Symmetric separable convex resource allocation problems with structured disjoint interval bound constraints

Motivated by the problem of scheduling electric vehicle (EV) charging with a minimum charging threshold in smart distribution grids, we introduce the resource allocation problem (RAP) with a symmetric separable convex objective function and disjoint interval bound constraints. In this RAP, the aim is to allocate an amount of resource over a set of $n$ activities, where each individual allocation is restricted to a disjoint collection of $m$ intervals. This is a generalization of classical RAPs studied in the literature where in contrast each allocation is only restricted by simple lower and upper bounds, i.e., $m=1$. We propose an exact algorithm that, for four special cases of the problem, returns an optimal solution in $O \left(\binom{n+m-2}{m-2} (n \log n + nF) \right)$ time, where the term $nF$ represents the number of flops required for one evaluation of the separable objective function. In particular, the algorithm runs in polynomial time when the number of intervals $m$ is fixed. Moreover, we show how this algorithm can be adapted to also output an optimal solution to the problem with integer variables without increasing its time complexity. Computational experiments demonstrate the practical efficiency of the algorithm for small values of $m$ and in particular for solving EV charging problems.Comment: 20 pages, 4 figure

Reliable landmarks as anchors for 3D face reconstruction

Abstract—A way to create a 3D reconstruction of a face is using a 5 camera setup. In this paper a method is presented for finding reliable landmarks in the five 2D images of the face that serve as anchor points to improve the 3D reconstruction. The method consist of four parts that all aid in finding more correct points. The four parts are: 1. Finding candidate points using the SURF algorithm, 2. Matching the points based on the global location of the point, 3. Rejecting poor-matched points by detecting outliers and by using the a multiscale Local Binary Pattern (LBP) algorithm, and 4. Combining all points in all five images. The best results are found for a number of candidate SURF point of 10, and a LBP threshold of D2 and D3: sufficient points are found and the points are well-distributed over the face. An increase in threshold results in both more correct and wrong points. The performance of the method depends also on the subject (facial hair, fair faces), the use of glasses and whether the subject is right in front of the camera or slightly skew. Further improvements on the performance can be achieved by improving parts of the method and optimizing other parameters