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To solve a large-scale unconstrained optimization problem, in this paper we propose a class of spectral three-term conjugate gradient methods. We indicate that the proposed class, in fact, generate...

This paper begins with a brief review of affine invariance and its significance for iterative algorithms. It then explores the existence of affine invariant descent directions for unconstrained min...

In this paper, based on the numerical efficiency of Hestenes–Stiefel (HS) method, a new modified HS algorithm is proposed for unconstrained optimization. The new direction independent of the line search satisfies in the sufficient descent condition. Motivated by theoretical and numerical features of three-term conjugate gradient (CG) methods proposed by Narushima et al., similar to Dai and Kou approach, the new direction is computed by minimizing the distance between the CG direction and the direction of the three-term CG methods proposed by Narushima et al. Under some mild conditions, we establish global convergence of the new method for general functions when the standard Wolfe line search is used. Numerical experiments on some test problems from the CUTEst collection are given to show the efficiency of the proposed method.

In this paper, we propose a non-monotone line search method for solving optimization problems on Stiefel manifold. The main novelty of our approach is that our method uses a search direction based on a linear combination of descent directions and a Barzilai–Borwein line search. The feasibility is guaranteed by projecting each iterate on the Stiefel manifold through SVD (singular value decomposition) factorizations. Some theoretical results for analysing the algorithm are presented. Finally, we provide numerical experiments for comparing our algorithm with other state-of-the-art procedures. The code is available online. The experimental results show that the proposed algorithm is competitive with other approaches and for particular problems, the computational performance is better than the state-of-the-art algorithms.

We suggest a modification of the descent splitting methods for decomposable composite optimization problems, which maintains the basic convergence properties, but enables one to reduce the computational expenses per iteration and to provide computations in a distributed manner. It consists of making coordinate-wise steps together with a special threshold control.

We consider the monotone composite variational inequality CVI where the underlying mapping is formed as the sum of two monotone mappings. We combine the forward–backward and descent direction ideas together, and thus present the unified algorithmic framework of forward–backward-based descent methods for solving the CVI. A new iterate of such a method is generated by a prediction–correction fashion, where the predictor is yielded by the forward–backward method and then the predictor is corrected by a descent step. We derive some implementable forward–backward-based descent algorithms for some concrete cases of the CVI, and verify their numerical efficiency via preliminary numerical experiments.

In [Zhou, A short note on the global convergence of the unmodified PRP method, Optim. Lett. doi: 10.1007/s11590-012-0511-7, to appear], Zhou showed that the classical unmodified Polak–Ribiere–Polyak PRP nonlinear conjugate gradient method converges globally in the sense that lim infk→∞‖∇fxk‖=0 for nonconvex minimization by the use of some nonmonotone line search. In this paper, we present a new non-descent backtracking type line search and show that the PRP method has strongly global convergence property i.e. lim k→∞‖∇ fxk‖=0 and locally R-linear convergence rate for nonconvex optimization with the proposed line search by suitably choosing the initial stepsize. Some numerical results compared with existing descent type line searches are reported.

In this paper, a modified Polak–Ribi‘ere–Polyak MPRP conjugate gradient method for smooth unconstrained optimization is proposed. This method produces at each iteration a descent direction, and this property is independent of the line search adopted. Under standard assumptions, we prove that the MPRP method using strong Wolfe conditions is globally convergent. The results of computational experiments are reported and show the effectiveness of the proposed method.

We present an algorithm to locally minimize nonsmooth, nonconvex functions. In order to find descent directions, the notion of quasisecants, introduced in this paper, is applied. We prove that the algorithm converges to Clarke stationary points. Numerical results are presented demonstrating the applicability of the proposed algorithm to a wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.

A globally convergent algorithm for equilibrium problems with differentiable bifunctions is proposed. The algorithm is based on descent directions of a suitable family of gap functions. The novelty of the approach is that assumptions which guarantee that the stationary points of the gap functions are global optima are not required.

This article presents a descent method for solving monotone variational inequalities with separate structures. The descent direction is derived from the well-known alternating directions method. The optimal step size along the descent direction also improves the efficiency of the new method. Some numerical results demonstrate that the new method is effective in practice.

In this paper, we consider the optimization method for monotone variational inequality probleln on polyhedral sets. First, we consider the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed and s desirable properties of the merit function are obtained. Under certain assumptions, we show tthat any stationary point of the merit function is a solution of the original problem. A descent metlpod for the optimization problem is proposed and the global convergence of the method is shown

In this paper, we consider structured quasi-Newton methods for finding a local solution to nonlinear least squares problems. This paper is concerned with the line search globalization method. Recently, factorized versions of the structured quasi-Newton methods have been studied by Sheng and Zou, and Yabe and Takahashi in order to obtain a descent search direction for the objective function. In this paper, we first generalize the update of Sheng and Zou, and propose a new factorized family corresponding to the Broyden family (SZ-Broyden family). Second, we suggest a relationship between the structured quasi-Newton updates and the factorized versions. We use this relationship to show that the factorized Broyden family proposed by Yabe and Yamaki corresponds to the Engels and Martinez family, and we further obtain a new structured quasi-Newton update which corresponds to the SZ-Broyden family in the sense of this relation. Finally, we apply sizing techniques to these methods and present some numerical experi...

We consider the application of the ABS algorithms, introduced by [I], mainly for the solution of linear algebraic systems, to constrained optimization problems with equality constraints. A general formulation of feasible direction algorithms is obtained and the ABS formulation is given of the reduced gradient method of Wolfe (1967) and of the projected gradient method of Rosen (1960)