and optimality of primal and dual weak accumulation points, are then established, assuming only existence of Karush-Kuhn-Tucker pairs. *The work of this author was partially supported by CNPq grant no. 301280/86 and by Marcel Dassault chair, Centro de Modelamiento Matemático, Universidad de Chile. [ABSTRACT FROM AUTHOR]

We present a new Bregman-function-based algorithm which is a modification of the generalized proximal point method for solving the variational inequality problem with a maximal monotone operator. The principal advantage of the presented algorithm is that it allows a more constructive error tolerance criterion in solving the proximal point subproblems. Furthermore, we eliminate the assumption of pseudomonotonicity which was, until now, standard in proving convergence for paramonotone operators. Thus we obtain a convergence result which is new even for exact generalized proximal point methods. Finally, we present some new results on the theory of Bregman functions. For example, we show that the standard assumption of convergence consistency is a consequence of the other properties of Bregman functions, and is therefore superfluous. [ABSTRACT FROM AUTHOR]