On the equilibrium prices of a regular locally Lipschitz exchange economy

We extend classical results by Debreu and Dierker about equilibrium prices of a regular economy with continuously differentiable demand functions/excess demand function to a regular exchange economy with these functions being locally Lipschitz. Our concept of a regular economy is based on Clarke’s concept of regular value and we show that such a regular economy … Read more

Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning

Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in … Read more

ODTlearn: A Package for Learning Optimal Decision Trees for Prediction and Prescription

ODTLearn is an open-source Python package that provides methods for learning optimal decision trees for high-stakes predictive and prescriptive tasks based on the mixed-integer optimization (MIO) framework proposed in Aghaei et al. (2019) and several of its extensions. The current version of the package provides implementations for learning optimal classification trees, optimal fair classification trees, … Read more

Delay-Resistant Robust Vehicle Routing with Heterogeneous Time Windows

We consider a robust variant of the vehicle routing problem with heterogeneous time windows (RVRP-HTW) with a focus on delay-resistant solutions. Here, customers have different availability time windows for every vehicle and must be provided with a preferably tight appointment window for the planned service. Different vehicles are a possibility to model different days on … Read more

Stability of Markovian Stochastic Programming

Multi-stage stochastic programming is notoriously hard, since solution methods suffer from the curse of dimensionality. Recently, stochastic dual dynamic programming has shown promising results for Markovian problems with many stages and a moderately large state space. In order to numerically solve these problems simple discrete representations of Markov processes are required but a convincing theoretical … Read more

Dual Conflict Analysis for Mixed-Integer Semidefinite Programs

Conflict analysis originally tried to exploit the knowledge that certain nodes in a relaxation-based branch-and-bound are infeasible. It has been extended to derive valid constraints also from feasible nodes. This paper adapts this approach to mixed-integer semidefinite programs. Using dual solutions, the primal constraints are aggregated and the resulting inequalities can be used at different … Read more

Solution Path of Time-varying Markov Random Fields with Discrete Regularization

\(\) We study the problem of inferring sparse time-varying Markov random fields (MRFs) with different discrete and temporal regularizations on the parameters. Due to the intractability of discrete regularization, most approaches for solving this problem rely on the so-called maximum-likelihood estimation (MLE) with relaxed regularization, which neither results in ideal statistical properties nor scale to … Read more

A Multicut Approach to Compute Upper Bounds for Risk-Averse SDDP

Stochastic Dual Dynamic Programming (SDDP) is a widely used and fundamental algorithm for solving multistage stochastic optimization problems. Although SDDP has been frequently applied to solve risk-averse models with the Conditional Value-at-Risk (CVaR), it is known that the estimation of upper bounds is a methodological challenge, and many methods are computationally intensive. In practice, this … Read more

A Tutorial on Solving Single-Leader-Multi-Follower Problems using SOS1 Reformulations

In this tutorial we consider single-leader-multi-follower games in which the models of the lower-level players have polyhedral feasible sets and convex objective functions. This situation allows for classic KKT reformulations of the separate lower-level problems, which lead to challenging single-level reformulations of MPCC type. The main contribution of this tutorial is to present a ready-to-use … Read more

Duality of upper bounds in stochastic dynamic programming

For multistage stochastic programming problems with stagewise independent uncertainty, dynamic programming algorithms calculate polyhedral approximations for the value functions at each stage.  The SDDP algorithm provides piecewise linear lower bounds, in the spirit of the L-shaped algorithm, and corresponding upper bounds took a longer time to appear.  One strategy uses the primal dynamic programming recursion … Read more