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Emily Speakman

Publications and source records attributed to Emily Speakman.

12 recordsLinked to original sources
arXiv2026

Volume Formulae for the Convex Hull of the Graph of a Trilinear Monomial: A Complete Characterization for General Box Domains

Solving difficult mixed-integer nonlinear programs via spatial branch-and-bound requires effective convex outer-approximations of nonconvex sets. In this framework, complex problem formulations are often decomposed into simpler library functions, whose relaxations are then composed to build relaxations of the overall problem. The trilinear monomial serves as one such fundamental library function, appearing frequently as a building block across diverse applications. By definition, its convex hull provides the tightest possible relaxation and thus serves as a benchmark for evaluating alternatives. Mixed volume techniques have yielded a parameterized volume formula for the convex hull of the graph of a trilinear monomial; however, existing results only address the case where all six bounds of the box domain are nonnegative. This restriction represents a notable gap in the literature, as variables with mixed-sign domains arise naturally in practice. In this work, we close the gap by extending to the general case via an exhaustive case analysis. We demonstrate that removing the nonnegative domain assumption alters the underlying structure of the convex hull polytope, leading to six distinct volume formulae that together characterize all possible parameter configurations.

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arXiv2026

On the convex hull of the graph of a simple monomial

Motivated by previous efforts toward mathematically analyzing the treatment of monomials in spatial branch-and-bound, we study the convex hull of the graph of a simple monomial on a nonnegative box domain in arbitrary dimension, where at most one of the variable lower bounds is positive. We give: (i) a description via linear inequalities, and (ii) a formula for the volume.

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arXiv2025

A scalable optimization approach for equitable facility location: Methodology and transportation applications

Efficient and equitable access to essential services, such as healthcare, food, and education, is an important goal in urban planning, public policy, and transport logistics. However, existing facility location models often do not scale well to large instances, or primarily focus on optimizing average accessibility, neglecting equity concerns, particularly for disadvantaged populations. This paper proposes a novel, scalable framework for equitable facility location, introducing a linearized proxy for the Kolm-Pollak Equally-Distributed Equivalent (EDE) metric to balance efficiency and fairness. Computational experiments demonstrate that our approach scales to extremely large problem instances, while being sensitive enough to account for inequity throughout the distribution, not merely via the maximum value. Moreover, optimal solutions represent significant improvements for the worst-off residents in terms of distance to an open amenity, while also attaining a near-optimal average experience for all users. An extensive real-world case study on supermarket access illustrates the practical applicability of the framework, with additional examples coming from polling applications. As such, the model is extended to handle real-world considerations such as capacity constraints, split demand assignments, and location-specific penalties. By bridging the gap between equity theory and practical optimization, this work offers a robust and versatile tool for researchers and practitioners in urban planning, transportation, and public policy.

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arXiv2025

Evacuation Planning on Time-Expanded Networks with Integrated Wildfire Information

We study the problem of evacuation planning for natural disasters, focusing on wildfire evacuations. By creating pre-planned evacuation routes that can be updated based on real-time data, we provide an easily adjustable approach to evacuation planning and implementation. Our method uses publicly available data and can be tailored for a particular region or circumstance. We formulate large-scale evacuations as maximum flow problems on time-expanded networks, in which we integrate hazard information given in the form of a shapefile. An initial flow and evacuation plan is found based on a predicted fire, and is then updated based on revised fire information received during the evacuation. We provide a proof of concept on three locations with historic deadly fires using data available through OpenStreetMaps, a basemap for a geographic information system (GIS), on a NetworkX Python script. The results validate viable running times and quality of information for application in practice. Particular strengths are the scalability and modularity of our approach and accompanying software package.

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arXiv2024

Hundreds of grocery outlets needed across the United States to achieve walkable cities

The notion of the $x$-minute city is again popular in urban planning, but the practical implications of developing walkable neighborhoods have not been rigorously explored. What is the scale of the challenge that cities needing to retrofit face? Where should new stores or amenities be located? For 500 cities in the United States, we explored how many additional supermarkets would be required to achieve various levels of $x$-minute access and where new stores should be located so that this access is equally-distributed. Our method is unique because it combines a novel measure of equality with a new model that optimally locates amenities for inequality-minimizing community access. We found that 25% of the studied cities could reach 15-minute access by adding five or fewer stores, while only 10% of the cities could even achieve 5-minute average access when using neighborhood centroids as potential sites; the cities that could, on average, required more than 100 stores each. This work provides a tool for cities to use evidenced-based planning to efficiently retrofit in order to enable active transport, benefiting both the climate and their residents' health. It also highlights the major challenge facing our cities due to the existing and ongoing car-dependent urban design that renders these goals unfeasible.

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arXiv2020

Constructing lattice-free gradient polyhedra in dimension two

Lattice-free gradient polyhedra can be used to certify optimality for mixed-integer convex minimization models. We consider how to construct these polyhedra for unconstrained models with two integer variables under the assumption that all level sets are bounded. A classic result of Bell, Doignon, and Scarf states that a lattice-free gradient polyhedron with at most four facets exists in this setting. We present an algorithm for creating a sequence of gradient polyhedra, each of which has at most four facets, that finitely converges to a lattice-free gradient polyhedron. Each update requires constantly many gradient evaluations. Our updates imitate the gradient descent algorithm, and consequently, it yields a gradient descent type of algorithm for problems with two integer variables.

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arXiv2020

Gaining or losing perspective

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[l,u]$, where $z$ is a binary indicatorof $x\in[l,u]$ ($u> \ell > 0$), and $y$ "captures" $f(x)$, which is assumed to be convex on its domain $[l,u]$, but otherwise $y=0$ when $x=0$. This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are convex. Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achieved via the "perspective reformulation" inequality $y \geq zf(x/z)$. We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when $f(x) := x^p$, for $p>1$, relaxations utilizing the inequality $yz^q \geq x^p$, for $q \in [0,p-1]$, are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with $f(x) := x^2$) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.

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arXiv2019

Computing the volume of the convex hull of the graph of a trilinear monomial using mixed volumes

Speakman and Lee (2017) gave a formula for the volume of the convex hull of the graph of a trilinear monomial, $y=x_1x_2x_3$, over a box in the nonnegative orthant, in terms of the upper and lower bounds on the variables. This was done in the context of using volume as a measure for comparing alternative convexifications to guide the implementation of spatial branch-and-bound for mixed integer nonlinear optimization problems. Here, we introduce an alternative method for computing this volume, making use of the rich theory of mixed volumes. This new method may lead to a natural approach for considering extensions of the problem.

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arXiv2018

On branching-point selection for trilinear monomials in spatial branch-and-bound: the hull relaxation

In Speakman and Lee (2017), we analytically developed the idea of using volume as a measure for comparing relaxations in the context of spatial branch-and-bound. Specifically, for trilinear monomials, we analytically compared the three possible "double-McCormick relaxations" with the tight convex-hull relaxation. Here, again using volume as a measure, for the convex-hull relaxation of trilinear monomials, we establish simple rules for determining the optimal branching variable and optimal branching point. Additionally, we compare our results with current software practice.

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arXiv2016

Quantifying Double McCormick

When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the bounds defining the box. In this manner, we precisely quantify the strength of the different possible relaxations defined by all three groupings, in addition to the trilinear hull itself. As a by product, we characterize the best double McCormick relaxation. We wish to emphasize that, in the context of spatial branch-and-bound for factorable formulations, our results do not only apply to variables in the input formulation. Our results apply to monomials that involve auxiliary variables as well. So, our results apply to the product of any three (possibly complicated) expressions in a formulation.

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arXiv2016

Experimental validation of volume-based comparison for double-McCormick relaxations

Volume is a natural geometric measure for comparing polyhedral relaxations of non-convex sets. Speakman and Lee gave volume formulae for comparing relaxations of trilinear monomials, quantifying the strength of various natural relaxations. Their work was motivated by the spatial branch-and-bound algorithm for factorable mathematical-programming formulations. They mathematically analyzed an important choice that needs to be made whenever three or more terms are multiplied in a formulation. We experimentally substantiate the relevance of their main results to the practice of global optimization, by applying it to difficult box cubic problems (boxcup). In doing so, we find that, using their volume formulae, we can accurately predict the quality of a relaxation for boxcups based on the (box) parameters defining the feasible region.

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