Bounds for Multistage Mixed-Integer Distributionally Robust Optimization

Multistage mixed-integer distributionally robust optimization (DRO) forms a class of extremely challenging problems since their size grows exponentially with the number of stages. One way to model the uncertainty in multistage DRO is by creating sets of conditional distributions (the so-called conditional ambiguity sets) on a finite scenario tree and requiring that such distributions remain … Read more

Robust and Distributionally Robust Optimization Models for Support Vector Machine

In this paper we present novel data-driven optimization models for Support Vector Machines (SVM), with the aim of linearly separating two sets of points that have non-disjoint convex closures. Traditional classi cation algorithms assume that the training data points are always known exactly. However, real-life data are often subject to noise. To handle such uncertainty, we … Read more

Multi-stage robust optimization problems: A sampled scenario tree based approach

In this paper, we consider multi-stage robust convex optimization problems of the minimax type. We assume that the total uncertainty set is the cartesian product of stagewise compact uncertainty sets and approximate the given problem by a sampled subproblem. Instead of looking for the worst case among the infinite and typically uncountable set of uncertain … Read more

Bounds for Probabilistic Programming with Application to a Blend Planning Problem

In this paper, we derive deterministic inner approximations for single and joint probabilistic constraints based on classical inequalities from probability theory such as the one-sided Chebyshev inequality, Bernstein inequality, Chernoff inequality and Hoeffding inequality (see Pinter 1989). New assumptions under which the bounds based approximations are convex allowing to solve the problem efficiently are derived. … Read more

A two-stage stochastic optimization model for the bike-sharing allocation and rebalancing problem

The Bike-sharing allocation and rebalancing problem is the problem of determining the initial daily allocation of bikes to stations in a bike-sharing system composed of one depot and multiple capacitated stations, in which bikes can be rebalanced at a point in time during the day. Due to the uncertain demand in each station, we propose … Read more

Bounds in multi-horizon stochastic programs

In this paper, we present bounds for multi-horizon stochastic optimization problems, a class of problems introduced in [16] relevant in many industry-life applications tipically involving strategic and operational decisions on two different time scales. After providing three general mathematical formulations of a multi-horizon stochastic program, we extend the definition of the traditional Expected Value problem … Read more

The Stochastic Multistage Fixed Charge Transportation Problem: Worst-Case Analysis of the Rolling Horizon Approach

We introduce the Stochastic multistage fixed charge transportation problem in which a producer has to ship an uncertain load to a customer within a deadline. At each time period, a fixed transportation price can be paid to buy a transportation capacity. If the transportation capacity is used, the supplier also pays an uncertain unit transportation … Read more

Guaranteed Bounds for General Non-discrete Multistage Risk-Averse Stochastic Optimization Programs

In general, multistage stochastic optimization problems are formulated on the basis of continuous distributions describing the uncertainty. Such ”infinite” problems are practically impossible to solve as they are formulated and finite tree approximations of the underlying stochastic processes are used as proxies. In this paper, we demonstrate how one can find guaranteed bounds, i.e. finite … Read more


Consider (typically large) multistage stochastic programs, which are defined on scenario trees as the basic data structure. It is well known that the computational complexity of the solution depends on the size of the tree, which itself increases typically exponentially fast with its height, i.e. the number of decision stages. For this reason approximations which … Read more