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Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…
It is well known that the most challenging question in optimization and discrete geometry is whether there is a strongly polynomial time simplex algorithm for linear programs (LPs). This paper gives a positive answer to this question by…
Given a polytope P in $\mathbb{R}^n$, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the linear projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. If a polytope P…
Linear model trees are regression trees that incorporate linear models in the leaf nodes. This preserves the intuitive interpretation of decision trees and at the same time enables them to better capture linear relationships, which is hard…
We present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters.…
We consider the polyhedral properties of two spanning tree problems with additional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less…
We investigate the geography of Hilbert schemes parametrizing closed subschemes of projective space with specified Hilbert polynomials. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their…
In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the…
We present an explicit closed-form formula for the vertices of the classical cut polytope $\operatorname{CUT}(n)$, defined as the convex hull of cut vectors of the complete graph $K_n$. Our derivation proceeds via a related polytope,…
Polyhedral estimate is a generic efficiently computable nonlinear in observations routine for recovering unknown signal belonging to a given convex compact set from noisy observation of signal's linear image. Risk analysis and optimal…
Neural Networks and Decision Trees: two popular techniques for supervised learning that are seemingly disconnected in their formulation and optimization method, have recently been combined in a single construct. The connection pivots on…
This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled…
In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to…
The Maximum Weight Independent Set of Polygons problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the 2-dimensional plane, the goal is to find a set of pairwise non-overlapping polygons with…
Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial…
A multiobjective optimization problem is simplicial if the Pareto set and front are homeomorphic to a simplex and, under the homeomorphisms, each face of the simplex corresponds to the Pareto set and front of a subproblem. In this paper, we…
In the discrete Tempotron learning problem a neuron receives time varying inputs and for a set of such input sequences ($\mathcal S_-$ set) the neuron must be sub-threshold for all times while for some other sequences ($\mathcal S_+$ set)…
We use matrix iteration theory to characterize acceleration in smooth games. We define the spectral shape of a family of games as the set containing all eigenvalues of the Jacobians of standard gradient dynamics in the family. Shapes…
We propose an algorithm for exploring the entire regularization path of asymmetric-cost linear support vector machines. Empirical evidence suggests the predictive power of support vector machines depends on the regularization parameters of…
This work presents novel algorithms for learning Bayesian network structures with bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed-integer linear programming formulations for structure…