A method with inertial extrapolation step for convex constrained monotone equations

In recent times, various algorithms have been incorporated with the inertial extrapolation step to speed up the convergence of the sequence generated by these algorithms. As far as we know, very few results exist regarding algorithms of the inertial derivative-free projection method for solving convex constrained monotone nonlinear equations. In this article, the convergence analysis of a derivative-free iterative algorithm (Liu and Feng in Numer. Algorithms 82(1):245–262, 2019) with an inertial extrapolation step for solving large scale convex constrained monotone nonlinear equations is studied. The proposed method generates a sufficient descent direction at each iteration. Under some mild assumptions, the global convergence of the sequence generated by the proposed method is established. Furthermore, some experimental results are presented to support the theoretical analysis of the proposed method.

Distance-guided protein folding based on generalized descent direction

Advances in the prediction of the inter-residue distance for a protein sequence have increased the accuracy to predict the correct folds of proteins with distance information. Here, we propose a distance-guided protein folding algorithm based on generalized descent direction, named GDDfold, which achieves effective structural perturbation and potential minimization in two stages. In the global stage, random-based direction is designed using evolutionary knowledge, which guides conformation population to cross potential barriers and explore conformational space rapidly in a large range. In the local stage, locally rugged potential landscape can be explored with the aid of conjugate-based direction integrated into a specific search strategy, which can improve exploitation ability. GDDfold is tested on 347 proteins of a benchmark set, 24 FM targets of CASP13 and 20 FM targets of CASP14. Results show that GDDfold correctly folds (TM-score ≥ 0.5) 316 out of 347 proteins, where 65 proteins have TM-scores that are greater than 0.8, and significantly outperforms Rosetta-dist (distance-assisted fragment assembly method) and L-BFGSfold (distance geometry optimization method). On CASP FM targets, GDDfold is comparable with five state-of-the-art full-version methods, namely, Quark, RaptorX, Rosetta, MULTICOM and trRosetta in the CASP 13 and 14 server groups.

Superlinear Convergence of a Modified Newton's Method for Convex Optimization Problems With Constraints

We consider the constrained optimization problem  defined by: $$f (x^*) = \min_{x \in  X} f(x)\eqno (1)$$ where the function  f : \pmb{\mathbb{R}}^{n} → \pmb{\mathbb{R}} is convex  on a closed bounded convex set X. To solve problem (1), most methods transform this problem into a problem without constraints, either by introducing Lagrange multipliers or a projection method. The purpose of this paper is to give a new method to solve some constrained optimization problems, based on the definition of a descent direction and a step while remaining in the X convex domain. A convergence theorem is proven. The paper ends with some numerical examples.

Quantum Natural Gradient

A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of the Quantum Geometric Tensor (QGT), also known as the Fubini-Study metric tensor. An efficient algorithm is presented for computing a block-diagonal approximation to the Fubini-Study metric tensor for parametrized quantum circuits, which may be of independent interest.

Use of prismatic waves in full-waveform inversion with the exact Hessian

The truncated Newton method uses information contained in the exact Hessian in full-waveform inversion (FWI). The exact Hessian physically contains information regarding doubly scattered waves, especially prismatic events. These waves are mainly caused by the scattering at steeply dipping structures, such as salt flanks and vertical or nearly vertical faults. We have systematically investigated the properties and applications of the exact Hessian. We begin by giving the formulas for computing each term in the exact Hessian and numerically analyzing their characteristics. We show that the second term in the exact Hessian may be comparable in magnitude to the first term. In particular, when there are apparent doubly scattered waves in the observed data, the influence of the second term may be dominant in the exact Hessian and the second term cannot be neglected. Next, we adopt a migration/demigration approach to compute the Gauss-Newton-descent direction and the Newton-descent direction using the approximate Hessian and the exact Hessian, respectively. In addition, we determine from the forward and the inverse perspectives that the second term in the exact Hessian not only contributes to the use of doubly scattered waves, but it also compensates for the use of single-scattering waves in FWI. Finally, we use three numerical examples to prove that by considering the second term in the exact Hessian, the role of prismatic waves in the observed data can be effectively revealed and steeply dipping structures can be reconstructed with higher accuracy.

A descent derivative-free algorithm for nonlinear monotone equations with convex constraints

In this paper, we present a derivative-free algorithm for nonlinear monotone equations with convex constraints. The search direction is a product of a positive parameter and the negation of a residual vector. At each iteration step, the algorithm generates a descent direction independent from the line search used. Under appropriate assumptions, the global convergence of the algorithm is given. Numerical experiments show the algorithm has advantages over the recently proposed algorithms by Gao and He (Calcolo 55 (2018) 53) and Liu and Li (Comput. Math. App. 70 (2015) 2442–2453).