Analysis of a quadratic programming decomposition algorithm

We analyze a decomposition algorithm for minimizing a quadratic objective function, separable in x1 and x2, subject to the constraint that x1 and x2 are orthogonal vectors on the unit sphere. Our algorithm consists of a local step where we minimize the objective function in either variable separately, while enforcing the constraints, followed by a global step where we minimize over a subspace generated by solutions to the local subproblems. We establish a local convergence result when the global minimizers nondegenerate. Our analysis employs necessary and sufficient conditions and continuity properties for a global optimum of a quadratic objective function subject to a sphere constraint and a linear constraint. The analysis is connected with a new domain decomposition algorithm for electronic structure calculations

Analysis of a quadratic programming decomposition algorithm

We analyze a decomposition algorithm for minimizing a quadratic objective function, separable in x1 and x2, subject to the constraint that x1 and x2 are orthogonal vectors on the unit sphere. Our algorithm consists of a local step where we minimize the objective function in either variable separately, while enforcing the constraints, followed by a global step where we minimize over a subspace generated by solutions to the local subproblems. We establish a local convergence result when the global minimizers nondegenerate. Our analysis employs necessary and sufficient conditions and continuity properties for a global optimum of a quadratic objective function subject to a sphere constraint and a linear constraint. The analysis is connected with a new domain decomposition algorithm for electronic structure calculations

On a new conformal functional for simplicial surfaces

We introduce a smooth quadratic conformal functional and its weighted version $$W_2=\sum_e \beta^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)\beta^2(e),$$ where $\beta(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of the mesh sharing the edge $e=(ij)$ and $n_i$ is the valence of vertex $i$. Besides minimizing the squared local conformal discrete Willmore energy $W$ this functional also minimizes local differences of the angles $\beta$. We investigate the minimizers of this functionals for simplicial spheres and simplicial surfaces of nontrivial topology. Several remarkable facts are observed. In particular for most of randomly generated simplicial polyhedra the minimizers of $W_2$ and $W_{2,w}$ are inscribed polyhedra. We demonstrate also some applications in geometry processing, for example, a conformal deformation of surfaces to the round sphere. A partial theoretical explanation through quadratic optimization theory of some observed phenomena is presented.Comment: 14 pages, 8 figures, to appear in the proceedings of "Curves and Surfaces, 8th International Conference", June 201

Generalized conditional entropy optimization for qudit-qubit states

We derive a general approximate solution to the problem of minimizing the conditional entropy of a qudit-qubit system resulting from a local projective measurement on the qubit, which is valid for general entropic forms and becomes exact in the limit of weak correlations. This entropy measures the average conditional mixedness of the post-measurement state of the qudit, and its minimum among all local measurements represents a generalized entanglement of formation. In the case of the von Neumann entropy, it is directly related to the quantum discord. It is shown that at the lowest non-trivial order, the problem reduces to the minimization of a quadratic form determined by the correlation tensor of the system, the Bloch vector of the qubit and the local concavity of the entropy, requiring just the diagonalization of a $3\times 3$ matrix. A simple geometrical picture in terms of an associated correlation ellipsoid is also derived, which illustrates the link between entropy optimization and correlation access and which is exact for a quadratic entropy. The approach enables a simple estimation of the quantum discord. Illustrative results for two-qubit states are discussed.Comment: 11 pages, 6 figures. Final published versio

The Complexity of Optimizing over a Simplex, Hypercube or Sphere: A Short Survey

We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere.These relatively simple optimization problems have many applications.We review known approximation results as well as negative (inapproximability) results from the recent literature.computational complexity;global optimization;linear and semidefinite programming;approximation algorithms

A Feature-Based Analysis on the Impact of Set of Constraints for e-Constrained Differential Evolution

Different types of evolutionary algorithms have been developed for constrained continuous optimization. We carry out a feature-based analysis of evolved constrained continuous optimization instances to understand the characteristics of constraints that make problems hard for evolutionary algorithm. In our study, we examine how various sets of constraints can influence the behaviour of e-Constrained Differential Evolution. Investigating the evolved instances, we obtain knowledge of what type of constraints and their features make a problem difficult for the examined algorithm.Comment: 17 Page

Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions

In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability.Comment: 24 pages, 17 figure