## Invited Session Wed.3.H 2033

#### Wednesday, 15:15 - 16:45 h, Room: H 2033

**Cluster 11: Integer & mixed-integer programming** [...]

### Some bridges between algebra and integer programming

**Chair: Justo Puerto**

**Wednesday, 15:15 - 15:40 h, Room: H 2033, Talk 1**

**Víctor Blanco**

Applications of discrete optimization to numerical semigroups

**Coauthor: Justo Puerto**

**Abstract:**

In this talk we will show some connections between discrete optimization and commutative algebra. In particular we analyze some problems in numerical semigroups, which are sets of nonnegative integers, closed under addition and such that their complement is finite. In this algebraic framework, we will prove that some computations that are usually performed by applying brute force algorithms can be improved by formulating the problems as (single or multiobjective) linear integer programming. For instance, computing the omega invariant of a numerical semigroup (a measure of the primality of the algebraic object), decompositions into irreducible numerical semigroups (special semigroups with simple structure), homogeneus numerical semigroups, or the Kunz-coordinates vector of a numerical semigroups can be done efficiently by formulating the equivalent discrete optimization problem.

**Wednesday, 15:45 - 16:10 h, Room: H 2033, Talk 2**

**José-María Ucha**

Algebraic tools for nonlinear integer programming problems 1: Getting started.

**Coauthors: F. J. Castro, J. Gago, M. I. Hartillo, J. Puerto**

**Abstract:**

In this first talk we revisit a classical approach for obtaining exact solutions of some nonlinear integer problems. We treat the case of linear objective function with linear and nonlinear constraints.

Besides the test-set of some linear subpart of the problem, calculated via Gröbner bases (sometimes obtained explicitly without computation), we propose some extra ingredients. We show how to use information from the continuous relaxation of the problem, add quasi-tangent hyperplanes and use penalty functions as a guide in the search process.

**Wednesday, 16:15 - 16:40 h, Room: H 2033, Talk 3**

**Maria Isabel Hartillo**

Algebraic tools for nonlinear integer programming problems 2: Applications

**Coauthors: Jesus Gago, Justo Puerto, Jose M. Ucha**

**Abstract:**

In this second talk of the series we present how the methodology works in some real problems, namely construction of integer portfolios and redundancy allocation problems in series-parallel systems. Only in the first case the nonlinear part is of convex type. We analyse how the ideas introduced in the first talk provide promising results in computational experiments. On the other side, the combination of using test-sets and heuristics techniques opens a new approach for getting good solutions in facing huge problems.