MIPaaL: Mixed Integer Program as a Layer

Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems from specific classes with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a Mixed Integer Linear Program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, which is an exact algorithm that iteratively adds constraints to a continuous relaxation of the problem until an integral solution is found. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and prescription separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP

Complexity of optimizing over the integers

In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and "complexity" in the context of general mathematical optimization, avoiding context dependent definitions which is one of the sources of difference in the treatment of complexity within continuous and discrete optimization. In the second part of the paper, we employ the language developed in the first part to study information theoretic and algorithmic complexity of {\em mixed-integer convex optimization}, which contains as a special case continuous convex optimization on the one hand and pure integer optimization on the other. We strive for the maximum possible generality in our exposition. We hope that this paper contains material that both continuous optimizers and discrete optimizers find new and interesting, even though almost all of the material presented is common knowledge in one or the other community. We see the main merit of this paper as bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide. In fact, our motivation behind Part I of the paper is to provide a common language for both communities

Topics in discrete optimization: models, complexity and algorithms

In this dissertation we examine several discrete optimization problems through the perspectives of modeling, complexity and algorithms. We first provide a probabilistic comparison of split and type 1 triangle cuts for mixed-integer programs with two rows and two integer variables in terms of cut coefficients and volume cutoff. Under a specific probabilistic model of the problem parameters, we show that for the above measure, the probability that a split cut is better than a type 1 triangle cut is higher than the probability that a type 1 triangle cut is better than a split cut. The analysis also suggests some guidelines on when type 1 triangle cuts are likely to be more effective than split cuts and vice versa. We next study a minimum concave cost network flow problem over a grid network. We give a polytime algorithm to solve this problem when the number of echelons is fixed. We show that the problem is NP-hard when the number of echelons is an input parameter. We also extend our result to grid networks with backward and upward arcs. Our result unifies the complexity results for several models in production planning and green recycling including the lot-sizing model, and gives the first polytime algorithm for some problems whose complexities were not known before. Finally, we examine how much complexity randomness will bring to a simple combinatorial optimization problem. We study a problem called the sell or hold problem (SHP). SHP is to sell k out of n indivisible assets over two stages, with known first-stage prices and random second-stage prices, to maximize the total expected revenue. Although the deterministic version of SHP is trivial to solve, we show that SHP is NP-hard when the second-stage prices are realized as a finite set of scenarios. We show that SHP is polynomially solvable when the number of scenarios in the second stage is constant. A max{1/2,k/n}-approximation algorithm is presented for the scenario-based SHP.Ph.D