Given a set S of n distinct points { (xi'Yi) 1 0 5 i < n] 9 the convex hull problem is to determine the vertices of the convex hull H(S) . All the known algorithms for solving this problem have a worst-case running time of cn log n or higher, and employ only quadratic tests, i.e., tests of ‘the form f(Xo’YO ⌧19 Yl> l l l ☺ ⌧nBl’ Y, 1> : 0 with f being any polynomial of degree not exceeding 2 . ...

The method Promethee II has produced attractive results in the choice of the most satisfactory optimal solution of convex multiobjective problems. However, according to the current literature, it may not work properly with nonconvex problems. A modified version of this method, called multiplicative Promethee, is proposed in this paper. Both versions are applied to some analytical problems, prev...

On a fixed finite set {1, . . . , n}, we consider the set of metrics for which the metric space can be isometrically embedded in the real line. To understand the geometry of this set, we study its convex hull, Qn, and the closure of its convex hull, Qn. We first show how the set of metrics is contained in its convex hull and characterize all unbounded one-dimensional extreme subsets of Qn combi...

Based on concurrent algorithms and on the convex combination of one slow and one fast CMA (Constant Modulus Algorithm), we propose a convex combination of two blind equalizers adapted respectively by CMA and the modified SDD (Soft Decision-Directed) algorithm for recovering of QAM (Quadrature Amplitude Modulation) signals. For high signal-to-noise ratio, the performance of the proposed scheme i...

In this paper, the authors solve the two stage stochastic programming with separable objective by obtaining convex polynomial approximations to the convex objective function with an arbitrary accuracy. Our proposed method will be valid for realistic applications, for example, the convex objective can be either non-differentiable or only accessible by Monte Carlo simulations. The resulting polyn...

In this work we develop a new algorithm for regularized empirical risk minimization. Our method extends recent techniques of Shalev-Shwartz [02/2015], which enable a dual-free analysis of SDCA, to arbitrary mini-batching schemes. Moreover, our method is able to better utilize the information in the data defining the ERM problem. For convex loss functions, our complexity results match those of Q...

We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte’s celebrated “springembedding” theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result gen...

We show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined in 2010). We extend this result to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables if the coefficients of the integer variables form a matrix of rank 2. We also present an alterna...

We consider the Retail Planning Problem in which the retailer chooses suppliers, and determines the production, distribution and inventory planning for products with uncertain demand in order to minimize total expected costs. This problem is often faced by large retail chains that carry private label products. We formulate this problem as a convex mixed integer program and show that it is stron...

This report shows that the performance of deep convolutional neural network can be improved by incorporating convex optimization techniques. First, we find that the sub-models learned by dropout can be more effectively combined by solving a convex problem. Also, we generalize this idea to models that are not trained by dropout. Compared to traditional methods, we get an improvement of 0.22% and...