arXiv subjects

Leonardo Lozano

Publications and source records attributed to Leonardo Lozano.

8 recordsLinked to original sources

arXiv2026

Convexification of mixed-integer quadratic optimization via decision diagrams

We study mixed-integer quadratic optimization (MIQO) problems with indicator variables. We propose a unified framework, based on decision diagrams, that serves both to solve the associated optimization problems and to construct ideal conic quadratic extended formulations of the closure of the convex hull of the underlying mixed-integer set. The construction applies to arbitrary quadratics and to any combinatorial constraints admitting a tractable dynamic programming representation. The resulting diagrams and the ensuing convex hull descriptions are of polynomial size when the quadratic is low-rank, or when the support graph of the Hessian or of its inverse is a tree, recovering and generalizing several results from the literature. For structured sparse and inverse-sparse quadratics, we show that approximate decision diagrams have size linear in the dimension while yielding solutions with arbitrarily low optimality gap. Computational experiments demonstrate the effectiveness of the proposed approach.

math.OC↗

arXiv2024

The Sensitivity of the U.S. Presidential Election to Coordinated Voter Relocation

U.S. presidential elections are decided by the Electoral College, established in 1789, and designed to mitigate potential risks arising from the collusion of large groups of citizens. A statewide winner-take-all popular voting system for electors is implemented in all but two states, which has led to instances where narrow victories in key states were decisive in several recent elections. Small groups of voters can significantly impact the election, for example, through voter turnout. However, another dynamic can also influence this: a surprisingly small number of dedicated voters moving short distances across state lines. The extent to which the election's outcome is sensitive to small and well-coordinated movements of people has not been investigated in detail. Using a combination of forecasting, simulation, and optimization, we show that a candidate's probability of winning can be increased by 1% through the strategic relocation of approximately 10,000 people no farther than 100 miles from their current county of residence, less than 0.006% of the eligible voting population. Moreover, an 8% probability increase can be realized by a mere 50,000 voters relocating across state lines, or 0.03% of the voting population. Given the remarkably small number of people involved and the fact that establishing electoral residence in many states takes about a month, this coordinated relocation of voters is not nearly as challenging as previously thought. As it stands, U.S. presidential elections may be vulnerable to the exploitation of the aforementioned loophole. Therefore, we anticipate our findings will have direct consequences on policymaking and campaign strategy, as well as motivate new operations research methods within the political sciences.

physics.soc-ph↗

arXiv2024

Network Relaxations for Discrete Bilevel Optimization under Linear Interactions

We investigate relaxations for a class of discrete bilevel programs where the interaction constraints linking the leader and the follower are linear. Our approach reformulates the upper-level optimality constraints by projecting the leader's decisions onto vectors that map to distinct follower solution values, each referred to as a state. Based on such a state representation, we develop a network-flow linear program via a decision diagram that captures the convex hull of the follower's value function graph, leading to a new single-level reformulation of the bilevel problem. We also present a reduction procedure that exploits symmetry to identify the reformulation of minimal size. For large networks, we introduce parameterized relaxations that aggregate states by considering tractable hyperrectangles based on lower and upper bounds associated with the interaction constraints, and can be integrated into existing mixed-integer bilevel linear programming (MIBLP) solvers. Numerical experiments suggest that the new relaxations, whether used within a simple cutting-plane procedure or integrated into state-of-the-art MIBLP solvers, significantly reduce runtimes or solve additional benchmark instances. Our findings also highlight the correlation between the quality of relaxations and the properties of the interaction matrix, underscoring the potential of our approach in enhancing solution methods for structured bilevel optimization instances.

math.OC↗

arXiv2024

Constrained Shortest-Path Reformulations via Decision Diagrams for Structured Two-stage Optimization Problems

Many discrete optimization problems are amenable to constrained shortest-path reformulations in an extended network space, a technique that has been key in convexification, bound strengthening, and search. In this paper, we propose a constrained variant of these models for two challenging classes of discrete two-stage optimization problems, where traditional methods (e.g., dualize-and-combine) are not applicable compared to their continuous counterparts. Specifically, we propose a framework that models problems as decision diagrams and introduces side constraints either as linear inequalities in the underlying polyhedral representation, or as state variables in shortest-path dynamic programming models. For our first structured class, we investigate two-stage problems with interdiction constraints. We show that such constraints can be formulated as indicator functions in the arcs of the diagram, providing an alternative single-level reformulation of the problem via a network-flow representation. Our second structured class is classical robust optimization, where we leverage the decision diagram network to iteratively identify label variables, akin to an L-shaped method. We evaluate these strategies on a competitive project selection problem and the robust traveling salesperson with time windows, observing considerable improvements in computational efficiency as compared to general methods in the respective areas.

math.OC↗

arXiv2024

Real-time solution of quadratic optimization problems with banded matrices and indicator variables

We consider mixed-integer quadratic optimization problems with banded matrices and indicator variables. These problems arise pervasively in statistical inference problems with time-series data, where the banded matrix captures the temporal relationship of the underlying process. In particular, the problem studied arises in monitoring problems, where the decision-maker wants to detect changes or anomalies. We propose to solve these problems using decision diagrams. In particular we show how to exploit the temporal dependencies to construct diagrams with size polynomial in the number of decision variables. We also describe how to construct the convex hull of the set under study from the decision diagrams, and how to deploy the method online to solve the problems in milliseconds via a shortest path algorithm.

math.OC↗

arXiv2022

Constraint Learning to Define Trust Regions in Predictive-Model Embedded Optimization

There is a recent proliferation of research on the integration of machine learning and optimization. One expansive area within this research stream is predictive-model embedded optimization, which proposes the use of pre-trained predictive models as surrogates for uncertain or highly complex objective functions. In this setting, features of the predictive models become decision variables in the optimization problem. Despite a recent surge in publications in this area, only a few papers note the importance of incorporating trust region considerations in this decision-making pipeline, i.e., enforcing solutions to be similar to the data used to train the predictive models. Without such constraints, the evaluation of the predictive model at solutions obtained from optimization cannot be trusted and the practicality of the solutions may be unreasonable. In this paper, we provide an overview of the approaches appearing in the literature to construct a trust region, and propose three alternative approaches. Our numerical evaluation highlights that trust-region constraints learned through isolation forests, one of the newly proposed approaches, outperform all previously suggested approaches, both in terms of solution quality and computational time.

cs.LG↗

arXiv2022

Optimizing over an ensemble of neural networks

We study optimization problems where the objective function is modeled through feedforward neural networks with rectified linear unit (ReLU) activation. Recent literature has explored the use of a single neural network to model either uncertain or complex elements within an objective function. However, it is well known that ensembles of neural networks produce more stable predictions and have better generalizability than models with single neural networks, which motivates the investigation of ensembles of neural networks rather than single neural networks in decision-making pipelines. We study how to incorporate a neural network ensemble as the objective function of an optimization model and explore computational approaches for the ensuing problem. We present a mixed-integer linear program based on existing popular big-M formulations for optimizing over a single neural network. We develop a two-phase approach for our model that combines preprocessing procedures to tighten bounds for critical neurons in the neural networks with a Lagrangian relaxation-based branch-and-bound approach. Experimental evaluations of our solution methods suggest that using ensembles of neural networks yields more stable and higher quality solutions, compared to single neural networks, and that our optimization algorithm outperforms (the adaption of) a state-of-the-art approach in terms of computational time and optimality gaps.

cs.LG↗

arXiv2021

Optimizing the expected maximum of two linear functions defined on a multivariate Gaussian distribution

We study stochastic optimization problems with objective function given by the expectation of the maximum of two linear functions defined on the component random variables of a multivariate Gaussian distribution. We consider random variables that are arbitrarily correlated, and we show that the problem is NP-hard even if the space of feasible solutions is unconstrained. We exploit a closed-form expression for the objective function from the literature to construct a cutting-plane algorithm that can be seen as an extension of the integer L-shaped method for a highly nonlinear function, which includes the evaluation of the c.d.f and p.d.f of a standard normal random variable with decision variables as part of the arguments. To exhibit the model's applicability, we consider two featured applications. The first is daily fantasy sports, where the algorithm identifies entries with positive returns during the 2018-2019 National Football League season. The second is a special case of makespan minimization for two parallel machines and jobs with uncertain processing times; for the special case where the jobs are uncorrelated, we prove the equivalence between its deterministic and stochastic versions and show that our algorithm can deliver a constant-factor approximation guarantee for the problem. The results of our computational evaluation involving synthetic and real-world data suggest that our discretization and upper bounding techniques lead to significant computational improvements and that the proposed algorithm outperforms sub-optimal solutions approaches.

math.OC↗