We propose to use input convex neural networks (ICNN) to build convex approximations of non-convex feasible sets of optimization problems, in the form of a set of linear equalities and inequalities in a lifted space.Our approach may be tailored to yield both inner-and outer-approximations, or to maximize its accuracy in regions closer to the minimum of a given objective function.We illustrate the method on twodimensional toy problems and motivate it by various instances of reliability management problems of large-scale electric power systems.

Motivated by the problem of bounding the number of iterations of the simplex algorithm, we investigate the possible lengths of monotone paths followed inside the oriented graphs of polyhedra (oriented by the objective function). We consider both the shortest and the longest monotone paths and estimate the monotone diameter and height of polyhedra. Our analysis applies to transportation polytopes, matroid polytopes, matching polytopes, shortest-path polytopes, and the traveling salesman polytope, among others. We begin by showing that combinatorial cubes have monotone diameter and Bland simplex height upper bounded by their dimension and that in fact all monotone paths of zonotopes are no larger than the number of edge directions of the zonotope. We later use this to show that several polytopes have polynomial-size monotone diameter. In contrast, we show that for many well-known combinatorial polytopes, the height is at least exponential. Surprisingly, for some famous pivot rules, e.g., greatest improvement and steepest edge, these same polytopes have polynomial-size simplex paths.

We present in this paper a new generic approach to variable branching in branch and bound for mixed-integer linear problems. Our approach consists in imitating the decisions taken by a good branching strategy, namely strong branching, with a fast approximation. This approximated function is created by a machine learning technique from a set of observed branching decisions taken by strong branching. The philosophy of the approach is similar to reliability branching. However, our approach can catch more complex aspects of observed previous branchings to take a branching decision. The experiments performed on randomly generated and MIPLIB problems show promising results.

To increase their competitiveness, many industrial companies monitor their production process, collecting large amount of measurements. This paper describes a technique using this data to improve the performance of a monitored process. In particular we wish to find a set of rules, i.e. intervals on a reduced number of parameters, for which an output value is maximized. The model-free optimization problem to solve is to find a box, restricted on a limited amount of dimensions, with the maximum mean value of the included points. This article compares a machine learning-based heuristic to the solution computed by a mixed-integer linear program on real-life databases from steel and glass manufacturing. Computational results show that the heuristic obtains comparable solutions to the mixed integer linear approach. However, the exact approach is computationally too expensive to tackle real life databases. Results show that the restriction of five process parameters, on these databases, may improve the quality of the process by 50%.