Efficient MIP techniques for computing the relaxation complexity

The relaxation complexity rc(X) of the set of integer points X contained in a polyhedron is the minimal number of inequalities needed to formulate a linear optimization problem over X without using auxiliary variables. Besides its relevance in integer programming, this concept has interpretations in aspects of social choice, symmetric cryptanalysis, and machine learning. We employ efficient mixed-integer programming techniques to compute a robust and numerically more practical variant of the relaxation complexity. Our proposed models require row or column generation techniques and can be enhanced by symmetry handling and suitable propagation algorithms. Theoretically, we compare the quality of our models in terms of their LP relaxation values. The performance of those models is investigated on a broad test set and is underlined by their ability to solve challenging instances that could not be solved previously

A Unified Framework for Symmetry Handling

Handling symmetries in optimization problems is essential for devising efficient solution methods. In this article, we present a general framework that captures many of the already existing symmetry handling methods (SHMs). While these SHMs are mostly discussed independently from each other, our framework allows to apply different SHMs simultaneously and thus outperforming their individual effect. Moreover, most existing SHMs only apply to binary variables. Our framework allows to easily generalize these methods to general variable types. Numerical experiments confirm that our novel framework is superior to the state-of-the-art SHMs implemented in the solver SCIP

The Impact of Symmetry Handling for the Stable Set Problem via Schreier-Sims Cuts

Symmetry handling inequalities (SHIs) are an appealing and popular tool for handling symmetries in integer programming. Despite their practical application, little is known about their interaction with optimization problems. This article focuses on Schreier-Sims (SST) cuts, a recently introduced family of SHIs, and investigate their impact on the computational and polyhedral complexity of optimization problems. Given that SST cuts are not unique, a crucial question is to understand how different constructions of SST cuts influence the solving process. First, we observe that SST cuts do not increase the computational complexity of solving a linear optimization problem over any polytope $P$. However, separating the integer hull of $P$ enriched by SST cuts can be NP-hard, even if $P$ is integral and has a compact formulation. We study this phenomenon more in-depth for the stable set problem, particularly for subclasses of perfect graphs. For bipartite graphs, we give a complete characterization of the integer hull after adding SST cuts based on odd-cycle inequalities. For trivially perfect graphs, we observe that the separation problem is still NP-hard after adding a generic set of SST cuts. Our main contribution is to identify a specific class of SST cuts, called stringent SST cuts, that keeps the separation problem polynomial and a complete set of inequalities, namely SST clique cuts, that yield a complete linear description. We complement these results by giving SST cuts based presolving techniques and provide a computational study to compare the different approaches. In particular, our newly identified stringent SST cuts dominate other approaches

Integrality of Linearizations of Polynomials over Binary Variables using Additional Monomials

Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an integral relaxation polytope, generalizing work by Del Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and Rodr\'iguez-Heck (European Journal of Operations Research, 2019). We also present an algorithm that finds these extra monomials for a given polynomial to yield an integral relaxation polytope or determines that no such set of extra monomials exists. In the former case, our approach yields an algorithm to solve the given polynomial optimization problem as a compact LP, and we complement this with a purely combinatorial algorithm.Comment: 27 pages, 11 figure

Mixed-integer programming techniques for the minimum sum-of-squares clustering problem

The minimum sum-of-squares clustering problem is a very important problem in data mining and machine learning with very many applications in, e.g., medicine or social sciences. However, it is known to be NP-hard in all relevant cases and to be notoriously hard to be solved to global optimality in practice. In this paper, we develop and test different tailored mixed-integer programming techniques to improve the performance of state-of-the-art MINLP solvers when applied to the problem—among them are cutting planes, propagation techniques, branching rules, or primal heuristics. Our extensive numerical study shows that our techniques significantly improve the performance of the open-source MINLP solver SCIP. Consequently, using our novel techniques, we can solve many instances that are not solvable with SCIP without our techniques and we obtain much smaller gaps for those instances that can still not be solved to global optimality

A simple method for convex optimization in the oracle model

We give a simple and natural method for computing approximately optimal solutions for minimizing a convex function f over a convex set K given by a separation oracle. Our method utilizes the Frank–Wolfe algorithm over the cone of valid inequalities of K and subgradients of f . Under the assumption that f is L-Lipschitz and that K contains a ball of radius r and is contained inside the origin centered ball of radius R, using O( (RL)^2/ε^2 · R^2/r^2 ) iterations and calls to the oracle, our main method outputs a point x ∈ K satisfying f (x) ≤ ε + minz∈K f (z). Our algorithm is easy to implement, and we believe it can serve as a useful alternative to existing cutting plane methods. As evidence towards this, we show that it compares favorably in terms of iteration counts to the standard LP based cutting plane method and the analytic center cutting plane method, on a testbed of combinatorial, semidefinite and machine learning instance

A simple method for convex optimization in the oracle model

We give a simple and natural method for computing approximately optimal solutions for minimizing a convex function f over a convex set K given by a separation oracle. Our method utilizes the Frank–Wolfe algorithm over the cone of valid inequalities of K and subgradients of f. Under the assumption that f is L-Lipschitz and that K contains a ball of radius r and is contained inside the origin centered ball of radius R, using O((RL)2ε2·R2r2) iterations and calls to the oracle, our main method outputs a point x∈ K satisfying f(x) ≤ ε+ min z∈Kf(z) . Our algorithm is easy to implement, and we believe it can serve as a useful alternative to existing cutting plane methods. As evidence towards this, we show that it compares favorably in terms of iteration counts to the standard LP based cutting plane method and the analytic center cutting plane method, on a testbed of combinatorial, semidefinite and machine learning instances

Strong IP formulations need large coefficients

The development of practically well-behaved integer programming formulations is an important aspect of solving linear optimization problems over a set X⊆{0,1} n. In practice, one is often interested in strong integer formulations with additional properties, e.g., bounded coefficients to avoid numerical instabilities. This article presents a lower bound on the size of coefficients in any strong integer formulation of X and demonstrates that certain integer sets X require (exponentially) large coefficients in any strong integer formulation. We also show that strong integer formulations of X⊆{0,1} n may require exponentially many inequalities while linearly many inequalities may suffice in weak formulations