Second-and higher-order derivatives are required by applications in scientiic computation, especially for optimization algorithms. The two complementary concepts of interpolating partial derivatives from univariate Taylor series and preaccumulating of \local" derivatives form the mathematical foundations for accurate, eecient computation of second-and higher-order partial derivatives for large ...

This paper studies recent modi cations of the limited memory BFGS (L-BFGS) method for solving large scale unconstrained optimization problems. Each modi cation technique attempts to improve the quality of the L-BFGS Hessian by employing (extra) updates in certain sense. Because at some iterations these updates might be redundant or worsen the quality of this Hessian, this paper proposes an upda...

The book [7] emphasizes three relevant aspects: first, the fact that the notion of convexity is strongly metric-dependent either through geodesics or through the Riemannian connection; second that Riemannian convexity of functions is a coordinate-free concept, and consequently it can be easily connected with symbolic computation; third, that the Riemannian structure is involved essentially in f...

In this paper, we give necessary and suucient optimality conditions which are easy veriied for the local solution of Celis-Dennis-Tapia sub-problem (CDT subproblem) where the Hessian at this local solution has one negative eigenvalue. If CDT subproblem has no global solution with Hessian of Lagrangian positive semi-deenite, the Hessian of Lagrangian has at least one negative eigenvalue. It is v...

One of the most commonly used means to characterize potential energy surfaces of reactions and chemical systems is the Hessian calculation, whose analytic evaluation is computationally and memory demanding. A new scalable distributed data analytic Hessian algorithm is presented. Features of the distributed data parallel coupled perturbed Hartree-Fock (CPHF) are (a) columns of density-like and F...

This paper studies recent modifications of the limited memory BFGS (L-BFGS) method for solving large scale unconstrained optimization problems. Each modification technique attempts to improve the quality of the L-BFGS Hessian by employing (extra) updates in a certain sense. Because at some iterations these updates might be redundant or worsen the quality of this Hessian, this paper proposes an ...

In this paper we will determine the multiple point manifolds of certain self-transverse immersions in Euclidean spaces. Following the triple points, these immersions have a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to Dold manifold $V^5$ or a boundary. We will show there is an immersion of $S^7times P^2$ in $mathbb{R}^{1...

Let φ be a polynomial over K (a field of characteristic 0) such that the Hessian of φ is a nonzero constant. Let φ̄ be the formal Legendre Transform of φ. Then φ̄ is well-defined as a formal power series over K. The Hessian Conjecture introduced here claims that φ̄ is actually a polynomial. This conjecture is shown to be true when K = R and the Hessian matrix of φ is either positive or negative de...

Stable techniques are considered for updating the reduced Hessian matrix that arises in a null{space active set method for Quadratic Programming when the Hessian matrix itself may be indeenite. A scheme for deening and updating the null-space basis matrix is described which is adequately stable and allows advantage to be taken of sparsity. A new canonical form for the reduced Hessian matrix is ...

Potential flow pressure matching is a classical inverse design aerodynamic problem. The resulting loss of regularity during the optimization poses challenges for shape optimization with normal perturbation of the surface mesh nodes. Smoothness is not enforced by the parameterization but by a proper choice of the scalar product based on the shape Hessian, which is derived in local coordinates fo...