Abstract Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or not guarantee protection. In this work, we develop an algorithm convex optimization problems where a key feature is method does rely specific of inner worst-case (adversarial) and allows latter to be no...

Minimizing a convex function over the integral points of a bounded convex set is polynomial in fixed dimension [6]. We provide an alternative, short, and geometrically motivated proof of this result. In particular, we present an oraclepolynomial algorithm based on a mixed integer linear optimization oracle.

In the last decade our capability of solving integer programming problems has increased dramatically due to the effectiveness of cutting plane methods based on polyhedral investigations. Polyhedral cutting planes have become central features in optimization software packages for integer programming. Here we present some of the important polyhedral methods used in discrete optimization. We discu...

We consider the constrained assortment optimization problem under the mixed multinomial logit model. Even moderately sized instances of this problem are challenging to solve directly using standard mixed-integer linear optimization formulations. This has motivated recent research exploring customized optimization strategies and approximation techniques. In contrast, we develop a novel conic qua...

We consider N-fold 4-block decomposable integer programs, which simultaneously generalize N-fold integer programs and two-stage stochastic integer programs with N scenarios. In previous work [R. Hemmecke, M. Köppe, R. Weismantel, A polynomial-time algorithm for optimizing over N-fold 4block decomposable integer programs, Proc. IPCO 2010, Lecture Notes in Computer Science, vol. 6080, Springer, 2...

Mixed-integer optimization problems arise in various application fields, such as chemical engineering and the optimization of medical image processing pipelines. Stochastic optimization algorithms, such as evolution strategies and estimation of distribution algorithms, can be used as solution methods for solving these problems approximately. Especially for real-world problems they often prove t...

Following from part I, which presents a retrospective on optimization, we focus here on areas that are recent active research topics and are likely to strongly influence the future of optimization algorithms and formulations. First, we discuss recent developments in deterministic global optimization algorithms applied to both nonlinear programs and mixed-integer programs. Second, we discuss log...