Over the last two decades, it has been observed that using the gradient vector as a search direction in large-scale optimization may lead to efficient algorithms. The effectiveness relies on choosing the step lengths according to novel ideas that are related to the spectrum of the underlying local Hessian rather than related to the standard decrease in the objective function. A review of these so-called spectral projected gradient methods for convex constrained optimization is presented. To illustrate the performance of these low-cost schemes, an optimization problem on the set of positive definite matrices is described.

https://www.jstatsoft.org/article/view/v060i03/v60i03.pdf

Spectral Projected Gradient Methods: Review and Perspectives

The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every iteration of the method requires only one evaluation of a monotone operator F and a proximal mapping g. The operator F need not be Lipschitz continuous, which also makes the algorithm interesting in the area of composite minimization. The method exhibits an ergodic O(1 / k) convergence rate and R-linear rate under an error bound condition. We discuss possible applications of the method to fixed point problems as well as its different generalizations.

Golden ratio algorithms for variational inequalities

In this work, we introduce self-adaptive methods for solving variational inequalities with Lipschitz continuous and quasimonotone mapping(or Lipschitz continuous mapping without monotonicity) in real Hilbert space. Under suitable assumptions, the convergence of algorithms are established without the knowledge of the Lipschitz constant of the mapping. The results obtained in this paper extend some recent results in the literature. Some preliminary numerical experiments and comparisons are reported.

Weak convergence of iterative methods for solving quasimonotone variational inequalities

The paper presents a fully explicit algorithm for monotone variational inequalities. The method uses variable stepsizes that are computed using two previous iterates as an approximation of the local Lipschitz constant without running a linesearch. Thus, each iteration of the method requires only one evaluation of a monotone operator $F$ and a proximal mapping $g$. The operator $F$ need not be Lipschitz-continuous, which also makes the algorithm interesting in the area of composite minimization where one cannot use the descent lemma. The method exhibits an ergodic $O(1/k)$ convergence rate and $R$-linear rate, if $F, g$ satisfy the error bound condition. We discuss possible applications of the method to fixed point problems. We discuss possible applications of the method to fixed point problems as well as its different generalizations.

Golden Ratio Algorithms for Variational Inequalities

Nonnegative matrix factorization and its graph regularized extensions have received significant attention in machine learning and data mining. However, existing approaches are sensitive to outliers and noise due to the utilization of the squared loss function in measuring the quality of graph regularization and data reconstruction. In this paper, we present a novel robust graph regularized NMF model (RGNMF) to approximate the data matrix for clustering. Our assumption is that there may exist some entries of the data corrupted arbitrarily, but the corruption is sparse. To address this problem, an error matrix is introduced to capture the sparse corruption. With this sparse outlier matrix, a robust factorization result could be obtained since a much cleaned data could be reconstructed. Moreover, the $$\ell _{1}$$
 -norm function is used to alleviate the influence of unreliable regularization which is incurred by unexpected graphs. That is, the sparse error matrix alleviates the impact of noise and outliers, and the $$\ell _{1}$$
 -norm function leads to a faithful regularization since the influence of the unreliable regularization errors can be reduced. Thus, RGNMF is robust to unreliable graphs and noisy data. In order to solve the optimization problem of our method, an iterative updating algorithm is proposed and its convergence is also guaranteed theoretically. Experimental results show that the proposed method consistently outperforms many state-of-the-art methods.

Robust graph regularized nonnegative matrix factorization for clustering

In this paper, we introduce two new algorithms for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient...

Modified subgradient extragradient algorithms for solving monotone variational inequalities

In this paper, we investigate and analyze classical variational inequalities with Lipschitz continuous and monotone mapping in real Hilbert space. The projected reflected gradient method, with varying step size, requires at most two projections onto the feasible set and one value of the mapping per iteration. We modify the method with a simple structure; a weak convergence theorem for our algorithm is proved without any requirement of additional projections and the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Preliminary results from numerical experiments are performed.

A Modified Projected Gradient Method for Monotone Variational Inequalities

We present a new infeasible-interior-point method, based on a wide neighborhood, for symmetric cone programming. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. Moreover, we derive the complexity bound of the wide neighborhood infeasible interior-point methods that coincides with the currently best known theoretical complexity bounds for the short step path-following algorithm.

A New Wide Neighborhood Primal–Dual Infeasible-Interior-Point Method for Symmetric Cone Programming

In this paper, we introduce a new concept of approximate optimal stepsize for gradient method, use it to interpret the Barzilai-Borwein (BB) method, and present an efficient gradient method with approximate optimal stepsize for large unconstrained optimization. If the objective function f is not close to a quadratic on a line segment between the current iterate x
 k
 and the latest iterate x
 k−1, we construct a conic model to generate the approximate optimal stepsize for gradient method if the conic model is suitable to be used. Otherwise, we construct a new quadratic model or two other new approximation models to generate the approximate optimal stepsize for gradient method. We analyze the convergence of the proposed method under some suitable conditions. Numerical results show the proposed method is very promising.

An efficient gradient method with approximate optimal stepsize for large-scale unconstrained optimization

In this paper, a self-adjust conjugate gradient method is proposed for solving unconstrained problems, which can generate sufficient descent directions at each iteration. Different from the existent methods, a dynamical adjustment of conjugacy condition in our proposed method is developed, which can be regarded as the inheritance and development of properties of standard Hestenes---Stiefel method. Under mild condition, we show the proposed method convergent globally even if the objective function is nonconvex. Numerical results illustrate that our method can efficiently solve the test problems and therefore is promising.

A Self-Adjusting Conjugate Gradient Method with Sufficient Descent Condition and Conjugacy Condition

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