Timm Oertel

Mirror-Descent Methods in Mixed-Integer Convex Optimization

In this paper, we address the problem of minimizing a convex function f over a convex set, with the extra constraint that some variables must be integer. This problem, even when f is a piecewise linear function, is NP-hard. We study an algorithmic approach to this problem, postponing its hardness to the realization of an oracle. If this oracle can be realized in polynomial time, then the problem can be solved in polynomial time as well. For problems with two integer variables, we show with a novel geometric construction how to implement the oracle efficiently, that is, in \(\mathcal {O}(\ln(B))\) approximate minimizations of f over the continuous variables, where B is a known bound on the absolute value of the integer variables. Our algorithm can be adapted to find the second best point of a purely integer convex optimization problem in two dimensions, and more generally its k-th best point. This observation allows us to formulate a finite-time algorithm for mixed-integer convex optimization.

Integer convex minimization by mixed integer linear optimization

Minimizing a convex function over the integral points of a bounded convex set is polynomial in fixed dimension (Grotschel et?al., 1988). We provide an alternative, short, and geometrically motivated proof of this result. In particular, we present an oracle-polynomial algorithm based on a mixed integer linear optimization oracle.

A note on non-degenerate integer programs with small sub-determinants

The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by AZmn and find an algorithm to solve such problems in polynomial-time provided that both the largest absolute value of an entry in A and m are constant.Then, this is applied to solve integer programs in inequality form in polynomial-time, where the absolute values of all maximal sub-determinants of A lie between 1 and a constant.

Centerpoints: A Link between Optimization and Convex Geometry

We introduce a concept that generalizes several different notions of a “centerpoint” in the literature. We develop an oracle-based algorithm for convex mixed-integer optimization based on centerpoints. Further, we show that algorithms based on centerpoints are “best possible” in a certain sense. Motivated by this, we establish structural results about this concept and provide efficient algorithms for computing these points. Our main motivation is to understand the complexity of oracle-based convex mixed-integer optimization.

Note on the complexity of the mixed-integer hull of a polyhedron

We study the complexity of computing the mixed-integer hull conv ( P ? ( Z n i? R d ) ) of a polyhedron P . Given an inequality description, with one integer variable, the mixed-integer hull can have exponentially many vertices and facets in d . For n , d fixed, we give an algorithm to find the mixed-integer hull in polynomial time. Given a finite set V ? Q n + d , with n fixed, we compute a vertex description of the mixed-integer hull of conv ( V ) in polynomial time and give bounds on the number of vertices of the mixed-integer hull.

Sparse solutions of linear Diophantine equations

We present structural results on solutions to the Diophantine system

Duality for mixed-integer convex minimization

Ay = b

A Polyhedral Frobenius Theorem with Applications to Integer Optimization

,

Integrality Gaps of Integer Knapsack Problems

y \in \mathbb{Z}^t_{\ge 0}

Convex integer minimization in fixed dimension

that have the smallest number of nonzero entries. Our tools are algebraic and number theoretic in nature and include Siegels lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.