arXiv subjects
Publications and source records attributed to Daphne Skipper.
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.
Non-convex, nonlinear gas network optimization models are used to determine the feasibility of flows on existing networks given constraints on network flows, gas mixing, and pressure loss along pipes. This work improves two existing gas network models: a discrete mixed-integer nonlinear program (MINLP) that uses binary variables to model positive and negative flows, and a continuous nonlinear program (NLP) that implements complementarity constraints with continuous variables. We introduce cuts to expedite the MINLP and we formulate two new pressure loss models that leverage the flow-splitting variables: one that is highly accurate and another that is simpler but less accurate. In computational tests using the global solver BARON our cuts and accurate pressure loss improves: (1) the average run time of the MINLP by a factor of 35, (2) the stability of the MINLP by solving every tested instance within 2.5 minutes (the baseline model timed out on 25% of instances), (3) the stability of the NLP by solving more instances than the baseline. Our simpler pressure loss model further improved run times in the MINLP (by a factor of 48 versus the baseline MINLP), but was unstable in the context of the NLP.
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.
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.
We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[\ell,u]$, where $z$ is a binary indicator of $x\in[\ell,u]$ ($0 \leq \ell 1$.
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.
Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling number $π(G)$ is the smallest number such that, for any distribution of $π(G)$ pebbles to the vertices of $G$ and choice of root vertex $r$ of $G$, there exists a sequence of pebbling moves that places a pebble on $r$. Computing $π(G)$ is provably difficult, and recent methods for bounding $π(G)$ have proved computationally intractable, even for moderately sized graphs. Graham conjectured that $π(G ~\square~ H) \leq π(G) π(H)$, where $G ~\square~ H$ is the Cartesian product of $G$ and $H$ (1989). While the conjecture has been verified for specific families of graphs, in general it remains open. This study combines the focus of developing a computationally tractable, IP-based method for generating good bounds on $π(G ~\square~ H)$, with the goal of shedding light on Graham's conjecture.We provide computational results for a variety of Cartesian-product graphs, including some that are known to satisfy Graham's conjecture and some that are not. Our approach leads to a sizable improvement on the best known bound for $π(L ~\square~ L)$, where $L$ is the Lemke graph, and $L ~\square~ L$ is among the smallest known potential counterexamples to Graham's conjecture.
Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation $\mathscr{Q}(G)$ of the boolean quadric polytope (also known as the (full) correlation polytope) $\mathscr{P}(G)$ of the complete graph $G=K_n$. We extend this work to structured sparse graphs, giving: (i) an efficient algorithm for $vol(\mathscr{Q}(G))$ when $G$ has bounded tree width, (ii) closed-form expressions (and asymptotic behaviors) for $vol(\mathscr{Q}(G))$ for all stars, paths, and cycles, and (iii) a closed-form expression for $vol(\mathscr{P}(G))$ for all cycles. Further, we demonstrate that when $G$ is a cycle, the simple relaxation $\mathscr{Q}(G)$ is a very close model for the much more complicated $\mathscr{P}(G)$. Additionally, we give some computational results demonstrating that this behavior of the cycle seems to extend to more complicated graphs. Finally, we speculate on the possibility of extending some of our results to cactii or even series-parallel graphs.
In the context of global optimization of mixed-integer nonlinear optimization formulations, we consider smoothing univariate functions $f$ that satisfy $f(0)=0$, $f$ is increasing and concave on $[0,+\infty)$, $f$ is twice differentiable on all of $(0,+\infty)$, but $f'(0)$ is undefined or intolerably large. The canonical examples are root functions $f(w):=w^p$, for $0 0$, then replacing the part of $f$ on $[0,δ]$ with the unique homogeneous cubic, matching $f$, $f'$ and $f''$ at $δ$. The parameter $δ$ is used to control (i.e., upper bound) the derivative at 0 (which controls it on all of $[0,+\infty)$ when $g$ is concave). Our main results: (i) we weaken an earlier sufficient condition to give a necessary and sufficient condition for the piecewise function $g$ to be increasing and concave; (ii) we give a general sufficient condition for $g'(0)$ to be decreasing in the smoothing parameter $δ$; under the same condition, we demonstrate that the worst-case error of $g$ as an estimate of $f$ is increasing in $δ$; (iii) we give a general sufficient condition for $g$ to underestimate $f$; (iv) we give a general sufficient condition for $g$ to dominate the simple `shift smoothing' $h(w):=f(w+λ)-f(λ)$ ($λ>0$), when the parameters $δ$ and $λ$ are chosen `fairly' --- i.e., so that $g'(0)=h'(0)$. In doing so, we solve two natural open problems of Lee and Skipper (2016), concerning (iii) and (iv) for root functions.
In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D'Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics, aimed at functions having some limited non-smoothness. Our results pertain to root functions ($w^p$ with $0<p<1$) and their increasing concave relatives. We provide (i) a sufficient condition (which applies to functions more general than root functions) for our smoothing to be increasing and concave, (ii) a proof that when $p=1/q$ for integers $q\geq 2$, our smoothing lower bounds the root function, (iii) substantial progress (i.e., a proof for integers $2\leq q\leq 10,000$) on the conjecture that our smoothing is a sharper bound on the root function than the natural and simpler "shifted root function", and (iv) for all root functions, a quantification of the superiority (in an average sense) of our smoothing versus the shifted root function near 0.