Related papers: Integer Polynomial Optimization in Fixed Dimension
Integer linear programs $\min\{c^T x : A x = b, x \in \mathbb{Z}^n_{\ge 0}\}$, where $A \in \mathbb{Z}^{m \times n}$, $b \in \mathbb{Z}^m$, and $c \in \mathbb{Z}^n$, can be solved in pseudopolynomial time for any fixed number of constraints…
In this paper, we solve a maximization problem where the objective function is quadratic and the constraints set is the reachable values set of a stable discrete-time affine system. This problem is equivalent to solve an infinite number of…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…
In bi-objective integer optimization the optimal result corresponds to a set of non-dominated solutions. We propose a generic bi-objective branch-and-bound algorithm that uses a problem-independent branching rule exploiting available…
For a given set of intervals on the real line, we consider the problem of ordering the intervals with the goal of minimizing an objective function that depends on the exposed interval pieces (that is, the pieces that are not covered by…
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…
This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained…
We consider optimal route planning when the objective function is a general nonlinear and non-monotonic function. Such an objective models user behavior more accurately, for example, when a user is risk-averse, or the utility function needs…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…
We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible…
Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…
We study a class of projective transformations of spectraplexes associated with self-dual cones and, on this basis, propose a polynomial-time algorithm for convex feasibility problems with positive definite constraints. At each iteration of…
This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We…
We study the integrality gap of convex mixed-integer programs, that is, the difference between the optimal value of such a problem and the optimal value of its continuous relaxation. We study classes of convex sets whose associated…
We consider the problem of projecting a convex set onto a subspace, or equivalently formulated, the problem of computing a set obtained by applying a linear mapping to a convex feasible set. This includes the problem of approximating convex…
We introduce a fixed point iteration process built on optimization of a linear function over a compact domain. We prove the process always converges to a fixed point and explore the set of fixed points in various convex sets. In particular,…
We consider the NP-hard problem of minimizing a separable concave quadratic function over the integral points in a polyhedron, and we denote by D the largest absolute value of the subdeterminants of the constraint matrix. In this paper we…
This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide…
We tackle the Optimal Experiment Design Problem, which consists of choosing experiments to run or observations to select from a finite set to estimate the parameters of a system. The objective is to maximize some measure of information…
Absolute value linear programming problems is quite a new area of optimization problems, involving linear functions and absolute values in the description of the model. In this paper, we consider interval uncertainty of the input…