Search

We consider convex optimization problems with a possibly nonsmooth objective function in the form of a mathematical expectation. The proposed framework (AN-SPS) employs Sample Average Approximations (SAA) to approximate the objective function, which is either unavailable or too costly to compute. The sample size is chosen in an adaptive manner, which eventually pushes the SAA error to zero almost surely (a.s.). The search direction is based on a scaled subgradient and a spectral coefficient, both related to the SAA function. The step size is obtained via a nonmonotone line search over a predefined interval, which yields a theoretically sound and practically efficient algorithm. The method retains feasibility by projecting the resulting points onto a feasible set. The a.s. convergence of AN-SPS method is proved without the assumption of a bounded feasible set or bounded iterates. Preliminary numerical results on Hinge loss problems reveal the advantages of the proposed adaptive scheme. In addition, a study of different nonmonotone line search strategies in combination with different spectral coefficients within AN-SPS framework is also conducted, yielding some hints for future work. [ABSTRACT FROM AUTHOR]

This study analyzes the conditional gradient method for constrained multiobjective optimization problems, also known as the Frank–Wolfe method. We assume that the objectives are continuously differentiable, and the constraint set is convex and compact. We employ an average-type nonmonotone line search, which takes the average of the recent objective function values. The asymptotic convergence properties without convexity assumptions on the objective functions are established. We prove that every limit point of the sequence of iterates that is obtained by the proposed method is a Pareto critical point. An iteration-complexity bound is provided regardless of the convexity assumption on the objective functions. The effectiveness of the suggested approach is demonstrated by applying it to several benchmark test problems. In addition, the efficiency of the proposed algorithm in generating approximations of the entire Pareto front is compared to the existing Hager–Zhang conjugate gradient method, the steepest descent method, the monotone conditional gradient method, and a nonmonotone conditional gradient method. In finding empirical comparison, we utilize two commonly used performance matrices—inverted generational distance and hypervolume indicators. [ABSTRACT FROM AUTHOR]

Robust approach is a special scalarization method to deal with multiobjective optimization problems in the worst-case. In this paper, we propose a new non-monotone gradient type algorithm for solving unconstrained multiobjective optimization problems by the conjugate technique and the robust approach. The proposed method has a memory gradient property since the search direction is constructed by using the current descent direction and the past multi-step iterative descent directions. For this, the search direction is called a memory gradient search direction. The step-size is computed by the non-monotone linear search. A lower bound of the stepsize is presented under some mild conditions. Then the iterative sequence generated by the proposed method is proved to be convergent to a Pareto critical point of the multiobjective optimization problem under some mild conditions. Numerical experiments are reported to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inexact projected gradient method equipped with a nonmonotone line search rule for smooth constrained multiobjective optimization. In this method, a new nonmonotone line search technique proposed here is employed and the relative errors on the search direction is admitted. We demonstrate that this method is well-defined. Then, we prove that each accumulation point of the sequence generated by this method is Pareto stationary and analyze the convergence rate of the algorithm. When the objective function is convex, the convergence of the sequence to a weak Pareto optimal point of the problem is established. [ABSTRACT FROM AUTHOR]

Here, we propose a practical method for solving nonsmooth convex problems by using conjugate gradient-type methods. The conjugate gradient method is one of the most remarkable methods to solve smooth and large-scale optimization problems. As a result of this fact, We present a modified HS conjugate gradient method. In the case that we have a nonsmooth convex problem, by the Moreau-Yosida regularization, we convert the nonsmooth objective function to a smooth function and then we use our method, by making use of a nonmonotone line search, for solving a nonsmooth convex optimization problem. We prove that our algorithm converges to an optimal solution under standard condition. Our algorithm inherits the performance of HS conjugate gradient method. [ABSTRACT FROM AUTHOR]

In this paper, we first develop an active set identification technique, and then we suggest a modified nonmonotone line search rule, in which a new parameter formula is introduced to control the degree of the nonmonotonicity of line search. By using the modified line search and the active set identification technique, we propose a global convergent method to solve the NMF based on the alternating nonnegative least squares framework. In addition, the larger step size technique is exploited to accelerate convergence. Finally, a large number of numerical experiments are carried out on synthetic and image datasets, and the results show that our presented method is effective in calculating speed and solution quality. [ABSTRACT FROM AUTHOR]

We concentrate on the general form of the weighted complementarity problem, which serves as a generalization of the nonlinear complementarity problem and finds extensive applications in various fields, including economics, sciences, engineering, atmospheric chemistry, and multibody dynamics. We introduce a novel Fischer-Burmeister-based one-parameter smoothing complementarity function. The WCP is then reformulated as a smoothing system of equations, and a new smoothing Newton method is devised to solve the problem efficiently on the new one-parameter smoothing complementarity function. To ensure global convergence, we introduce a new line search rule. The new method exhibits both global and local quadratic convergence properties under appropriate conditions, as demonstrated through several numerical experiments that confirm its effectiveness and stability. [ABSTRACT FROM AUTHOR]

Based on the memoryless BFGS (Broyden–Fletcher–Goldfarb–Shanno) updating formula of a recent well-structured diagonal approximation of the Hessian, we propose an improved proximal method for solving the minimization problem of nonsmooth composite functions. More exactly, a diagonally scaled matrix is iteratively used to approximate Hessian of the smooth ingredient of the cost function, which leads to straightly determining the search directions in each iteration. Afterward, in light of the Zhang–Hager nonmonotone scheme, a nonmonotone technique for performing the line search for the unconstrained optimization models with composite cost functions is devised. What is more, we address convergence of the suggested proximal algorithm. We close the discussion by empirically studying performance of the proposed algorithm on some large-scale compressive sensing and sparse logistic regression problems. [ABSTRACT FROM AUTHOR]

The current study provides two modified adaptive cubic regularized methods for the unconstrained optimization problems. The basis of these methods are based on a combination of the adaptive cubic regularized method (ARC) with the nonmonotone Armijo-type line search. The step lengths are calculated in different paths for the modified algorithms. Moreover, we analyse the global convergence properties for the proposed methods. The obtained results illustrate the capability of the modified algorithms compared to the other provided algorithms. [ABSTRACT FROM AUTHOR]

Considering a minimization problem according to the Byrd-Nocedal measure function together with the secant equation, a diagonal quasi-Newton updating formula is suggested. To find the optimal elements of the updating matrix, the well-known algorithm of the alternating direction method of multipliers (ADMM) is employed. Moreover, convergence analysis is conducted based on a modified nonmonotone Armijo line search incorporating the simulated annealing strategy. Lastly, performance of the method is numerically tested on a set of CUTEr functions and on a smooth transcendental approximation of the compressive sensing problem. Across the computational spectrum, the given method turns out to be successful. [ABSTRACT FROM AUTHOR]

Considering a minimization problem according to the Byrd-Nocedal measure function together with the secant equation, a diagonal quasi-Newton updating formula is suggested. To find the optimal elements of the updating matrix, the well-known algorithm of the alternating direction method of multipliers (ADMM) is employed. Moreover, convergence analysis is conducted based on a modified nonmonotone Armijo line search incorporating the simulated annealing strategy. Lastly, performance of the method is numerically tested on a set of CUTEr functions and on a smooth transcendental approximation of the compressive sensing problem. Across the computational spectrum, the given method turns out to be successful.

Founded upon a sparse estimation of the Hessian obtained by a recent diagonal quasi-Newton update, a conjugacy condition is given, and then, a class of conjugate gradient methods is developed, being modifications of the Hestenes–Stiefel method. According to the given sparse approximation, the curvature condition is guaranteed regardless of the line search technique. Convergence analysis is conducted without convexity assumption, based on a nonmonotone Armijo line search in which a forgetting factor is embedded to enhance probability of applying more recent available information. Practical advantages of the method are computationally depicted on a set of CUTEr test functions and also, on the well-known signal processing problems such as sparse recovery and nonnegative matrix factorization. [ABSTRACT FROM AUTHOR]

Founded upon the scaled memoryless symmetric rank-one updating formula, we propose an approximation of the Newton-type proximal strategy for minimizing the nonsmooth composite functions. More exactly, to approximate the inverse Hessian of the smooth part of the objective function, a symmetric rank-one matrix is employed to straightly compute the search directions for a special category of well-known functions. Convergence of the given algorithm is argued with a nonmonotone backtracking line search adjusted for the corresponding nonsmooth model. Also, its practical advantages are computationally depicted in the two well-known real-world models. [ABSTRACT FROM AUTHOR]

In this paper, we consider the composite optimization problems over the Stiefel manifold. A successful method to solve this class of problems is the proximal gradient method proposed by Chen et al. (SIAM J Optim 30:210–239, 2020. https://doi.org/10.1137/18M122457X). Motivated by the proximal Newton-type techniques in the Euclidean space, we present a Riemannian proximal quasi-Newton method, named ManPQN, to solve the composite optimization problems. The global convergence of the ManPQN method is proved and iteration complexity for obtaining an ϵ -stationary point is analyzed. Under some mild conditions, we also establish the local linear convergence result of the ManPQN method. Numerical results are encouraging, which shows that the proximal quasi-Newton technique can be used to accelerate the proximal gradient method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of inequalities associated with circular cones. By constructing a new smoothing function, the problem is reformulated as a system of parameterized smooth equations. In addition, we suggest a Newton-type algorithm for solving the smooth equations so that a solution of the problem concerned is found. In particular, the algorithm is proved to be globally and locally quadratically convergent under suitable conditions. The preliminary numerical results demonstrate that the algorithm is effective. [ABSTRACT FROM AUTHOR]

In this paper, we first develop an active set identification technique, and then we suggest a modified nonmonotone line search rule, in which a new parameter formula is introduced to control the degree of the nonmonotonicity of line search. By using the modified line search and the active set identification technique, we propose a global convergent method to solve the NMF based on the alternating nonnegative least squares framework. In addition, the larger step size technique is exploited to accelerate convergence. Finally, a large number of numerical experiments are carried out on synthetic and image datasets, and the results show that our presented method is effective in calculating speed and solution quality.

Smoothing Newton methods, which usually inherit local quadratic convergence rate, have been successfully applied to solve various mathematical programming problems. In this paper, we propose an accelerated smoothing Newton method (ASNM) for solving the weighted complementarity problem (wCP) by reformulating it as a system of nonlinear equations using a smoothing function. In spirit, when the iterates are close to the solution set of the nonlinear system, an additional approximate Newton step is computed by solving one of two possible linear systems formed by using previously calculated Jacobian information. When a Lipschitz continuous condition holds on the gradient of the smoothing function at two checking points, this additional approximate Newton step can be obtained with a much reduced computational cost. Hence, ASNM enjoys local cubic convergence rate but with computational cost only comparable to standard Newton's method at most iterations. Furthermore, a second-order nonmonotone line search is designed in ASNM to ensure global convergence. Our numerical experiments verify the local cubic convergence rate of ASNM and show that the acceleration techniques employed in ASNM can significantly improve the computational efficiency compared with some well-known benchmark smoothing Newton method. [ABSTRACT FROM AUTHOR]

This work proposes a matrix-free algorithm incorporating structured spectral parameters with a nonmonotone line search for solving nonlinear least squares problems. The structured Hessian matrix associated with the Newton's-like direction is approximated with a scalar multiple of identity. The scalar is a convex combination of the spectral parameters such that the structured quasi-Newton condition is approximately satisfied. Using a nonmonotone line search with some appropriate conditions, the global convergence of the sequence generated by the proposed algorithm is presented. The numerical result shows that the algorithm is competitive with the existing algorithms in the literature. [ABSTRACT FROM AUTHOR]

According to a minimization problem founded upon the Byrd–Nocedal measure function coupled with a penalty term of the secant equation, a diagonal quasi–Newton updating formula is given. Then, the proposed update is applied as the initial (inverse) Hessian estimation of the limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) method. An improved version of the method is also given based on the Li–Fukushima modified secant equation. Convergence analysis for the general functions is conducted based on a nonmonotone Armijo line search. Eventually, practical advantages of the methods are computationally appraised through some CUTEr functions and also, on the compressive sensing problem. [ABSTRACT FROM AUTHOR]

In an effort to make modification on the classical Fletcher–Reeves method, Jiang and Jian suggested an efficient nonlinear conjugate gradient algorithm which possesses the sufficient descent property when the line search fulfills the strong Wolfe conditions. Here, we develop a scaled modified version of the method which satisfies the sufficient descent condition independent of the line search. Also, a nonmonotone backtracking Armijo-type line search is proposed under which the global convergence of the method is established without convexity assumption. Performance of the method is evaluated by computational experiments on a set of CUTEr test functions and also, on the image reconstruction as a case study. The results show numerical efficiency of the method. [ABSTRACT FROM AUTHOR]

This paper proposes a nonmonotone spectral gradient method for solving large-scale unconstrained optimization problems. The spectral parameter is derived from the eigenvalues of an optimally sized memoryless symmetric rank-one matrix obtained under the measure defined as a ratio of the determinant of updating matrix over its largest eigenvalue. Coupled with a nonmonotone line search strategy where backtracking-type line search is applied selectively, the spectral parameter acts as a stepsize during iterations when no line search is performed and as a milder form of quasi-Newton update when backtracking line search is employed. Convergence properties of the proposed method are established for uniformly convex functions. Extensive numerical experiments are conducted and the results indicate that our proposed spectral gradient method outperforms some standard conjugate-gradient methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonmonotone gradient method for smooth constrained multiobjective optimization problems. Under mild assumptions, we demonstrate the Pareto stationarity of the accumulation point of the sequence generated by this method, while the convergence of the full sequence to a weak Pareto optimal solution of the problem is proven when the function is convex. Further, by imposing some assumptions on the gradients of the objective functions and the search directions, the linear convergence of the function value sequence to the optimal value is provided. The initial point in the convergence results established here can be any one in the constraint set. [ABSTRACT FROM AUTHOR]

In this paper, a nonmonotone Broyden family method is presented for the unconstrained optimization problems. The proposed line search technique is designed based on a nonmonotone line search proposed by Huang, Wan and Zhang (J. Comput. Appl. Math. 330: 586–604, 2018). The new line search technique can overcome the shortcoming that the nonmonotone line search proposed by Huang et. al only can be applied to conjugate gradient (CG) method, not Broyden family (including BFGS-type) method. The proposed method in this paper possesses some good properties: (i) a new nonmonotone line search technique is presented, (ii) the proposed nonmonotone line search technique can be applied not only to CG method but also to Broyden family (including BFGS-type) method, (iii) global convergence of the Broyden family method has been obtained with the proposed nonmonotone line search for nonconvex unconstrained minimization. In addition, numerical performance shows that the nonmonotone Broyden family method is more competitive versus the classical Broyden family method. [ABSTRACT FROM AUTHOR]

We propose structured spectral gradient algorithms for solving nonlinear least squares problems based on a modified structured secant equation. The idea was to integrate more details of the Hessian of the objective function into the standardized spectral parameters with the goal of improving numerical efficiency. We safeguard the structured spectral parameters to avoid negative curvature search direction. The sequence of the search direction generated by the respective proposed algorithm satisfies the sufficient descent condition. Using a nonmonotone line search strategy, we establish the global convergence of the proposed algorithms under some standard assumptions. Numerical experiments on some benchmark test problems show that the proposed algorithms are efficient and outperform some existing competitors. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Xinyang Normal University Natural Science Edition is the property of Journal of Xinyang Normal University Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider a projected gradient method equipped with the nonmonotone line search procedure for convex constrained multiobjective optimization problems. Under mild assumptions, we show the convergence of the full sequence generated by the algorithm to a weak Pareto optimal point. Furthermore, under some appropriate Lipschitz continuity assumption of the gradients of objective functions, a linear convergence result for this method is also established. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of derivative-free nonmonotone line search methods for solving nonlinear systems of equations, which includes the method N-DF-SANE proposed in Cheng and Li (IMA J Numer Anal 29:814–825, 2009). These methods correspond to derivative-free optimization methods applied to the minimization of a suitable merit function. Assuming that the mapping defining the system of nonlinear equations has Lipschitz continuous Jacobian, we show that the methods in the referred class need at most O | log (ϵ) | ϵ - 2 function evaluations to generate an ϵ -approximate stationary point to the merit function. For the case in which the mapping is strongly monotone, we present two methods with evaluation-complexity of O | log (ϵ) | . [ABSTRACT FROM AUTHOR]

In this paper, two new subspace minimization conjugate gradient methods based on p-regularization models are proposed, where a special scaled norm in p-regularization model is analyzed. Different choices of special scaled norm lead to different solutions to the p-regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. R-linear convergence of the proposed methods is also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J. Optim. 16(1):170–192, 2005), Dai and Kou (SIAM J. Optim. 23(1):296–320, 2013), Liu and Liu (J. Optim. Theory. Appl. 180(3):879–906, 2019) and Li et al. (Comput. Appl. Math. 38(1):2019), respectively. [ABSTRACT FROM AUTHOR]

This article is concerned with the problem of minimizing a smooth function over the Stiefel manifold. In order to address this problem, we introduce two adaptive scaled gradient projection methods that incorporate scaling matrices that depend on the step-size and a parameter that controls the search direction. These iterative algorithms use a projection operator based on the QR factorization to preserve the feasibility in each iteration. However, for some particular cases, the proposals do not require the use of any projection operator. In addition, we consider a Barzilai and Borwein-like step-size combined with the Zhang–Hager nonmonotone line-search technique in order to accelerate the convergence of the proposed procedures. We proved the global convergence for these schemes, and we evaluate their effectiveness and efficiency through an extensive computational study, comparing our approaches with other state-of-the-art gradient-type algorithms. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a globalized version of the Newton method for finding a singularity of the nonsmooth vector fields. Basically, the new method combines a version of nonsmooth Newton method with a nonmonotone line search strategy. The global convergence analysis of the proposed method as well as results on its rate are established under mild assumptions. Finally, numerical experiments illustrating the practical advantages of the proposed scheme are reported. [ABSTRACT FROM AUTHOR]

Graph matching problem has been widely used in many fields. Due to the discrete property of graph matching problem, the continuous relaxation will result in a bad influence on the solution. Thus, we propose a hypergraph matching model with an entropy barrier function considering the discrete property of the assignment matrix. The model is solved by a nonmonotone active set projected Newton method whose nonmonotone line search technique has a good ability to accept the projected Newton stepsize. We evaluate our method through several experiments, which indicates that our method has better matching results than others.

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we present two nonmonotone versions of adaptive cubic regularized (ARC) method for unconstrained optimization problems. The proposed methods are a combination of the ARC algorithm with the nonmonotone line search methods introduced by Zhang and Hager [A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (2004), pp. 1043–1056] and Ahookhosh et al. [A nonmonotone trust-region line search method for large-scale unconstrained optimization, Appl. Math. Model. 36 (2012), pp. 478–487]. The global convergence analysis for these iterative algorithms is established under suitable conditions. Several numerical examples are given to illustrate the efficiency and robustness of the newly suggested methods. The obtained results show the satisfactory performance of the proposed algorithms when compared to the basic ARC algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient method for nonnegative matrix factorization (NMF) based on the alternating nonnegative least-squares framework. To solve the nonnegativity constrained least-squares problems efficiently, we propose an extrapolated quadratic regularization projected Barzilai–Borwein (EQRPBB) method utilizing the extrapolation technique and a modified nonmonotone line search. The efficiency of the proposed method is demonstrated through experiments on synthetic and image datasets. We observe that our method significantly outperform existing ones in terms of computational speed. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new algorithmic framework to solve black-box problems with integer variables. The strategy included in the framework makes use of specific search directions (so called primitive directions) and a suitably developed nonmonotone line search, thus guaranteeing a high level of freedom when exploring the integer lattice. First, we describe and analyze a version of the algorithm that tackles problems with only bound constraints on the variables. Then, we combine it with a penalty approach in order to solve problems with simulation constraints. In both cases we prove finite convergence to a suitably defined local minimum of the problem. We report extensive numerical experiments based on a test bed of both bound-constrained and generally-constrained problems. We show the effectiveness of the method when compared to other state-of-the-art solvers for black-box integer optimization. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a filter QP-free infeasible method with nonmonotone line search is proposed for minimizing a smooth optimization problem with smooth inequality constraints. This proposed method is based on the solution of nonsmooth equations, which are obtained by the Lagrangian multiplier method and the function of the nonlinear complementarity problem for the Karush–Kuhn–Tucker optimality conditions. Especially, each iteration of this method can be viewed as a perturbation of a Newton or quasi-Newton iteration on both the primal and dual variables for the solution of the Karush–Kuhn–Tucker optimality conditions. What is more, it is considered to use the function of the nonlinear complementarity problem in the filter, which makes the proposed algorithm avoid the incompatibility. Then the global convergence of the proposed method is given. And under some mild conditions, the superlinear convergence rate can be obtained. Finally, some preliminary numerical results are shown to illustrate that the proposed filter QP-free infeasible method is quite promising.

In this paper, we present a new derivative-free spectral residual method for solving large-scale systems of equations. Our algorithm is equipped with a dwindling multidimensional nonmonotone filter in which whose envelope is dwindling as the step-length of line search is decreasing. The proposed algorithm is also combined with a relaxed nonmonotone line search technique which allows the algorithm to enjoy the nonmonotone property from scratch. Under some standard assumptions, the global convergence property of the proposed algorithm is established. Numerical results on some test problems show the efficiency and effectiveness of the new algorithm in practice.

In this paper, we investigate nonlinear semidefinite programming and propose a modified sequential semidefinite programming (SSDP for short) algorithm without a penalty function or a filter. At each iteration, the search direction is yielded by solving a linear semidefinite programming subproblem and a quadratic semidefinite programming subproblem. The nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced. Under some appropriate conditions, the global convergence of the proposed algorithm is shown. Some preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

In this paper we study the nonlinear weighted complementarity problem (denoted by NWCP). We first introduce a smoothing function which can be used to reformulate the NWCP as a system of smooth nonlinear equations. Then we propose a new smoothing-type algorithm to solve the NWCP which adopts a nonmonotone line search scheme. In each iteration, our algorithm solves one linear system of equations and performs one line search. Under suitable assumptions, we prove that the proposed algorithm is globally and locally quadratically convergent. Some numerical results are reported. [ABSTRACT FROM AUTHOR]

A new coordinate search method for bound constrained optimization is introduced. The proposed algorithm employs coordinate directions, in a suitable way, with a nonmonotone line search for accepting the new point, without using derivatives of the objective function. The main global convergence results are strongly based on the relationship between the step length and a stationarity measure. Also, a detailed benchmark study comparing different line search strategies is presented using a well known set of test problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a QP-free algorithm without a penalty function or a filter for nonlinear semideflnite programming. At each iteration, two systems of linear equations with the same coefficient matrix are solved to determine search direction; the nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced. There is no feasibility restoration phase in our algorithm, which is necessary for traditional filter methods. The proposed algorithm is globally convergent under some mild conditions. Preliminary numerical results indicate that the proposed algorithm is comparable. [ABSTRACT FROM AUTHOR]

This work concerns the numerical solution of high-dimensional systems of nonlinear equations, when derivatives are not available for use, but assuming that all functions defining the problem are continuously differentiable. A hybrid approach is taken, based on a derivative-free iterative method, organized in two phases. The first phase is defined by derivative-free versions of a fixed-point method that employs spectral parameters to define the steplength along the residual direction. The second phase consists on a matrix-free inexact Newton method that employs the Generalized Minimal Residual algorithm to solve the linear system that computes the search direction. This second phase will only take place if the first one fails to find a better point after a predefined number of reductions in the step size. In all stages, the criterion to accept a new point considers a nonmonotone decrease condition upon a merit function. Convergence results are established and the numerical performance is assessed through experiments in a set of problems collected from the literature. Both the theoretical and the experimental analysis support the feasibility of the proposed hybrid strategy. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient nonmonotone adaptive cubic regularization method with line search for solving large-scale unconstrained optimization problem. Its global convergence is analyzed. Numerical experiments are performed to show the efficiency and effectiveness of the proposed method and its superiority to the other existing methods. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of structured nonsmooth difference-of-convex (DC)minimization in which the first convex component is the sum of a smooth and a nonsmooth function while the second convex component is the supremum of finitely many convex smooth functions. The existing methods for this problem usually have weak convergence guarantees or exhibit slow convergence. Due to this, we propose two nonmonotone enhanced proximal DC algorithms for solving this problem. For possible acceleration, one uses a nonmonotone line-search scheme in which the associated Lipschitz constant is adaptively approximated by some local curvature information of the smooth function in the first convex component, and the other employs an extrapolation scheme. It is shown that every accumulation point of the solution sequence generated by them is a D-stationary point of the problem. These methods may, however, become inefficient when the number of convex smooth functions in defining the second convex component is large. To remedy this issue, we propose randomized counterparts for them and show that every accumulation point of the generated solution sequence is a D-stationary point of the problem almost surely. Some preliminary numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this work we consider an extension of the projected gradient method (PGM) to constrained multiobjective problems. The projected gradient scheme for multiobjective optimization proposed by Graña Drummond and Iusem and analyzed by Fukuda and Graña Drummond is extended to include a nonmonotone line search based on the average of the successive previous functions values instead of the traditional Armijo-like rules. Under standard assumptions, stationarity of the accumulation points is established. Moreover, under standard convexity assumptions, we prove full convergence to weakly Pareto optimal solutions of any sequence produced by the proposed algorithm. [ABSTRACT FROM AUTHOR]

In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the objective function values decrease at each iteration of the algorithms. In this work, we consider nonmonotone line searches, i.e., we allow the increase of objective function values in some iterations. Two well-known types of nonmonotone line searches are considered here: the one that takes the maximum of recent function values, and the one that takes their average. We also propose a new nonmonotone technique specifically for multiobjective problems. Under reasonable assumptions, we prove that every accumulation point of the sequence produced by the nonmonotone version of the steepest descent and Newton methods is Pareto critical. Moreover, we present some numerical experiments, showing that the nonmonotone technique is also efficient in the multiobjective case. [ABSTRACT FROM AUTHOR]