Margarida P. Mello

Validation of an Augmented Lagrangian Algorithm with a Gauss-Newton Hessian Approximation Using a Set of Hard-Spheres Problems

An Augmented Lagrangian algorithm that uses Gauss-Newton approximations of the Hessian at each inner iteration is introduced and tested using a family of Hard-Spheres problems. The Gauss-Newton model convexifies the quadratic approximations of the Augmented Lagrangian function thus increasing the efficiency of the iterative quadratic solver. The resulting method is considerably more efficient than the corresponding algorithm that uses true Hessians. A comparative study using the well-known package LANCELOT is presented.

Box-constrained minimization reformulations of complementarity problems in second-order cones

Reformulations of a generalization of a second-order cone complementarity problem (GSOCCP) as optimization problems are introduced, which preserve differentiability. Equivalence results are proved in the sense that the global minimizers of the reformulations with zero objective value are solutions to the GSOCCP and vice versa. Since the optimization problems involved include only simple constraints, a whole range of minimization algorithms may be used to solve the equivalent problems. Taking into account that optimization algorithms usually seek stationary points, a theoretical result is established that ensures equivalence between stationary points of the reformulation and solutions to the GSOCCP. Numerical experiments are presented that illustrate the advantages and disadvantages of the reformulations.

Realizability of the Morse polytope

In this article we will show that, in general, for each integral point (γ0,...,γn) in the Morse polytope,Pκ(ho, ...,hn), one can associate an abstract Lyapunov graphL(ho, ...,hn,κ) withntd-labelling and realize a corresponding flow onMn, where the Betti numbers ofMn satisfy βj(Mn)=βn−j(Mn)=γj for all 0

Computing the sparsity pattern of Hessians using automatic differentiation

We compare two methods that calculate the sparsity pattern of Hessian matrices using the computational framework of automatic differentiation. The first method is a forward-mode algorithm by Andrea Walther in 2008 which has been implemented as the driver called hess_pat in the automatic differentiation package ADOL-C. The second is edge_push_sp, a new reverse mode algorithm descended from the edge_pushing algorithm for calculating Hessians by Gower and Mello in 2012. We present complexity analysis and perform numerical tests for both algorithms. The results show that the new reverse algorithm is very promising.

Poincaré-Hopf inequalities

In this article the main theorem establishes the necessity and sufficiency of the Poincare-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.

Dynamical and Topological Aspects of Lyapunov Graphs

In this survey we present the interplay between topological dynamical systems theory with network flow theory in order to obtain a continuation result for abstract Lyapunov graphsL(h0, …, hn, k) in dimensionn with cycle numberk. We also show that an abstract Lyapunov graph satisfies the Poincaré-Hopf inequalities if and only if it satisfies the Morse inequalities and the first Betti number γ1 is greater than or equal tok. We define the Morse polytope determined by the Morse inequalities and describe some of its geometrical properties.

Mixed Nonlinear Complementarity Problems via Nonlinear Optimization: Numerical Results on Multi-Rigid-Body Contact Problems with Friction

Abstract In this work we show that the mixed nonlinear complementarity problem may be formulated as an equivalent nonlinear bound-constrained optimization problem that preserves the smoothness of the original data. One may thus take advantage of existing codes for bound-constrained optimization. This approach is implemented and tested by means of an extensive set of numerical experiments, showing promising results. The mixed nonlinear complementarity problems considered in the tests arise from the discretization of a motion planning problem concerning a set of rigid 3D bodies in contact in the presence of friction. We solve the complementarity problem associated with a single time frame, thus calculating the contact forces and accelerations of the bodies involved.

Beyond domes, umbrellas and tents

Do the following objects shown on the cover belong together? We will argue that they do bear a certain kinship, sharing the common gene of cylindrical intersection. In fact, we hope this essay will awaken in the reader the ability to discern several other members of this family in the world around him or her. Cylinder intersection is used widely in construction. Beautiful examples abound in old world architecture, for instance the famous Florence cathedral in Figure 1.