Polyhedral approximation strategies for nonconvex mixed-integer nonlinear programming in SHOT

Different versions of polyhedral outer approximation are used by many algorithms for mixed-integer nonlinear programming (MINLP). While it has been demonstrated that such methods work well for convex MINLP, extending them to solve nonconvex problems has traditionally been challenging. The Supporting Hyperplane Optimization Toolkit (SHOT) is a solver based on polyhedral approximations of the nonlinear feasible set of MINLP problems. SHOT is an open source COIN-OR project, and is currently one of the most efficient global solvers for convex MINLP. In this paper, we discuss some extensions to SHOT that significantly extend its applicability to nonconvex problems. The functionality include utilizing convexity detection for selecting the nonlinearities to linearize, lifting reformulations for special classes of functions, feasibility relaxations for infeasible subproblems and adding objective cuts to force the search for better feasible solutions. This functionality is not unique to SHOT, but can be implemented in other similar methods as well. In addition to discussing the new nonconvex functionality of SHOT, an extensive benchmark of deterministic solvers for nonconvex MINLP is performed that provides a snapshot of the current state of nonconvex MINLP.

Polyhedral approximations of the semidefinite cone and their application

We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

A CONVEX SURFACE WITH FRACTAL CURVATURE

We construct a surface that is obtained from the octahedron by pushing out four of the faces so that the curvature is supported in a copy of the Sierpinski gasket (SG) in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.

On Polyhedral Approximations of Polytopes for Learning Bayesian Networks

The motivation for this paper is the geometric approach to statistical learning Bayesiannetwork (BN) structures. We review three vector encodings of BN structures. The first one hasbeen used by Jaakkola et al. [9] and also by Cussens [4], the other two use special integral vectorsformerly introduced, called imsets [18, 20]. The topic is the comparison of outer polyhedral approximationsof the corresponding polytopes. We show how to transform the inequalities suggested byJaakkola et al. [9] into the framework of imsets. The result of our comparison is the observationthat the implicit polyhedral approximation of the standard imset polytope suggested in [21] givesa tighter approximation than the (transformed) explicit polyhedral approximation from [9]. Asa consequence, we confirm a conjecture from [21] that the above-mentioned implicit polyhedralapproximation of the standard imset polytope is an LP relaxation of that polytope. In the end,we review recent attempts to apply the methods of integer programming to learning BN structuresand discuss the task of finding suitable explicit LP relaxation in the imset-based approach.