In this paper, a new nonmonotone line search rule is proposed,which is verified to be an improved version of the nonmonotone line search technique proposed by Zhang and Hager. Unlike the Zhang and Hager's method, our nonmonotone line search is proved to own a nice property similar to the standard Armijo line search. In virtue of such a property, global convergence is established for the developed algorithm, where the search direction is supposed to satisfy some mild conditions and the stepsize is chosen by the new line search rule. R-linear convergence of the developed algorithm is proved for strongly convex objective functions. The developed algorithm is used to solve the test problems available in the CUTEr, the numerical results demonstrate that the new line search strategy outperforms the other similar ones.

A new nonmonotone line search technique for unconstrained optimization

A nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities

Randomized Kaczmarz methods for tensor complementarity problems

An extended nonmonotone line search technique for large-scale unconstrained optimization

Merit functions for semi-definite complementarity problems

The monotone line search schemes have been extensively used in the iterative methods for solving various optimization problems. It is well known that the non-monotone line search technique can improve the likelihood of finding a global optimal solution and the numerical performance of the methods, especially for some difficult nonlinear problems. The traditional non-monotone line search approach requires that a maximum of recent function values decreases. In this paper, we propose a new line search scheme which requires that a convex combination of recent function values decreases. We apply the new line search technique to solve unconstrained optimization problems, and show the proposed algorithm possesses global convergence and R-linear convergence under suitable assumptions. We also report the numerical results of the proposed algorithm for solving almost all the unconstrained testing problems given in CUTEr, and give numerical comparisons of the proposed algorithm with two famous non-monotone methods.

A Non-monotone Line Search Algorithm for Unconstrained Optimization

For any function φ from ℜr to ℜr, Tao and Gowda [Math. Oper. Res., 30 (2005), pp. 985–1004] introduced a corresponding nonlinear transformation Rφ over a Euclidean Jordan algebra (which is called a relaxation transformation) and established some useful relations between φ and Rφ. In this paper, we further investigate some interconnections between properties of φ and properties of Rφ, including the properties of continuity, (local) Lipschitz continuity, directional differentiability, (continuous) differentiability, semismoothness, monotonicity, the P0-property, and the uniform P-property. As an application, we investigate the symmetric cone complementarity problem with a relaxation transformation. A property of the solution set of this class of problems is given. We also investigate a smoothing algorithm for solving this class of problems and show that the algorithm is globally convergent under an assumption that the solution set of the problem concerned is nonempty.

Properties of a Class of Nonlinear Transformations Over Euclidean Jordan Algebras with Applications to Complementarity Problems

In this article, we consider the complementarity problem over a closed convex cone in a Hilbert space. We propose three classes of merit functions for solving such a problem, including natural residual merit functions, LT merit functions and implicit Lagrangian merit functions, and investigate their important properties related to the unconstrained reformation of the problem, including coerciveness properties of functions and the related error bound results. In addition, we also propose a necessary and sufficient condition of a stationary point of the unconstrained reformation being a solution to the original problem for implicit Lagrangian merit functions. These provide a theoretical basis for designing the merit function method for solving the problem concerned.

Three classes of merit functions for the complementarity problem over a closed convex cone

Solvability of Newton equations in smoothing-type algorithms for the SOCCP

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Papers

A Non-monotone Line Search Algorithm for Unconstrained Optimization

Properties of a Class of Nonlinear Transformations Over Euclidean Jordan Algebras with Applications to Complementarity Problems

Three classes of merit functions for the complementarity problem over a closed convex cone

Solvability of Newton equations in smoothing-type algorithms for the SOCCP