Logarithmic-quadratic Proximal Method Research Articles

New descent LQP alternating direction methods for solving a class of structured variational inequalities

In this paper, we propose two descent alternating direction methods based on a logarithmic-quadratic proximal method for structured variational inequalities. The first method can be viewed as an extension of the method proposed by Bnouhachem and Xu (Comput. Math. Appl. 67:671-680, 2014) by performing an additional step at each iteration. The second method generates the new iterate by searching the optimal step size along a new descent direction, which can be viewed as a refinement and improvement of the first one. Under certain conditions, the global convergence of the both methods is proved.

A descent LQP alternating direction method for solving variational inequality problems with separable structure

Based on the logarithmic–quadratic proximal method, a descent alternating direction method is introduced for structured variational inequalities in the paper. The proposal method generates the new iteration by searching the optimal step size along the integrated descent direction from two descent directions. Global convergence of the proposal method is proved under certain assumptions. To illustrate the proposal method and demonstrate its efficiency, some applications and their numerical results are also provided.

An inexact LQP alternating direction method for solving a class of structured variational inequalities

In this paper, we present an inexact alternating direction method. Each iteration of the proposed method contains a prediction and a correction, the predictor is obtained via solving the logarithmic-quadratic proximal (LQP) system approximately under significantly relaxed accuracy criterion and the new iterate is computed directly by an explicit formula derived from the original LQP method. Under certain conditions, the global convergence of the proposed method is proved.

Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints

In this paper, we propose two interior proximal algorithms inspired by the logarithmic-quadratic proximal method. The first method we propose is for general linearly constrained quasiconvex minimization problems. For this method, we prove global convergence when the regularization parameters go to zero. The latter assumption can be dropped when the function is assumed to be pseudoconvex. We also obtain convergence results for quasimonotone variational inequalities, which are more general than monotone ones.

An LQP-based descent method for structured monotone variational inequalities

This paper proposes a descent method to solve a class of structured monotone variational inequalities. The descent directions are constructed from the iterates generated by a prediction-correction method [B.S. He, Y. Xu, X.M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Comput. Optim. Appl. 35 (2006) 19–46], which is based on the logarithmic-quadratic proximal method. In addition, the optimal step-sizes along these descent directions are identified to accelerate the convergence of the new method. Finally, some numerical results for solving traffic equilibrium problems are reported.

A New Hybrid Inexact Logarithmic-Quadratic Proximal Method for Nonlinear Complementarity Problems

In this paper, the authors present and analyze a new hybrid inexact Logarithmic-Quadratic Proximal method for solving nonlinear complementarity problems. Each iteration of the new method consists of a prediction and a correction step. The predictor is produced using an inexact Logarithmic-Quadratic Proximal method, which is then corrected by the Proximal Point Algorithm. The new iterate is obtained by combining predictor and correction point at each iteration. In this paper, the authors prove the convergence of the new method under the mild assumptions that the function involved is continuous and monotone. Comparison to another existing method with numerical experiments on classical NCP instances demonstrates its superiority.

Continuous optimization and combinatorial optimization

Optimization is a classical yet fast-growing field. It has been playing vital roles in many areas of Science, Engineering, Business, and our daily life. In recent years, the vast applications of optimization and the tremendous advancement in computing speed (even parallel computers) have created many new research frontiers and challenges both theoretically and numerically for the researchers and practitioners in the field of optimization. In this special issue, papers in continuous optimization as well as combinatorial optimization cover theoretical investigations, numerical experiments, and practical applications. The paper by Xiaoling Fu and Bingsheng He studies a class of monotone variational inequalities with linear constraints. Strong theoretical achievement and some efficient numerical scheme have been obtained. Numerical experiments, including some traffic equilibrium problems, are presented. In the paper by Min Li and Xiao-Ming Yuan, they focus on some nonlinear complementarity problems; by using the logarithmic-quadratic proximal method, a new efficient computing scheme is proposed. Solid convergence and attractive numerical results are presented. The paper by Hongyu Zhang and Yiju Wang proposes a new type of solution methods for the split feasibility problem. Its global convergence is proved under mild conditions, and preliminary numerical results are reported to show the efficiency of the proposed method. The next three papers focus on the combinatorial optimization. The paper by Qizhi Fang, Rudolf Fleischer, Jian Li, and Xiaoxun Sun is devoted to core stability

A new logarithmic-quadratic proximal method for nonlinear complementarity problems

In this paper, we propose a new modified logarithmic-quadratic proximal (LQP) method for solving nonlinear complementarity problems (NCP). We suggest using a prediction-correction method to solve NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed accuracy criterion and the new iterate is computed by using a new step size α k . Under suitable conditions, we prove that the new method is globally convergent. We report preliminary computational results to illustrate the efficiency of the proposed method. This new method can be considered as a significant refinement of the previously known methods for solving nonlinear complementarity problems.

A new predicto-corrector method for pseudomonotone nonlinear complementarity problems

Based on the Logarithmic-Quadratic Proximal (LQP) method 2, we propose a new prediction-correction method for nonlinear complementarity problem (NCP). We obtain the predictor through a simplified inexact logarithmic-quadratic proximal method under a relaxed inexact criterion. The corrector is obtained by the improved extragradient method. Under certain conditions, the global convergence of the proposed method is proved. Preliminary numerical results indicate the efficiency of the proposed method.

Extended LQP Method for Monotone Nonlinear Complementarity Problems

To solve nonlinear complementarity problems (NCP), the logarithmic-quadratic proximal (LQP) method solves a system of nonlinear equations at each iteration. In this paper, the iterates generated by the original LQP method are extended by explicit formulas and thus an extended LQP method is presented. It is proved theoretically that the lower bound of the progress obtained by the extended LQP method is greater than that by the original LQP method. Preliminary numerical results are provided to verify the theoretical assertions and the effectiveness of both the original and the extended LQP method.

A new criterion for the inexact logarithmic-quadratic proximal method and its derived hybrid methods

To solve nonlinear complementarity problems, the inexact logarithmic-quadratic proximal (LQP) method solves a system of nonlinear equations (LQP system) approximately at each iteration. Therefore, the efficiencies of inexact-type LQP methods depend greatly on the involved inexact criteria used to solve the LQP systems. This paper relaxes inexact criteria of existing inexact-type LQP methods and thus makes it easier to solve the LQP system approximately. Based on the approximate solutions of the LQP systems, a descent method, and a prediction---correction method are presented. Convergence of the new methods are proved under mild assumptions. Numerical experiments for solving traffic equilibrium problems demonstrate that the new methods are more efficient than some existing methods and thus verify that the new inexact criterion is attractive in practice.

The prediction–correction approach to nonlinear complementarity problems

This paper presents a prediction–correction approach to solving the nonlinear complementarity problem (NCP). Each iteration of the new method consists of a prediction and a correction. The predictor is produced by an inexact Logarithmic-Quadratic Proximal method; and then it is corrected by the Proximal Point Algorithm. Convergence of the new method is proved under mild assumptions. Comparison to existing methods shows the superiority of the new method. Numerical experiments including the application to traffic equilibrium problems demonstrate that the new method is attractive in practice.

An self-adaptive LQP method for constrained variational inequalities

In this paper, we present a logarithmic-quadratic proximal (LQP) type prediction–correction methods for solving constrained variational inequalities VI ( S , f ) , where S is a convex set with linear constraints. The computational load in each iteration is quite tiny. However, the number of iterations is significantly dependent on a parameter which balances the primal and dual variables. We then propose a self-adaptive prediction–correction method that adjusts the scalar parameter automatically. Under certain conditions, the global convergence of the proposed method is established. In order to demonstrate the efficiency of the proposed method, we provide numerical results for a convex nonlinear programming and traffic equilibrium problems.

A hybrid inexact Logarithmic–Quadratic Proximal method for nonlinear complementarity problems

Inspired by the Logarithmic–Quadratic Proximal method [A. Auslender, M. Teboulle, S. Ben-Tiba, A logarithmic–quadratic proximal method for variational inequalities, Comput. Optim. Appl. 12 (1999) 31–40], we present a new prediction–correction method for solving the nonlinear complementarity problems. In our method, an intermediate point is produced by approximately solving a nonlinear equation system based on the Logarithmic–Quadratic Proximal method; and the new iterate is obtained by convex combination of the previous point and the one generated by the improved extragradient method at each iteration. The proposed method allows for constant relative errors and this yields a more practical Logarithmic–Quadratic Proximal type method. The global convergence is established under mild conditions. Preliminary numerical results indicate that the method is effective for large-scale nonlinear complementarity problems.

On inexact generalized proximal methods with a weakened error tolerance criterion

Two inexact versions of a Bregman-function-based proximal method for finding a zero of a maximal monotone operator, suggested in [J. Eckstein (1998). Approximate iterations in Bregman-function-based proximal algorithms. Math. Programming, 83, 113–123; P. da Silva, J. Eckstein and C. Humes (2001). Rescaling and stepsize selection in proximal methods using separable generalized distances. SIAM J. Optim., 12, 238–261], are considered. For a wide class of Bregman functions, including the standard entropy kernel and all strongly convex Bregman functions, convergence of these methods is proved under an essentially weaker accuracy condition on the iterates than in the original papers. 1 Concerning , they coincide with A1–A3 in Section 2. Also the error criterion of a logarithmic–quadratic proximal method, developed in [A. Auslender, M. Teboulle and S. Ben-Tiba (1999). A logarithmic-quadratic proximal method for variational inequalities. Computational Optimization and Applications, 12, 31–40], is relaxed, and convergence results for the inexact version of the proximal method with entropy-like distance functions are described. For the methods mentioned, like in [R.T. Rockafellar (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim., 14, 877–898] for the classical proximal point algorithm, only summability of the sequence of error vector norms is required.