Related papers: The Support of Integer Optimal Solutions
We consider optimal control problems with integer-valued controls and a total variation regularization penalty in the objective on domains of dimension two or higher. The penalty yields that the feasible set is sequentially closed in the…
We present the first linear lower bound for the number of bits required to be accessed in the worst case to increment an integer in an arbitrary space- optimal binary representation. The best previously known lower bound was logarithmic. It…
This work presents a novel algorithm for impulsive optimal control of linear time-varying systems with the inclusion of input magnitude constraints. Impulsive optimal control problems, where the optimal input solution is a sum of delta…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
The SOR-like iteration method for solving the absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $\nu = \|A^{-1}\|_2 < 1$ is investigated. The convergence conditions of the SOR-like iteration method…
In this paper, we present some new necessary and sufficient optimality conditions in terms of the Clarke subdifferentials for approximate Pareto solutions of a nonsmooth vector optimization problem which has an infinite number of…
A multidimensional optimization problem is formulated in the tropical mathematics setting as to maximize a nonlinear objective function, which is defined through a multiplicative conjugate transposition operator on vectors in a…
Solving integer programs of the form $\min \{\mathbf{x} \mid A\mathbf{x} = \mathbf{b}, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \mathbf{x} \in \mathbb{Z}^n \}$ is, in general, $\mathsf{NP}$-hard. Hence, great effort has been put into…
This note presents an analytic construction of the optimal unit-norm direction hat(x) = x/|x| that maximizes or minimizes the objective linear expression, B . hat(x), subject to a system of linear constraints of the form [A] . x = 0, where…
The combinatorial optimization problem is one of the important applications in neural network computation. The solutions of linearly constrained continuous optimization problems are difficult with an exact algorithm, but the algorithm for…
Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors $v_1,\ldots,v_m \in \mathbb{R}^d$ and a constraint family ${\cal B}\subseteq 2^{[m]}$, find a set $S \in \cal{B}$ that…
This work addresses inverse linear optimization where the goal is to infer the unknown cost vector of a linear program. Specifically, we consider the data-driven setting in which the available data are noisy observations of optimal…
Our first result is a statement of a somewhat general form of a non-substitution theorem for linear programming problems, along with a very easy proof of the same. Subsequently, we provide an easy proof of theorem 1 in a 1979 paper of Olvi…
We study the general integer programming (IP) problem of optimizing a separable convex function over the integer points of a polytope: $\min \{f(\mathbf{x}) \mid A\mathbf{x} = \mathbf{b}, \, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \,…
We establish some upper bounds for the number of integer solutions to the Thue inequality $|F(x , y)| \leq m$, where $F$ is a binary form of degree $n \geq 3$ and with non-zero discriminant $D$, and $m$ is an integer. Our upper bounds are…
We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…
Integer programming is concerned with solving linear systems of equations over the non-negative integers. The basic question is to find a solution which minimizes a given linear objective function for a fixed right hand side. Here we also…
We study algorithms for computing stable models of propositional logic programs and derive estimates on their worst-case performance that are asymptotically better than the trivial bound of O(m 2^n), where m is the size of an input program…
Consider a large linear system where $A_n$ is a $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ is a $n\times 1$ vector of ones and with unknown the $n\times 1$ vector $\boldsymbol{x}_n$…
We show that one can enumerate the vertices of the convex hull of integer points in polytopes whose constraint matrices have bounded and nonzero subdeterminants, in time polynomial in the dimension and encoding size of the polytope. This…