Let x(t) be a trajectory of the gradient of a real analytic function and suppose that x0 is a limit point of x(t). We prove the gradient conjecture of R. Thom which states that the secants of x(t) at x0 have a limit. Actually we show a stronger statement: the radial projection of x(t) from x0 onto the unit sphere has finite length.

The contribution presents defined problems of a control of nonlinear systems, processes with constraints and limitations. Optimality conditions for a control and iterative gradient methods of nonlinear systems optimization have been given. The defined problem with the Lagrangian criterion function is converted to an optimalization problem of the theory of games. In order to solve such a problem...

Human behavior recognition has become a hot research topic in computer vision. In this paper, we propose an effective method to recognize human behaviors from sequences of depth maps, which provide additional body shape and motion information for behavior recognition. In our approach, we construct a novel difference motion history image, and propose axonometric projection to capture the target ...

Anovel characteristic expandedmixed finite elementmethod is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term ∇ ⋅ (a(x, t)∇u) is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical H(div; Ω) space and the hyperbolic part d(x)(∂u/∂t) + c(x, t) ⋅ ∇u is handled by the characteristic method...

For lack of directivity in Total Variation (TV) which only uses x-coordinate and y-coordinate gradient transform as its sparse representation approach during the iteration process, this paper brought in Adaptive-weighted Diagonal Total Variation (AwDTV) that uses the diagonal direction gradient to constraint reconstructed image and adds associated weights which are expressed as an exponential f...

In this paper, we present a new conditional gradient type method for convex optimization by utilizing a linear optimization (LO) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time to time, and as a result, c...

OBJECTIVE: Spiral MRI image quality is degraded by timing misalignments between gradient transmission and data sampling. Even well compensated and calibrated MRI scanners can experience residual, sequence specific gradient delays. Several methods exist for measuring and correcting for deviations in the k-space trajectory. This work proposes a simple, fast method for measuring gradient delays at...

We propose two affine projection algorithms (APA) with variable regularization parameter. The proposed algorithms dynamically update the regularization parameter that is fixed in the conventional regularized APA (R-APA) using a gradient descent based approach. By introducing the normalized gradient, the proposed algorithms give birth to an efficient and a robust update scheme for the regulariza...

In this paper, I study the problem of phase retrieval with total variation (TV) smoothing and denoising. I construct an optimization model with an anisotropic TV penalty term, then show how to solve it using a proximal gradient method. This iterative-shrinkage-thresholding phase retrieval algorithm (ISTPRA) combines fixed-point iterations with a fast gradient solver for the convex subproblem. I...

Image reconstruction gives rise to some challenging large-scale constrained optimization problems. We consider a convex minimization problem with nonnegativity constraints that arises in astronomical imaging. To solve this problem, we use an efficient hybrid gradient projection-reduced Newton (active-set) method. By "reduced Newton," we mean that we take Newton steps only in the inactive variab...