key: cord-0486161-6b6qoiq0
authors: Smith, Kevin D.; Bullo, Francesco
title: Convex Optimization of the Basic Reproduction Number
date: 2021-09-16
journal: nan
DOI: nan
sha: 1cdc782624ee248cd77e31e15d57c90ab99a9321
doc_id: 486161
cord_uid: 6b6qoiq0

The basic reproduction number $R_0$ is a fundamental quantity in epidemiological modeling, reflecting the typical number of secondary infections that arise from a single infected individual. While $R_0$ is widely known to scientists, policymakers, and the general public, it has received comparatively little attention in the controls community. This note provides two novel characterizations of $R_0$: a stability characterization and a geometric program characterization. The geometric program characterization allows us to write $R_0$-constrained and budget-constrained optimal resource allocation problems as geometric programs, which are easily transformed into convex optimization problems. We apply these programs to a case study of allocating vaccines and antidotes, finding that targeting $R_0$ instead of the spectral abscissa of the Jacobian matrix (a common target in the controls literature) leads to qualitatively different solutions.

Perhaps the most important parameter in an epidemic is the basic reproduction number. This number, denoted R 0 , is the number of secondary infections that arise from a typical infected individual within an otherwise completely susceptible population. R 0 is a widely-known term, especially since 2020, when articles with "R 0 " in the title ran in mainstream publications like The New York Times and The Wall Street Journal. Since R 0 is an intuitive and widely-known quantity, one might also expect it to appear frequently in the controls literature on epidemics, but this is not the case.

Instead, the literature tends to focus on two other major approaches to epidemic control. First, in the optimal control framework, parameters or control inputs are chosen to minimize some cost function integrated along the model trajectory [13] , [8] , [6] , [14] . These trajectories seldom admit closedform solutions, so this approach generally requires modelspecific analysis and numerical solutions of Pontryagin's conditions [13] , [8] , potentially large-scale optimization to embed discrete-time dynamics [6] , or linearization and a discount factor to ensure convergence [14] . The second major approach is the spectral optimization framework, in which resources are allocated to minimize the spectral abscissa of the model's Jacobian matrix about some disease-free equilibrium [12] , [15] , [10] , [9] , [7] . If the Jacobian is stable, then the abscissa represents the rate at which the trajectory converges to this equilibrium, so minimizing the (negative) abscissa leads to a faster-decaying epidemic. Spectral optimization is based on a linear approximation of the model, but it is nonetheless an appealing framework for resource allocation, since the spectral abscissa can be directly evaluated from model parameters (without computing a trajectory).

The spectral abscissa is closely related to the basic reproduction number R 0 . They are both equivalent threshold parameters for whether the epidemic spreads or decays: in compartmental epidemic models (under reasonable assumptions), the epidemic enters an exponential growth phase if and only if the abscissa is positive, if and only if R 0 > 1 [3] . Furthermore, intuitively, both quantities reflect the rate at which the epidemic spreads or decays. But it is important to note that the abscissa and R 0 are different quantities. For example, the basic Kermack-McKendrick SIR model can realize any pair of values for the abscissa and R 0 > 0, so long as the abscissa and R 0 − 1 have the same sign. Thus, while the intuition for these two quantities is similar, minimizing the abscissa will generally lead to a different allocation of resources than minimizing R 0 directly.

The premise of this note is that the basic reproduction number R 0 is a more appealing parameter for resource allocation than the spectral abscissa. One important reason is communicability. Since optimal resource allocation during epidemics is ultimately meant to influence public policy, it is important to properly inform the public and clearly communicate the objective of this allocation. Arguably, the minimization of R 0 is a much easier objective to explain to a layperson than that of the spectral abscissa. A second advantage of R 0 is that it more directly reflects the rate at which new individuals are infected. Despite these advantages, to our knowledge there is no work in the literature that focuses on directly minimizing or constraining R 0 in the resource allocation problem.

Contributions: This note proposes a modification of the spectral optimization framework to operate on R 0 instead of on the spectral abscissa. We offer three primary contributions:

(i) We provide two novel characterizations of R 0 in compartmental epidemic models. One characterization relates R 0 to the stability of perturbations to the Jacobian matrix, and the other expresses R 0 as a geometric program, which can be transformed into a convex optimization problem. (ii) We define two R 0 -based optimal resource allocation problems: the R 0 -constrained allocation problem, which identifies the lowest-cost allocation to restrict R 0 below a given upper bound; and the budget-constrained allocation problem, which minimizes R 0 with a limited allowance for resource cost. We provide a geometric programming arXiv:2109.07643v1 [math.OC] 16 Sep 2021 transcription for both of these problems, allowing them to be solved efficiently with off-the-shelf software. (iii) We consider a case study based on a county-level multigroup SEIR model in California, parameterized using real-world cell phone mobility data. The case study examines the allocation of vaccines and antidotes, a classical problem in spectral optimization. We explain and emphasize the differences between the allocations based on R 0 and the corresponding allocations based on the abscissa. Organization: Section II introduces the general family of compartmental epidemic models that we consider ( §II-A), formally defines R 0 ( §II-B), briefly reviews geometric programming ( §II-C), and states three key lemmas about Metzler and Hurwitz matrices ( §II-D). Section III presents our main theoretical results, including the two new characterizations of R 0 ( §III-A), and the two R 0 -based optimal resource allocation problems and their geometric program transcriptions ( §III-B). Finally, Section IV presents the numerical case study.

Notation: The matrix A ∈ R n×n is Metzler if all its off-diagonal entries are non-negative and is Hurwitz if all its eigenvalues have negative real part. Let ρ(A) denote the spectral radius of A. Given A ∈ R n×n , let diag(A) denote the vector in R n composed of the diagonal elements of A. Given x ∈ R n , let diag(x) denote the diagonal matrix whose diagonal is x. Thus diag(diag(x)) = x, and diag(diag(A)) is a copy of A with all off-diagonal entries set to zero. Given a set S, we write cl(S) to denote the closure of S.

Compartmental models are a general and widely-used family of epidemic models that divide a population into compartments based on disease state and other demographic factors. This paper focuses on deterministic epidemic models, in which the expected number of individuals in each compartment is governed by a system of differential equations. Perhaps the most well-known example is Kermack and McKendrick's SIR model, which has three compartments (susceptible, infected, and recovered), but compartmental models can be arbitrarily complex to capture nuances in the spread of infection between different parts of the population in different disease states.

We consider the general compartmental model in [3] , with n infected compartments and m non-infected compartments. The components of this model are as follows. Let x ∈ R n be the expected numbers of individuals in each infected compartment, and let y ∈ R m be the expected numbers of non-infected individuals. The resulting dynamics iṡ

where f , v, and g are continuously differentiable and defined on non-negative domains. Assumption 1 (Regularity of f , v, and g). The vector fields f , v, and g have the following properties:

(i) f (x, y) ≥ 0 n for all x and y;

(ii) f (0 n , y) = 0 n and v(0 n , y) = 0 n for all y;

(iii) for all x, y, and i, x i = 0 implies that v i (x, y) ≥ 0; and (iv) for all x, y, and j, y j = 0 implies that g j (x, y) ≥ 0. thus every disease-free state is an equilibrium of (1a). Finally, conds. (iii) and (iv) reflect the fact that individuals cannot transition out from an empty compartment. We also assume that (1) admit a disease-free equilibrium point (0 n , y * ) that is locally asymptotically stable in the absence of new infections. That is, if new infections are "switched off" by dropping the vector field f from the dynamics, then the population will return to (0 n , y * ) even if a small number of infected individuals are introduced.

Assumption 2 (Existence of a Stable Equilibrium). There exists y * ≥ 0 m such that g(0 n , y * ) = 0 m and the following Jacobian matrix is Hurwitz:

The point (0 n , y * ) satisfying Assumption 2 is not necessarily unique, and while it is also an equilibrium point of the full model, it may be unstable when f is no longer ignored. Under Assumptions 1 and 2, linearizing the dynamics of (1a) about (0 n , y * ) decouples them from y, and we obtaiṅ

is Hurwitz and Metzler. We refer the reader to [3, Lemma 1] for the details of this linearization.

The basic reproduction number is well-known in epidemiology as the typical number of secondary infections that arise from a single infected individual, within an otherwise completely susceptible population. Diekmann, Heesterbeek, and Metz [4] introduced the next generation operator to compute this quantity in general models with structured populations. This approach was later applied by van den Driessche and Watmough [3] specifically to the compartmental model (1).

For a compartmental epidemic model (1a)-(1b) satisfying Assumptions 1 and 2 and with linearization (2) about (0 n , y * ), the basic reproduction number is

We refer the reader to [4, §2] and [3, §3] for derivations of (3) from the epidemiological definition of R 0 .

Geometric programs are a family of generally non-convex optimization problems that can be transformed into convex optimization problems by a change of variables. Geometric programs enjoy a multitude of applications in engineering and control theory, including the design of optimal positive systems [11] , a problem which is closely related to the resource allocation considered in this note. We refer the reader to [1] as a standard introduction to geometric programming and briefly introduce the key concepts in what follows.

A

A posynomial function is a sum of monomial functions. Note that posynomials are closed under addition and multiplication, and that a posynomial divided by a monomial is a posynomial. Given a posynomial function f 0 , a set of posynomial functions f i , i ∈ {1, . . . , m}, and a set of monomial functions g i , i ∈ {1, . . . , p}, a geometric program in standard form is:

The problem becomes convex after the change of variables x i → e yi . Off-the-shelf software is available for geometric programs, including the CVX package in MATLAB [5] .

We now reproduce three lemmas regarding properties of Metzler and Hurwitz matrices that will be necessary for our main results. The first lemma is a standard result characterizing the stability of Metzler matrices (see [2, Theorem 15 .17]):

Lemma 4 (Metzler Hurwitz Lemma). Let M ∈ R n×n be a Metzler matrix. The following are equivalent:

We borrow the next two results from [3] ; the first is a slight restatement of [3, Lemma 5], so we do not include a proof. The second result is abstracted from the proof of [3, Theorem 2] and we include a self-contained proof.

Lemma 6 (Stability of Perturbed Metzler Matrices). Let H ∈ R n×n be Metzler and Hurwitz, and let E ∈ R n×n ≥0 be a nonnegative perturbation matrix. The following are equivalent:

the Perron-Frobenius theorem guarantees that its dominant eigenvalue is real and non-negative, so −(I n + EH −1 ) has an eigenvalue with non-negative real part.

The main theoretical result of this paper is the following theorem, which provides two novel characterizations of R 0 :

The following are characterizations of the basic reproduction number:

(i) Stability characterization:

(ii) Geometric program characterization:

Proof. To prove that (4) follows from (3), we compute

where the first step follows from Lemma 6. We now use (4) to prove (5) .

In the last step, we note that the V w ≤ 0 n constraint is implied by (F +rV )w ≤ 0 n , so we are free to remove it. Manipulating the (F + rV )w ≤ 0 n constraint into the standard form for geometric programming yields (5). 

But there is no feasible point w > 0 2 that satisfies the inequality constraint with r = 1. Thus, in general, we cannot replace the infimum in (5) with a minimum.

The geometric program characterization (5) sets us up to efficiently optimize model parameters to either minimize or constrain R 0 . In a manner analogous to [12] , we consider two forms of the resource allocation problem: R 0 -constrained allocation, and budget-constrained allocation. In both forms of the resource allocation problem, we suppose that the model parameters F , V od , and V d depend on a vector of "resources" θ ≥ 0 k , and that the cost of a particular allocation of resources is given by a cost function c(θ). Furthermore, the resources must satisfy some collection of constraints h(θ) ≤ 1 q . The dependence on θ must obey the following conditions: Conds. (i) and (ii) are necessary to transcribe the allocation problem as a geometric program, while cond. (iii) ensures that the matrix parameters F , V od , and V d satisfy the antecedent of Theorem 7 for any feasible allocation. Cond. (iii) also ensures the feasible θ are confined to a compact set. Under these assumptions, for all θ ∈ h −1 ≤ (1 q ), the resource dependence of R 0 can be written as

Additional resources will typically reduce the rate of new infections or increase the rate at which existing infections are removed. This property is not included in Assumption 9, since it is not needed for any of the results in this section. However, if this property is true, then it is useful (albeit unsurprising) to note that R 0 (θ) is monotone decreasing in θ.

Lemma 10 (Monotonicity). Suppose that F (θ), V od (θ), and V d (θ) satisfy Assumption 9. If additionally F (θ) and V od (θ) are non-increasing and V d (θ) is non-decreasing in θ, then for

where the last inequality follows from Lemma 4 (ii), since V (θ) and V (θ ) are Hurwitz and Metzler, and thus

since the spectral radius is monotone increasing in the elements of a non-negative matrix.

We now define the two optimal allocation problems. In the R 0 -constrained allocation problem, we identify the cheapest allocation of resources to ensure that R 0 ≤ r max , where r max > 0 is some arbitrary threshold. In the budgetconstrained allocation problem, some budget c max > 0 is available to spend on resources, and we would like to deploy these limited resources to minimize R 0 .

Definition 11 (Optimal Allocation Problems). Let F (θ), V od (θ), V d (θ), c(θ), and h(θ) satisfy Assumption 9. We define the following optimization problems:

(i) Given r max > 0, we say that θ * is an optimal R 0constrained allocation if θ * is a minimizer of min θ≥0 k {c(θ) : h(θ) ≤ 1 q and R 0 (θ) ≤ r max } (7)

(ii) Given c max > 0, we say that θ * is an optimal budgetconstrained allocation if θ * is a minimizer of (6) is well-defined over the feasible sets; furthermore, R 0 (θ) is continuous, since the matrix inverse and spectral radius are continuous functions of the matrix elements. Thus the feasible sets are compact, so the minima of both problems exist. Using Theorem 7, we can construct a pair of geometric programs to solve for optimal R 0 -constrained and budgetconstrained allocations. For notational convenience, we define a map p : R >0 × R n >0 × R k ≥0 → R n >0 by p(r, w, θ) = diag(rV d (θ)w) −1 (F (θ) + rV od (θ))w (9)

Under Assumption 9, p(r, w, θ) is posynomial, so the following are geometric programs:

Problem 12 (R 0 -Constrained Allocation GP). Given r max > 0 and a precision parameter δ ≥ 0:

minimize : c(θ) variables : r > 0, w > 0 n , θ > 0 k subject to : p(r, w, θ) ≤ 1 n h(θ) ≤ 1 q r ≤ r max + δ Problem 13 (Budget-Constrained Allocation GP). Given c max > 0:

minimize : r variables : r > 0, w > 0 n , θ > 0 k subject to : p(r, w, θ) ≤ 1 n h(θ) ≤ 1 q c(θ) ≤ c max Theorem 14 (Geometric Program Transcription). Let θ * ≥ 0 k , r max > 0, and c max > 0. Let F 1 (δ) for δ > 0 and F 2 be the sets of feasible points (r, w, θ) for Problems 12 and 13.

The following are true:

(i) θ * is an optimal R 0 -constrained allocation if and only if the infimum of Problem 12 converges to c(θ * ) as δ → 0 + and there exists r * , w * such that (r * , w * , θ * ) ∈ cl(F 1 (δ)) for all δ > 0. (ii) θ * is an optimal budget-constrained allocation if and only if R 0 (θ * ) is the infimum of Problem 13 and there exists r * , w * such that (r * , w * , θ * ) ∈ cl(F 2 ).

See Appendix B for the proof. We note that Problem 12 is an arbitrarily accurate approximation of the R 0 -constrained allocation problem, controlled by the parameter δ ≥ 0. This approximation is necessary due to the closed inequality constraint on R 0 and the representation of R 0 by the infimum in (5), which is not always attained:

Remark 15 (Degenerate Cases, Pt. II). In some cases, Problem 12 may be infeasible when δ = 0, for example, if F (θ) = F and V (θ) = V are the matrices defined in Remark 8 and r max = 1. Fortunately, the feasible set is nonempty for all δ > 0, so we can still consider the limit of solutions to Problem 12 as δ → 0 + . This feasibility problem arises due to the constraint on R 0 , so it is not an issue in Problem 13.

In practice, the issue of an empty feasible set is not of significant concern, since numerical optimization already has inherently limited precision. We suggest solving Problem 12 with δ = 0 (and only using a small positive value if the solver reports primal infeasibility).

We adopt a standard multigroup SEIR model (with vital dynamics) for an epidemic in the state of California, where each group corresponds to one of the state's n = 58 counties. The SEIR model has two infected states (exposed and infectious) and two non-infected states (susceptible and recovered). Letting s, e, z, r ∈ R n ≥0 denote the expected number of people in each group and disease state, the model dynamics for each group i ∈ {1, . . . , n} arė

The model requires a matrix of inter-group contact rates A ∈ R n×n ≥0 , as well as parameters λ, β, µ, γ, δ > 0 n , described in Table I . We selected baseline values roughly in agreement with the COVID-19 pandemic in the United States. However, we emphasize that the intention of this section is to showcase potential applications of our resource allocation framework, not to make specific predictions or policy recommendations regarding the current pandemic.

We estimated A using data from SafeGraph. 1 In particular, we used the Social Distancing Metrics dataset to estimate a matrix P ∈ R n×n , where p ij is the daily fraction of people from county i who visited county j, averaged over each day in 2020. Then (P P T ) ij approximates the probability that two random individuals from counties i and j are co-located in the same county on a given day. For simplicity, we set A = αP P T , where the scalar α = 2.3667 × 10 −7 was chosen to ensure R 0 = 2.5 in the pre-intervention model.

It is clear that the model has a unique disease-free equilibrium (s * , 0 n , 0 n , 0 n ), where s * i = λ i /µ i for each i. One can also verify that this equilibrium point satisfies Assumption 2. Linearizing about this point, we obtain

Because the diag(β) diag(s * )A term is the only one corresponding to the creation of new infections, we decompose this Jacobian into the two matrices

where F is non-negative and V is Hurwitz and Metzler.

We consider the following optimal resource allocation scenario from [12] , in which there are two types of pharmaceutical interventions: vaccines, which reduce the local transmission rates β i ; and antidotes, which increase the local recovery rates δ i . By allocating vaccines to patch i, we can optimize the local transmission rate within a range β i ∈ [β i , β i ], where β i ≥ β i > 0. The cost of this vaccine allocation is, for all i,

Note that the most aggressive allocation has a cost of f i (β i ) = 1, while allocation of no vaccines at all has a cost f i (β i ) = 0.

The form of (10) ensures diminishing returns in the investment of vaccines at each patch. Similarly, by allocating antidotes to patch i, the local recovery rate can be optimized in the range

The cost of the antidote allocation is, for all i,

where the parametersδ i > δ i control the shape of the cost curve. The total cost, summing over the local costs of vaccines and antidotes over all patches, is constrained by a budget c max . Appendix C contains the details on writing this allocation problem as a budget-constrained geometric program. For this case study, we selected uniform parameters of β i = 0.01, β i = 0.1, δ i = 0.1, δ i = 0.5, andδ i = 1.

We performed the budget-constrained resource allocation to minimize R 0 and the abscissa for values of c max between 0.5 and 10. Figure 1 plots various quantities from these allocations. As the top two plots show, both the R 0 and the abscissa decrease as the budget increases, regardless of whether R 0 or the abscissa are targets for minimization. However, the bottom left plot reveals a major difference in how resources are allocated for these two different objectives: when minimizing R 0 , the majority of the budget is spent on vaccines, but when minimizing the abscissa, the majority of the budget is spent on antidotes (at least in this particular case study). The bottom right plot also highlights the difference in efficacy between these two allocation strategies, as minimizing R 0 leads to a smaller cumulative number of infections at the end of the epidemic (as computed from the linear model). All of the code used to generate these results (as well as additional plots) is available online. 2

In this note, we have established a new formula for the basic reproduction number of a compartmental epidemic model. We 2 The MATLAB script and functions used to generate these results is available at https://www.mathworks.com/matlabcentral/fileexchange/ 99354-geometric-programs-for-r0. Running the code requires an installation of CVX 2.2 and the MOSEK solver. have then applied this formula to resource allocation problems that minimize or constrain R 0 , transcribing these problems as geometric programs. We have argued that R 0 is a more desirable objective for resource allocation than the spectral abscissa, and we have provided a numerical case study to highlight that targeting R 0 instead of the abscissa can result in qualitatively different solutions. The possible applications of our optimization framework are broad, since it works for a general class of epidemic models and cost functions. Of course, policymakers should be aware of the limitations of optimal resource allocation: models (and linear models in particular) have limited accuracy, and mathematics does not address the complex social considerations of epidemic response. Nonetheless, we believe that this work and its future extensions-coupled with judicious choices of models and cost functions-can provide useful insight for epidemic preparedness and response.

Lemma 16 (A Relaxing Lemma). Let V ∈ R n×n be a Metzler and Hurwitz matrix, and let F be a non-negative matrix of the same shape (that has at least one positive element). Let W = {w > 0 n : V w < 0 n } andŴ = {w > 0 n : V w ≤ 0 n }, and define

Proof. It is obvious thatR 0 ≤ R 0 , so we need only show that R 0 ≥ R 0 . Before we embark on this task, we will construct useful expressions forR 0 and R 0 . Let I be the (possibly empty) set of indices for which F (i) = 0 n . For any w ∈Ŵ , observe that {r > 0 : (F + rV )w ≤ 0 n } is non-empty if and only if (V w) i = 0 implies that i ∈ I. Thus, we definē

where W ⊂W ⊂Ŵ . Then we can writê

It is straightforward to solve for r * (w):

Similar toR 0 , we have the following expression for R 0 :

The remainder of the proof is to show that R 0 is a lower bound onR. Letr ∈R, so thatr > 0 and (F +rV )ŵ ≤ 0 n for someŵ ∈W . Let x > 0 n such that V x < 0 n (which must exist because V is Hurwitz), and for all t ≥ 0, let w(t) = w + tx. We can also show that

Then for all t > 0,

Thus, given any > 0, we can choose t < (max i∈I c κ i ) −1 to ensure that |r * (w(t)) − r * (ŵ)| < . Because w(t) ∈ W for all t > 0, it is the case that r * (w(t)) ∈ R for all t > 0, so that every open ball around r * (ŵ) contains a point in R.

Then r * (ŵ) ∈ cl(R), which implies that r * (ŵ) ≥ R 0 . But r * (ŵ) ≤r, andr was chosen arbitrarily fromR, so R 0 is a lower bound onR. ButR 0 is the greatest such lower bound, so we conclude thatR 0 ≥ R 0 .

Let G 1 , G 2 be the sets of feasible points θ for (7) and (8), respectively. We define a Metlzer matrix

Since the determinant of M (r, θ) is a polynomial in r of degree n, for some scalars a 1 , a 2 , . . . , a n ∈ C, we can write |M (r, θ)| = (r − a 1 )(r − a 2 ) · · · (r − a n )

Due to (4) in Theorem 7, M (R 0 (θ), θ) must be singular, so R 0 (θ) is a root; then we can assign a 1 , a 2 , . . . , a = R 0 (θ) up to some multiplicity . Define a "pseudo-determinant" µ(r, θ) = (r − a +1 ) · · · (r − a n ) as the product of the remaining factors, which is real and nonzero for all r ≥ R 0 (θ). Then

, ∀r > R 0 (θ)

Now, pick z > 0 n arbitrarily, and let w(r, θ) = −(r − R 0 (θ)) M −1 (r, θ)z, ∀r > R 0 (θ) (13) and define

For any r > R 0 (θ), (4) in Theorem 7 implies that M (r, θ) is Hurwitz, so −M −1 (r, θ) ≥ 0, and thus w(r, θ) > 0 n . Furthermore, M (r, θ)w(r, θ) < 0 n , so expanding M (r, θ) with (12) and re-arranging, we obtain p(r, w(r, θ), θ) < 1 n . We now use w * (θ) to formally establish relationships between the feasible sets of both pairs of optimization problems:

Lemma 17 (Relating the Feasible Sets). For each δ > 0, let Θ 1 (δ) ⊂ R k be the set of θ such that (r, w, θ) ∈ cl(F 1 (δ)) for some r, w. Similarly, let Θ 2 ⊂ R k be the set of θ such that (r, w, θ) ∈ cl(F 2 ) for some r, w. The following are true: (i) θ ∈ G 1 implies that (R 0 (θ), w * (θ), θ) ∈ cl(F 1 (δ)) for all δ > 0, (ii) θ ∈ G 2 implies that (R 0 (θ), w * (θ), θ) ∈ cl(F 2 ),

Proof. To prove (i), let θ ∈ G 1 , so h(θ) ≤ 1 q and R 0 (θ) ≤ r max . Fix any δ > 0, and let > 0. By (14), we can choose r > R 0 (θ) such that ||w(r, θ) − w * (θ)|| < and |r − R 0 (θ)| < min{δ, }. Since p(r, w(r, θ), θ) ≤ 1 q and r < R 0 (θ) + δ ≤ r max + δ, we have (r, w(r, θ), θ) ∈ F 1 (δ), so every neighborhood of (R 0 (θ), w * (θ), θ) (by choice of ) contains a point in F 1 (δ). We prove (ii) by a similar argument (without δ), noting that θ ∈ G 2 implies c(θ) ≤ c max .

To prove (iii), we note that (i) implies that G 1 ⊆ δ>0 Θ 1 (δ). If θ ∈ Θ 1 (δ) for all δ > 0, then h(θ) ≤ 1 q and R 0 (θ) ≤ r max + δ for all δ > 0, which implies R 0 (θ) ≤ r max , and thus θ ∈ G 1 . Hence G 1 ⊇ δ>0 Θ 1 (δ) as well. Statement (iv) follows from a similar argument.

Proof of Theorem 14. In order to prove (i), we first define c * (δ) as the infimum of Problem 12 for all δ > 0, and we define c * as the minimum cost of the R 0 -constrained allocation problem. Then, noting that Θ 1 (δ) are nested downward as δ → 0, c * = min G 1 = min δ>0 Θ 1 (δ) = lim δ→0 + min Θ 1 (δ) = lim δ→0 + c * (δ) The second step is due to Lemma 17, and the third step is a general property of intersections of nested sets. Let θ * be an optimal R 0 -constrained allocation. Then θ * ∈ G 1 , so by Lemma 17, (R 0 (θ * ), w * (θ * ), θ * ) ∈ cl(F 1 (δ)) for all δ > 0, and we have shown that c * (δ) → c * = c(θ * ) as δ → 0 + . On the other hand, if there exist r * , w * such that (r * , w * , θ * ) ∈ cl(F 1 (δ)) for all δ > 0, then Lemma 17 guarantees θ * ∈ G, and c * (δ) → c(θ * ) implies that c(θ * ) = c * .

We now prove (ii). Let θ * be an optimal budget-constrained allocation. Then θ * ∈ G 2 , so Lemma 17 implies that (R 0 (θ * ), w * (θ * ), θ * ) ∈ cl(F 2 ). Consider any other point (r, w, θ) ∈ cl(F 2 ), and note that Lemma 17 also implies θ ∈ G 2 , so that R 0 (θ * ) ≤ R 0 (θ). But R 0 (θ) ≤ r by (5), so R 0 (θ * ) ≤ r. Thus R 0 (θ * ) is the min value of r over cl(F 2 ).

Finally, suppose that (R 0 (θ * ), w * (θ * ), θ * ) ∈ cl(F 2 ) and that R 0 (θ * ) is the infimum of Problem 13. Lemma 17 guarantees that θ * ∈ G 2 . Consider any other point θ ∈ G 2 , and note that (R 0 (θ), w * (θ), θ) ∈ cl(F 2 ) as well, so that R 0 (θ * ) ≤ R 0 (θ). Therefore θ * is a minimizer for (8), so it is an optimal budget-constrained allocation.

We must encode the following budget constraint in the standard form for geometric programming:

Note that g i have non-posynomial dependence on δ i , so we must replace 1 − δ i with auxiliary variables η i , constrained bỹ δ i − δ i ≤ η i ≤δ i − δ i . Then the budget constraint can be written in the standard form

where

is a positive constant. Next, noting that the dependence of V d on δ is non-monomial, we introduce another set of auxiliary variables σ i constrained by

and define V d (σ) = diag(µ + γ) 0 0 diag(σ)

Since V d (σ) is increasing in σ (and V od and F have no dependence), Lemma 10 ensures that R 0 is decreasing in σ, so minimizing R 0 will drive σ to the upper bound σ i = 1 + µ i − η i = µ i + δ i . Altogether, the resource vector is θ T = β T η T σ T , and h(θ) ≤ 1 q encodes the constraints (15) , and (16).

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