Inverse conic linear programs in Banach spaces∗

Archis GhateDepartment of Industrial & Systems Engineering

BOX 352650, University of Washington, Seattle, WA 98195, USAarchis@uw.edu

Abstract

Given the costs and a feasible solution for a finite-dimensional linear program (LP), inverseoptimization involves finding new costs that are close to the original and that also render thegiven solution optimal. Ahuja and Orlin employed the absolute sum norm and the maximumabsolute norm to quantify distances between cost vectors, and applied duality to establishthat the inverse LP problem can be formulated as another finite-dimensional LP. This wasrecently extended to semi-infinite LPs, countably infinite LPs, and finite-dimensional conicoptimization problems. These works provide sufficient conditions so that the inverse problemalso belongs to the same class as the forward problem. This paper extends this result to conicLPs in potentially infinite-dimensional Banach spaces. Moreover, the paper presents concretederivations for continuous conic LPs, whose special cases include continuous linear programsand continuous conic programs; normed cone programs in Banach spaces, which include second-order cone programs as a special case; and semi-definite programs in Hilbert spaces. Thesederivations reveal the sharper result that, in each case, the inverse problem belongs to thesame specific subclass as the forward problem. Instances where existing forward algorithmscan then be adapted to solve the inverse problems are identified. Results in this paper mayenable the application of inverse optimization to as yet unexplored areas such as continuous-time economic planning, continuous-time job-shop scheduling, continuous-time network flow,maximum flow with time-varying edge-capacities, and wireless optimization with time-varyingcoverage requirements.

Keywords: infinite-dimensional optimization; duality; Hilbert spaces; inverse optimization

1 Introduction

Inverse optimization was studied first by geophysicists [47, 48, 63, 71]. Tarantola [63] described itas follows: “To solve the inverse problem is to infer the values of the model parameters from givenobserved values of the observable parameters.” Applications of inverse optimization include trans-portation [27, 28], demand management [20], auctions [12], production planning [65], healthcare[10, 24, 29], and finance [14].

The mathematical programming community started working on inverse optimization when[18, 19] studied inverse shortest path problems. Since then, inverse problems have been stud-ied in finite-dimensional linear programs (LPs) [2, 22, 64] and conic programs [40], combinatorialoptimization [38], network optimization [72, 73], multi-objective optimization [21, 23, 33, 46, 57], in-teger programs [60], mixed-integer programs [67], separable convex programs with linear constraints[74], polynomial optimization [43], and machine learning/statistics [25, 62].

∗Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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This paper builds upon the seminal work of Ahuja and Orlin [2] on inverse optimization in finite-dimensional LPs. They studied the following problem: given a cost vector c and a feasible solutionx, find a cost vector h that is as close as possible to c and also makes x optimal. They showedusing LP duality that when distances between cost vectors are measured via either the weightedl1 or weighted l∞ metric, the inverse problem can be reformulated as another finite-dimensionalLP. More concrete results for network problems such as minimum cost flow and shortest pathwere recovered as special cases. Iyengar and Kang [40] were able to extend the Ahuja and Orlinresult to certain finite-dimensional conic programs. In particular, the inverse optimization problem(henceforth simply termed “the inverse problem”) for a finite-dimensional conic program can bereformulated as another finite-dimensional conic program under an affine-structure assumption onthe gradient of the objective function, by using the lq metric (q ≥ 1) to quantify distances betweenparameters. This included finite-dimensional LPs, second-order cone programs, and semi-definiteprograms as special cases.

The author’s recent efforts have focused on extending the Ahuja and Orlin framework to infinite-dimensional LPs. First, he provided sufficient conditions under which it can be generalized tocountably infinite LPs [34]. These are problems where both the number of constraints and thenumber of variables are countably infinite. Nourollahi and Ghate [49] then applied this theoreticalfoundation to minimum cost flow problems in networks with a countably infinite number of nodesand arcs. The author recently extended the Ahuja and Orlin framework to semi-infinite linearprograms [35]. There, the number of variables was finite, but the number of constraints was eithercountably or uncountably infinite. In each of these three papers, the key result was that the inverseproblem belongs to the same class as the forward.

This paper continues this stream of the author’s work in infinite-dimensional spaces. The paperwas inspired by the work of Shapiro [61], where he described a duality framework for conic LPsin topological vector spaces. The research objective of the present work is to apply methods fromfunctional and convex analysis to prove that the inverse problem for such conic LPs belongs to thesame class as the forward problem. This is achieved in an abstract setting in Section 3. The maincontributions of the paper appear in Sections 4-6, where concrete formulations of both the forwardand the inverse problems are developed in detail. To the best of the author’s knowledge, theseforward formulations themselves are not available in the literature and hence special care is neededto identify appropriate normed spaces, cones, and duals before presenting the inverse formulationin each case. This additional work on concrete special cases pays off with the stronger result thatthe inverse problem is not just a conic LP, but in fact, belongs to the same specific subclass asthe forward problem. That is, an inverse continuous conic LP is a continuous conic LP (Section4). An inverse normed cone program is a normed cone program; and as a special case, an inversesecond-order cone program in a Hilbert space is also a second-order cone program (Section 5).Finally, an inverse semi-definite program in a Hilbert space is also a semi-definite program (Section6).

2 A dual pair of conic linear programs

Relevant duality results on conic LPs are recalled here from Shapiro [61] in the present notation

and with added tailored exposition. The symboldef= means “is defined as”; vector spaces are written

using block letters such as X; convex cones are denoted by capital letters such as X; and elements(vectors) of vector spaces are written using small letters such as x.

Let X be a vector space over the field of real numbers. A subset X ⊆ X is called a cone if x ∈ Ximplies λx ∈ X for all nonnegative real numbers λ. If, in addition, x, y ∈ X implies x + y ∈ X,

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then X is called a convex cone. It is easy to check that a convex cone is a convex set, as the namesuggests. The possibility that λ = 0 implies that X includes the null vector 0X ∈ X.

Suppose X and X′ are two vector spaces. Let 〈·, ·〉 : X′ × X → < be a bilinear form. That is,〈x′, ·〉 is a linear function over X for each fixed x′ ∈ X′, and 〈·, x〉 is a linear function over X′ for eachfixed x ∈ X. Suppose that (i) for each 0X 6= x ∈ X, there exists an x′ ∈ X′ such that 〈x′, x〉 6= 0;and (ii) for each 0X′ 6= x′ ∈ X′, there exists an x ∈ X such that 〈x′, x〉 6= 0. Then (X,X′) is called adual pair, and so is (X′,X) by symmetry. Properties (i) and (ii) are summarized by saying that thedual pair separates points. Observe that linearity of 〈x′, ·〉 over X implies that 〈x′, 0X〉 = 0 for eachfixed x′ ∈ X′. Similarly, 〈0X′ , x〉 = 0 for each fixed x ∈ X. Perhaps one of the simplest examples is

the dual pair (<n,<n) with the bilinear form 〈x′, x〉 =n∑i=1

x′ixi. Another standard example is the

dual pair (`∞, `1) with the bilinear form 〈x′, x〉 =∞∑i=1

x′ixi. Anderson and Nash [6] discuss more

details about dual pairs of vector spaces in their Section 3.2. Also see Definition 5.90 of [3].Broadly speaking, an optimization problem where the decision variables belong to cones, other

constraints enforce affine (linear plus constant) mappings of variables to lie in cones, and theobjective function equals a bilinear form, are called conic LPs. Suppose (X,X′) is a dual pair ofvector spaces. Let Y be another vector space and A : X → Y be a linear mapping. Linearity of Aimplies that A(0X) = 0Y . Suppose b ∈ Y and c ∈ X′ are fixed vectors, and let X be a convex conein X. Similarly, let Y be a convex cone in Y . Then, the optimization problem

(P ) ν?(P )def= inf

x〈c, x〉

A(x) + b ∈ Yx ∈ Xx ∈ X

is called a conic LP in this paper. Problem (P ) is seen as the forward problem in this paper.Suppose Y ′ is a vector space such that (Y ,Y ′) forms a dual pair along with a bilinear form 〈·, ·〉.

Throughout this paper, the notation 〈·, ·〉 is employed to denote several bilinear forms regardless ofthe corresponding dual pair of vector spaces. The dual pair corresponding to a particular bilinearform will be clear from context. The following standard assumption is enforced throughout.

Assumption 1. For each y′ ∈ Y ′, ∃ x′ ∈ X′ such that 〈y′, A(x)〉 = 〈x′, x〉 ∀x ∈ X.

The x′ in the above assumption, when it exists, is unique. Suppose not. Then there are two distinctx′, x′ ∈ X′ such that 〈y′, A(x)〉 = 〈x′, x〉 and 〈y′, A(x)〉 = 〈x′, x〉 for all x ∈ X. Subtracting yields〈x′, x〉 − 〈x′, x〉 = 0 for all x ∈ X. That is, 〈x′ − x′, x〉 = 0 for all x ∈ X. But since x′ − x′ 6= 0, thiscontradicts the fact that the dual pair (X,X′) with the bilinear form 〈·, ·〉 separates points.

Assumption 1 delivers the following definition of an adjoint map A′ : Y ′ → X′. For each y′ ∈ Y ′,A′(y′) equals that unique x′ ∈ X′ for which 〈y′, A(x)〉 = 〈x′, x〉 for all x ∈ X. In other words,

〈y′, A(x)〉 = 〈A′(y′), x〉 ∀x ∈ X (1)

defines the adjoint. Conversely, if an adjoint that satisfies (1) exists, then it delivers an x′ = A′(y′)that satisfies Assumption 1. In this sense, the assumption is equivalent to the existence of an adjointthat satisfies Equation (1). As such, the assumption is essential for deriving a dual problem. Astandard argument shows that the adjoint is linear. Linearity of the adjoint implies that A′(0Y ′) =0X′ .

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Let X ′ ⊆ X′ denote the dual of the convex cone X. That is,

X ′def={x′ ∈ X′ : 〈x′, x〉 ≥ 0, ∀x ∈ X

}. (2)

This dual is easily shown to be a convex cone.Let Y ′ ⊆ Y ′ denote the dual of the convex cone Y defined similar to Equation (2), and −Y ′ be

its negative. That is, −Y ′ def= {−y′ : y′ ∈ Y ′}. This is also a convex cone.

The Lagrangian dual of (P ) can then be written as

(D) ν?(D)def= sup

y′〈b, y′〉

A′(y′) + c ∈ X ′

y′ ∈ −Y ′

y′ ∈ Y ′.

This is also a conic LP. Proofs of the next two results are standard and hence omitted (see Shapiro[61]; Bonnans and Shapiro [16], Section 2.5.6).

Theorem 1. [Weak duality] If x is feasible to (P ) and y′ is feasible to (D), then 〈c, x〉 ≥ 〈b, y′〉.Specifically, ν?(P ) ≥ ν?(D).

Weak duality has an immediate consequence.

Corollary 1. Suppose x is feasible to (P ) and y′ is feasible to (D) such that 〈c, x〉 = 〈b, y′〉. Then,x is optimal to (P ) and y′ is optimal to (D).

The inverse framework in the next section relies on these two duality results.

3 Inverse optimization with costs in a normed vector space

Now suppose that X′ is a normed space with norm || · ||X′ (Luenberger [44], Section 2.6). A fixedc ∈ X′, and an x ∈ X that is feasible to (P ) are given. For each h ∈ X′, let (Ph) denote the conicLP that is identical in form to (P ) but with cost coefficients h. This subscript h is included foremphasis. Similarly, (Dh) denotes the dual conic LP of (Ph). Weak duality as in Theorem 1 andits consequence in Corollary 1 hold for this primal-dual pair.

Consider the following inverse optimization problem: find the closest distance to c over allh ∈ X′ that make x optimal to (Ph) with no duality gap. That is, solve

(Q) ν?(Q)def= inf

h,y′||c− h||X′ (3)

〈b, y′〉 − 〈h, x〉 = 0 (4)

A′(y′) + h ∈ X ′ (5)

y′ ∈ −Y ′ (6)

y′ ∈ Y ′ (7)

h ∈ X′. (8)

Constraint (4) ensures that the objective values of x in (Ph) and y′ in (Dh) match (no duality gap);constraints (5-7) ensure that y′ is feasible to (Dh). Since x is given to be feasible to (Pc) and henceis feasible to (Ph), constraints in the above formulation guarantee by Corollary 1 that x is optimal

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to (Ph). The trivial solution h = 0X′ , y′ = 0Y ′ is feasible to this problem. That is, this solution

renders x optimal to (P0X′ ) and 0Y ′ optimal to (D0X′ ). The objective function is nonnegative andhence ν?(Q) ≥ 0. The feasible solution h = 0X′ yields the upper bound ν?(Q) ≤ ||c||X′ . These lastthree sentences hold regardless of the specific x that is given. In particular, although there maybe a duality gap between a conic LP and its dual in general, the existence of a (trivial) feasiblesolution to constraint (4) and hence to problem (Q) is guaranteed.

Formulation (3-8) can be converted into a conic LP via algebraic modifications. Substitute thevariable transformation c− h = u and rewrite (Q) as

(Q) ν?(Q) = infu,y′,r

r (9)

||u||X′ ≤ r (10)

〈b, y′〉+ 〈u, x〉 − 〈c, x〉 = 0 (11)

A′(y′)− u+ c ∈ X ′ (12)

y′ ∈ −Y ′ (13)

y′ ∈ Y ′ (14)

u ∈ X′ (15)

r ∈ <. (16)

Define the subset W of the vector space (X′ × <) as Wdef= {(u, r) ∈ (X′ × <) : ||u||X′ ≤ r}. This

subset is a convex cone of (X′×<). In this paper, such subsets will be called norm cones followingExample 2.25 of [17]; the phrase second-order cone/Lorentz cone/ice cream cone is reserved for thespecial case of L2 norms. Constraint (10) can be equivalently written in cone format as (u, r) ∈W .Finally, the singleton {0} is a trivial cone in <, and hence constraint (11) can be equivalentlywritten in cone format as 〈b, y′〉 + 〈u, x〉 − 〈c, x〉 ∈ {0}. In summary, problem (Q) is a conic LPthat reads

(Q) ν?(Q) = infu,y′,r

r (17)

(u, r) ∈W (18)

〈b, y′〉+ 〈u, x〉 − 〈c, x〉 ∈ {0} (19)

(12− 16). (20)

Note again that the solution u = c, y′ = 0Y ′ , r = ||c||X′ is feasible to this conic LP. Also, any(u, y′, r) that is feasible to this conic LP renders x optimal to problem (Ph) with h = c− u and y′

optimal to the corresponding dual problem (Dh).The next three sections provide detailed derivations of special cases of the abstract framework

here. In each case, a concrete conic LP and its dual are described along with appropriate choicesfor the dual pairs (X,X′), (Y ,Y ′), and also for convex cones X,X ′, Y, Y ′. In each case, it is shownthat the inverse conic LP belongs to the same specific subclass as the forward problem.

4 Continuous conic linear programs

Bellman [13] first studied a special type of continuous LP called the bottleneck problem. This wasmotivated by an economic planning application. Anderson [5] introduced a so-called “separated”continuous LP model for job-shop scheduling. Informally, in separated problems, constraints thatinclude time-integrals of variables do not include instantaneous values of those variables, and vice

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versa. Anderson et al. [7] investigated the structure of extreme points for separated continuousLPs, and Anderson and Philpott [8] utilized this structure to devise a simplex-type algorithm forLPs on continuous-time networks. Anderson and Nash studied LPs on continuous-time networksand separated continuous LPs in Chapters 6 and 7 of their book [6]. Other works on separatedand nonseparated continuous LPs include [11, 31, 36, 39, 45, 50, 51, 52, 53, 54, 55, 56, 69]. Wanget al. [68] studied a generalization of the separated continuous LP described on page 135 of theAnderson and Nash book [6], and in Section 5 of Shapiro [61]. They called it a separated continuousconic program, which was motivated by an application from dynamic control of resources in a fluidnetwork with quadratic costs.

As a further generalization of the separated continuous LP from Anderson and Nash [6] and ofthe separated continuous conic program from Wang et al. [68], this section considers problems ofthe form

(CCLP ) infx

1∫0

c>(t)x(t)dt

t∫0

M(s, t)x(s)ds+ L(t)x(t) + b(t) ∈ K1, for a. e. t ∈ [0, 1]

x(t) ∈ K2, for a. e. t ∈ [0, 1].

Here, c(t) is a column vector in <n for each t ∈ [0, 1]; b(t) is a column vector in <m for eacht ∈ [0, 1]; M is an m× n matrix for each (s, t) ∈ [0, 1]× [0, 1]; and L(t) is an m× n matrix for eacht ∈ [0, 1]. For each fixed i, j, for i = 1, . . . ,m and j = 1, . . . , n, component Mij(s, t) of this matrixis a Lebesgue measurable and integrable function of (s, t) ∈ [0, 1] × [0, 1]. Similarly, for each fixedi, j, for i = 1, . . . , k and j = 1, . . . , n, component Lij(t) of matrix L is a Lebesgue measurable andintegrable function of t ∈ [0, 1]. The decision variables are column vectors x(t) ∈ <n. All integralsare Lebesgue and “a.e.” means “almost every” with respect to the Lebesgue measure (Rudin [58],Chapter 11). Convex cone K1 is in <m; convex cone K2 is in <n. This problem is not necessarilyseparated because time-integrals of variables x appear in the same constraint as their instantaneousvalues.

Let L∞[0, 1] denote the Banach space of all essentially bounded Lebesgue measurable functionson the interval [0, 1]; Ln∞[0, 1] is the n-dimensional Cartesian product of these vector spaces. Whenequipped with the natural maximum (over the n component norms) norm, Ln∞[0, 1] becomes aBanach space. Decision variables x are embedded in this Banach space Ln∞[0, 1]. Let L1[0, 1]denote the Banach space of all Lebesgue measurable and integrable functions on the interval [0, 1];Ln1 [0, 1] is the n-dimensional Cartesian product of these vector spaces. Again, when equipped withthe natural maximum norm, Ln1 [0, 1] becomes a Banach space. The cost function c is embeddedin this Banach space Ln1 [0, 1]. Note here that Ln∞[0, 1] is the normed dual of Ln1 [0, 1], and thus itis appropriate to consider X ≡ Ln∞[0, 1] and X′ ≡ Ln1 [0, 1] as a dual pair with the bilinear form

〈x′, x〉 def=

1∫0

[x′(t)]>[x(t)]dt =1∫0

( n∑j=1

x′j(t)xj(t))dt. Similarly, the constraint function b is embedded

in the Banach space Y ≡ Lm1 [0, 1] with its natural maximum norm. This is paired with the Banachspace of dual variables Y ′ ≡ Lm∞[0, 1], with its natural maximum norm. The corresponding bilinear

form is 〈y′, y〉 def=

1∫0

[y′(t)]>[y(t)]dt.

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Consider a mapping A given by

A(x) =

t∫0

M(s, t)x(s)ds+ L(t)x(t), t ∈ [0, 1]. (21)

For any x ∈ Ln∞[0, 1], the output A(x) produced by this mapping belongs to Lm1 [0, 1]. Specifically,

for any i = 1, . . . ,m, componentt∫

0

( n∑j=1

Mij(s, t)xj(s))ds+

n∑j=1

Lij(t)xj(t) is in L1[0, 1]. To see why

this holds, observe that

1∫0

∣∣∣ t∫0

( n∑j=1

Mij(s, t)xj(s)ds)

+

n∑j=1

Lij(t)xj(t)∣∣∣dt

≤1∫

0

( t∫0

( n∑j=1

∣∣Mij(s, t)∣∣∣∣xj(s)∣∣ds)+

n∑j=1

∣∣Lij(t)∣∣∣∣xj(t)∣∣)dt≤ B

( 1∫0

t∫0

n∑j=1

∣∣Mij(s, t)∣∣dsdt+

1∫0

n∑j=1

∣∣Lij(t)∣∣dt), by factoring out a bound on x

= B( n∑j=1

1∫0

t∫0

∣∣Mij(s, t)∣∣dsdt+

n∑j=1

1∫0

∣∣Lij(t)∣∣dt), interchange integration and sum

≤ B( n∑j=1

1∫0

1∫0

∣∣Mij(s, t)∣∣dsdt+

n∑j=1

1∫0

∣∣Lij(t)∣∣dt), including extra nonnegative terms

<∞.

This derivation uses the fact that there exists an upper bound Bj such that |xj(s)| ≤ Bj for almost

every (in the Lebesgue measure sense) s ∈ [0, 1], and hence xj(s) ≤ Bdef= max

j=1,...,nBj for almost

every s ∈ [0, 1]. It also utilizes the fact that Mij(s, t) is Lebesgue integrable on (s, t) ∈ [0, 1]× [0, 1].Similarly for Lij(t) on t ∈ [0, 1]. In summary, the mapping A is an operator from Ln∞[0, 1] intoLm1 [0, 1]. It is not difficult to verify that this is a linear operator.

The adjoint of A is next derived to later help write the dual of (CCLP ). The adjoint A′ shouldsatisfy

〈A(x), y′〉 = 〈x,A′(y′)〉, ∀x ∈ Ln∞[0, 1], y′ ∈ Lm∞[0, 1]. (22)

The next lemma delivers such an adjoint.

Lemma 1. The adjoint A′ of the linear operator A defined in (21) is given by

A′(y′) =

s∫0

M>(s, t)y′(t)dt+ L>(s)y′(s). (23)

Proof. The definition of the bilinear form 〈·, ·〉 on the left hand side of (22) yields

〈A(x), y′〉 =

1∫0

t∫0

M(s, t)x(s)ds+ L(t)x(t)

> y′(t)dt7

=

1∫0

t∫0

x>(s)M>(s, t)ds

y′(t)dt+

1∫0

x>(t)L>(t)y′(t)dt

=

1∫0

x>(s)

s∫0

M>(s, t)y′(t)dt

ds+

1∫0

x>(s)L>(s)y′(s)ds

=

1∫0

x>(s)

s∫0

M>(s, t)y′(t)dt+ L>(s)y′(s)

ds=

1∫0

x>(s)

s∫0

M>(s, t)y′(t)dt+ L>(s)y′(s)

ds = 〈x,A′(y′)〉.

Here, the interchange in the order of integration in the third equality is allowed by Fubini’s Theorem(Billingsley [15], page 231). Thus, the adjoint proposed in (23) is correct as per Equation (22).

Problem (CCLP ) utilizes the convex cones

X = {x ∈ Ln∞[0, 1] : x(t) ∈ K2, for a. e. t ∈ [0, 1]} , (24)

andY = {y ∈ Lm1 [0, 1] : y(t) ∈ K1, for a. e. t ∈ [0, 1]} (25)

in the primal variable and constraint spaces, respectively. The next two lemmas provide intuitivecharacterizations of the duals of these two cones.

Lemma 2. The dual cone of X in Equation (24) is given by

X ′ = {x′ ∈ Ln1 [0, 1] : x′(t) ∈ K′2, for a. e. t ∈ [0, 1]}. (26)

Proof. By definition, the dual cone equals

X ′ ={x′ ∈ Ln1 [0, 1] :

1∫0

[x′(t)]>[x(t)]dt ≥ 0, ∀x ∈ X}. (27)

For any x′ ∈ X ′, it is immediate that1∫0

[x′(t)]>[x(t)]dt ≥ 0 for each x ∈ X, because [x′(t)]>[x(t)] ≥ 0

for almost every t ∈ [0, 1] by definition of the dual K′2 of K2. Thus, X ′ ⊆ X ′.The other inclusion that X ′ ⊆ X ′ is established next by contrapositive. Suppose x′ /∈ X ′. That

is, there is a set S ⊆ [0, 1] of positive Lebesgue measure such that x′(t) /∈ K′2 for any t ∈ S. Thatis, for each t ∈ S, there is a 0 6= x(t) ∈ K2 such that [x′(t)]>[x(t)] < 0. Then there must exist anε > 0 and a corresponding set T ⊆ [0, 1] of positive Lebesgue measure such that, for each t ∈ Tthere is a 0 6= x(t) with [x′(t)]>[x(t)] ≤ −ε. Define a set-valued mapping Ψ : [0, 1] ⇒ <n as

Ψ(t) =

{{x(t) ∈ K2 : [x′(t)]>[x(t)] ≤ −ε

}, if t ∈ T ,

{0}, if t ∈ [0, 1] \ T .

For each t ∈ [0, 1], Ψ(t) is nonempty and closed in the usual topology of <n. Since x′ is Lebesguemeasurable, so is Ψ (Aubin and Frankowska [9], Theorem 8.2.9). Consequently, there exists a

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measurable selection x : [0, 1] → <n of Ψ (Aubin and Frankowska [9], Definition 8.1.2, Theorem8.1.3). Use this selection to construct a function x∗ that takes values in <n such that

x∗(t) =

x(t)

maxj=1,...,n

|xj(t)| , if t ∈ T

0, if t ∈ [0, 1] \ T .

Observe that x∗ is in Ln1 [0, 1]. This holds because, for any j = 1, . . . , n, the integral1∫0

|x∗j (t)|dt

equals∫T

|xj(t)|max

j=1,...,n|xj(t)|dt ≤

∫Tdt, which in turn equals the (finite) Lebesgue measure of T . Since

x∗(t) is formed by scaling x(t) ∈ K2, for t ∈ S, x∗(t) is also in K2. Furthermore, x∗(t) = 0 for

t ∈ [0, 1]\T . Thus, x∗ is in X by definition of X. Then,1∫0

[x′(t)]>[x∗(t)]dt =∫T

[x′(t)]>[x∗(t)]dt < 0.

Thus, x′ /∈ X ′.This completes the proof that X ′ = X ′.

Lemma 3. The dual cone of Y is given by

Y ′ = {y′ ∈ Lm∞[0, 1] : y′(s) ∈ K′1, for a. e. s ∈ [0, 1]}. (28)

Proof. Similar to Lemma 2, hence omitted.

The discussion thus far implies that the dual of (CCLP ) can be written as

(DCCLP ) supy′

1∫0

b>(s)y′(s)ds (29)

s∫0

M>(s, t)y′(t)dt+ L>(s)y′(s) + c(s) ∈ K′2, for a. e. s ∈ [0, 1] (30)

y′(s) ∈ −K′1, for a. e. s ∈ [0, 1] (31)

y′ ∈ Lm∞[0, 1]. (32)

The generic inverse optimization problem (Q) thus specializes to the conic LP

(QCCLP ) infu,y′,r

r (33)

||u||Ln1 [0,1] ≤ r (34)

1∫0

b>(s)y′(s)ds+

1∫0

x>(t)u(t)dt−1∫

0

c>(t)x(t)dt ∈ {0} (35)

(30)− (32) (36)

u ∈ Ln1 [0, 1] (37)

r ∈ <. (38)

Note that

||u||Ln1 [0,1] ≤ r ⇔ maxj=1,...,n

||uj ||L1[0,1] ≤ r ⇔1∫

0

|uj(t)|dt ≤ r, j = 1, . . . , n.

9

Make the nonnegative variable substitution zj(t) = |uj(t)| in (QCCLP ), and include additionalconstraints −zj(t) ≤ uj(t) ≤ zj(t), for j = 1, . . . , n and all t ∈ [0, 1], to obtain the problem

(QCCLPz) infu,y′,z,r

r

(35− 38)

1∫0

zj(t)dt ≤ r, j = 1, . . . , n

−zj(t) ≤ uj(t) ≤ zj(t), j = 1, . . . , n, t ∈ [0, 1]

z(t) ≥ 0, t ∈ [0, 1]

z ∈ Ln1 [0, 1].

Suppose u, y′, z, r is a feasible solution to (QCCLPz). Then, 0 ≤ |uj(t)| ≤ zj(t). Thus,1∫0

|uj(t)|dt ≤1∫0

zj(t)dt ≤ r and hence using the alternative values of zj(t) = |uj(t)| has no effect on the objec-

tive value while maintaining feasibility. Problem (QCCLPz) is therefore equivalent to problem(QCCLP ). Problem (QCCLPz) can be shown to be a continuous conic LP itself via additionalalgebraic reorganization of constraints after including appropriate blocks of null matrix functions.This discussion yields

Proposition 1. The inverse problem (QCCLPz) belongs to the same subclass as the forward prob-lem (CCLP ) — both are continuous conic LPs.

Algorithms to solve special cases of such problems are available in the literature on continuousLPs and continuous conic programs that was reviewed at the beginning of this section.

5 Normed cone programs in Banach spaces

Let X,Z be Banach spaces over the field of real numbers with their norms denoted by || · ||X and|| · ||Z. Let F : X→ Z be a bounded linear operator and g be a vector in Z. Let X′ be the normeddual of X with the corresponding norm denoted by || · ||X′ . Let d be a vector in X′ and e be a real

number. The normed convex cone in the product space Ydef= (Z×<) is defined as

Ydef= {(z, t) ∈ (Z×<) : ||z||Z ≤ t}. (39)

Suppose c is a vector in X′. Consider the optimization problem

(CP ) infx〈c, x〉

(F (x) + g, 〈d, x〉+ e) ∈ Y,x ∈ X.

Problems of this form (and also those that include additional affine constraints of the form G(x) +f = 0V , where V is a Banach space, G : X → V is a bounded linear operator, and f ∈ V ) arecalled normed cone programs in this paper. This problem can be viewed as a special case of (P )by defining an operator A : X→ Y and a vector b ∈ Y as

A(x) =

[F (x)〈d, x〉

], b =

[ge

].

10

To write the dual of this problem, first observe that the dual cone of X is given by the set{x′ ∈ X′ : 〈x′, x〉 ≥ 0, ∀x ∈ X}. This dual cone therefore equals the singleton {0X′}. It takes morework to characterize the dual cone of Y in the next lemma.

Lemma 4. The dual of the normed convex cone described in (39) is given by

Y = {(z′, s) ∈ (Z′ ×<) : ||z′||∗ ≤ s}, (40)

where || · ||∗ is the dual norm of || · ||Z that is defined as

||z′||∗def= sup

z

{〈z′, z〉 : ||z||Z ≤ 1

}. (41)

Proof. The normed dual of Y = (Z× <) is Y ′ = (Z′ × <). Thus, as per the generic definition (2),

the dual of the convex cone described in (39) is given by Y ′def= {(z′, s) ∈ (Z′ × <) : 〈z′, z〉 + st ≥

0, ∀(z, t) ∈ Y }. The proof of Y = Y ′ includes two steps that are identical to the finite-dimensionalapproach in Example 2.25 of [17] and is hence omitted.

The dual problem thus reads

(DCP ) supz′,s〈g, z′〉+ es (42)

F ′(z′) + sd+ c ∈ {0X′} (43)

(z′, s) ∈ −Y (44)

(z′, s) ∈ (Z′ ×<). (45)

The generic inverse optimization problem (Q) formulated in (17-20) thus specializes in this contextto the conic LP

(QCP ) infu,z′,s,r

r

(u, r) ∈W〈g, z′〉+ es+ 〈u, x〉 − 〈c, x〉 ∈ {0}

(43− 45)

u ∈ X′

r ∈ <,

where W is the norm cone {(u, r) ∈ (X′ ×<) : ||u||X′ ≤ r}. This discussion yields

Proposition 2. The inverse problem (QCP ) belongs to the same subclass as the forward problem(CP ) — both are normed cone programs.

The author is not aware of any existing algorithms to solve the forward problem (CP ) andhence the inverse problem (QCP ). A special case of this framework that may be of particularinterest is described briefly in the remainder of this section.

Consider a special case of the above framework, where X is a Hilbert space over the field of realnumbers. Its inner product is denoted by the dot symbol •, and the corresponding norm is givenby ||x||X = (x •x)1/2 for all x ∈ X. Let X′ be its normed dual. By the Riesz-Frechet representationtheorem (Luenberger [44], page 109), the bilinear form 〈x′, x〉 : X′ × X → < has the followingproperty: every x′ ∈ X can be identified with some f ∈ X such that 〈x′, x〉 = f • x, for all x ∈ X.

11

Thus, informally, the normed dual X′ can be identified with X itself (Aliprantis and Border [3],Corollary 6.55). Similarly, let Z be a Hilbert space over the field of real numbers with the innerproduct also denoted by the • symbol and the corresponding norm ||z||Z = (z • z)1/2. Again, thenormed dual Z′ can be identified with Z by the Riesz-Frechet representation theorem. The normedcone program (CP ) then reduces to a second-order cone program where the convex cone Y in (39)is written as {(z, t) ∈ (Z×<) :

√z • z ≤ t}.

The key point of interest now is that the dual cone Y defined in (40) reduces to Y itself; thatis, Y is self-dual. To see this, observe that the dual norm (41) in this case reduces to ||z′||Z.This holds by the Cauchy-Schwarz inequality (Luenberger [44], Lemma 1 on page 47) becausez′ • z ≤ |z′ • z| ≤ ||z′||Z||z||Z. Thus, the supremum of z′ • z over ||z||Z ≤ 1 is attained whenz = z′

||z′||Z . In other words, ||z′||∗ = ||z′||Z. This means that

Y = {(z′, s) ∈ (Z×<) : ||z′||Z ≤ s} = Y, (46)

as claimed. This discussion yields

Proposition 3. The inverse second-order cone program is also a second-order cone program.

A polynomial-time primal-dual algorithm for infinite-dimensional second-order cone programsis available in [30].

6 Semi-definite programs in Hilbert spaces

In a semi-definite program (SDP), the goal is to minimize a linear function under the constraint thatan affine combination of symmetric matrices is positive semi-definite (PSD). Motivated by a broadrange of applications, a rich theory and computational methods have been developed for SDPs inthis matrix setting (see [17, 66, 70]; and Section 7.4.3 in [16]). Ahmadi and El Khadir [1] recentlyintroduced SDPs where the feasible set and objective were functions of time, and developed asolution algorithm. Their methodology was applied to a maximum-flow problem with time-varyingedge capacities, and a wireless optimization problem with time-varying coverage requirements.These SDPs were a generalization of continuous LPs and were also related to continuous conicprograms. This section studies inverse optimization for a generalization of such time-varying SDPs.

Let X be a Hilbert space over the field of real numbers. Its inner product is denoted by the dotsymbol •, and the corresponding norm is given by ||x||X = (x •x)1/2, for all x ∈ X. Let Z also be aHilbert space over the field of real numbers, and B(Z) be the set of all bounded linear operators fromZ to itself. It is easy to verify that this is a vector space. Let {eθ : θ ∈ Θ} be any orthonormal basisfor Z. Existence of an orthonormal basis is guaranteed by Zorn’s Lemma (Gudder [37], Theorem2.1). Note that the index set Θ might be uncountable since Z is not assumed to be separable.Throughout this paper, sums over uncountable index sets are interpreted as in page 72 of Conway[26]. An operator T ∈ B(Z) is said to be Hilbert-Schmidt (Conway [26], Definitions 18.3, 18.4) if∑

θ∈Θ

[T (eθ)] • [T (eθ)] <∞. (47)

The value of the left hand side of the above inequality is independent (over the extended realnumbers) of the choice of an orthonormal basis (Conway [26], Corollary 18.2). Consequently,whether an operator in B(Z) is Hilbert-Schmidt or not is also independent of the choice of anorthonormal basis.

12

Let B(Z) be the subset formed by all Hilbert-Schmidt operators in B(Z). If operators T1, T2

are in B(Z), then it is possible to define the inner product

T1 • T2def= Trace(T ′1T2)

def=∑θ∈Θ

[T ′1T2(eθ)] • (eθ) =∑θ∈Θ

[T2(eθ)] • [T1(eθ)] (48)

=∑θ∈Θ

(eθ) • [T ′2T1(eθ)] = Trace(T ′2T1).

This is called the Hilbert-Schmidt inner product. Then B(Z) becomes a Hilbert space with thisHilbert-Schmidt inner product and the induced norm ||T ||HS = (T • T )1/2 (Conway [26], Theorem18.14). Any Hilbert-Schmidt operator T ∈ B(Z) is compact (Rudin [59], page 103; and Conway[26], Corollary 18.7). That is, for every bounded sequence {zn} in Z, the sequence {T (zn)}∞n=1 inZ has a convergent subsequence. This compactness will be helpful later while characterizing dualcones.

Let S(Z) be a subset of B(Z) that only includes all self-adjoint operators with respect to theinner product of Z; that is, operators T ∈ B(Z) such that T ′ = T . It is easy to verify that thisis a subspace of B(Z). In fact, it is a closed subspace; that is, if {Tn} is a sequence of self-adjointoperators in S(Z) that converges in the ||·||HS norm to an operator T ∈ B(Z), then T is self-adjoint.To see this, observe that

||T − T ′||HS = ||T − Tn + Tn − T ′n + T ′n − T ′||HS

≤ ||T − Tn||HS + ||Tn − T ′n||HS︸ ︷︷ ︸0

+||T ′n − T ′||HS

= ||T − Tn||HS + ||(Tn − T )′||HS = ||T − Tn||HS + ||Tn − T ||HS

= 2||T − Tn||HS.

Here, the first inequality follows from the triangle inequality (Luenberger [44], page 22) and T ′n = Tnbecause Tn is self-adjoint. The second line follows because it is easy to verify that (Tn−T )′ = T ′n−T ′by linearity of operators, and because the norm of an operator equals the norm of its adjoint(Luenberger [44], Theorem 1 on page 151). Finally, taking the limit as n→∞ yields that the righthand side is 0 because Tn → T . Thus the left hand side, which is known to be nonnegative, mustalso be 0. In other words, T ′ = T . Since S(Z) is a closed subspace of the Hilbert space B(Z), itis itself a Hilbert space with the inherited Hilbert-Schmidt inner product and norm. This followsfrom the more general result that a closed subset of a complete metric space is complete (Rudin[58], page 54).

Let S+(Z) be a subset of S(Z) that only includes PSD operators with respect to the innerproduct of Z; that is, operators T ∈ S(Z) such that [T (z)] • z ≥ 0, for all z ∈ Z, where the •here denotes the inner product of Z. This set is a cone because, for any λ ≥ 0, [λT (z)] • (z) =λ[T (z)]•z, and hence T ∈ S+(Z) implies λT ∈ S+(Z). It is convex because, for any T1, T2 ∈ S+(Z),[(T1 + T2)(z)] • z = [T1(z) + T2(z)] • z = [T1(z)] • z + [T2(z)] • z ≥ 0, for al z ∈ Z.

Now let F : X → S(Z) be any bounded linear operator over the Hilbert space X, and letG ∈ S(Z) be another operator. Then, for any x ∈ X, the operator F (x) +G belongs to S(Z). Letc ∈ X be a fixed vector. Consider the optimization problem

(SDP ) infx

c • x

F (x) +G ∈ S+(Z)

x ∈ X.

13

In this paper, problems of this form (and also those that include additional affine constraints ofthe form A(x) + b = 0V , where V is a Hilbert space, A : X → V is a bounded linear operator, andb ∈ V ) are called semi-definite programs (SDPs).

To write the dual of (SDP ), first observe that the normed dual [S(Z)]′ of Hilbert space S(Z)can be identified with S(Z) itself via the Riesz-Frechet representation theorem. Thus, the bilinearform for this dual pair is replaced with the Hilbert-Schmidt inner product of S(Z). The dual coneof X is given by the set {f ∈ X : f • x ≥ 0, ∀x ∈ X}. This dual cone therefore equals the singleton{0X}. It takes more work to characterize the dual cone of S+(Z) in the next lemma.

Lemma 5. The dual of S+(Z) equals S+(Z). That is, the convex cone S+(Z) is self-dual.

Proof. The dual cone of S+(Z) is defined as

[S+(Z)]′def= {T ∈ S(Z) : T • V ≥ 0 ∀V ∈ S+(Z)}. (49)

Claim 1. Suppose T ∈ [S+(Z)]′. Then T ∈ S+(Z). That is, [S+(Z)]′ ⊆ S+(Z).Begin Proof of Claim 1. Suppose T is in S(Z) but T /∈ S+(Z). Then, there exists a z ∈ Z suchthat

0 > [T (z)] • z =∑θ∈Θ

(T (z) • eθ)(z • eθ) =∑θ∈Θ

(z • T (eθ))(z • eθ) (50)

=∑θ∈Θ

Vz(eθ) • T (eθ) = Trace(TVz), (51)

where Vz : Z → Z is an operator defined as Vz(σ) = (z • σ)z, for each σ ∈ Z. This operator islinear because of the linearity of the inner product • of Z. It is bounded because

||Vz|| = sup||σ||Z=1

||Vz(σ)||Z = sup||σ||Z=1

||(z • σ)z||Z ≤ ||z||2Z.

Here, the last inequality is Cauchy-Schwarz. The operator is Hilbert-Schmidt because∑θ∈Θ

[Vz(eθ)] • [Vz(eθ)] =∑θ∈Θ

[(z • eθ)z] • [(z • eθ)z] = ||z||2Z∑θ∈Θ

(z • eθ)(z • eθ)

= ||z||2Z||z||2Z = ||z||4Z <∞.

The operator is self-adjoint because

Vz(σ) • ρ = (z • σ)(z • ρ) = (σ • z)(z • ρ) = σ • Vz(ρ), ∀σ, ρ ∈ Z.

Finally, the operator is PSD because

Vz(σ) • σ = (z • σ)(z • σ) = (z • σ)2 ≥ 0, ∀σ ∈ Z.

Thus, Vz ∈ S+(Z) but T • Vz < 0 by (51). Consequently, T /∈ [S+(Z)]′ by definition (49). Thiscompletes the proof of Claim 1 by contrapositive.End Proof of Claim 1.Claim 2. Suppose T ∈ S+(Z). Then T ∈ [S+(Z)]′. That is, S+(Z) ⊆ [S+(Z)]′.Begin Proof of Claim 2. Let V ∈ S+(Z) be any operator. According to definition (49), it suffices toshow that Trace(TV ) ≥ 0. Since V is self-adjoint, all its eigenvalues are real numbers (this is notdifficult to establish; see Kreyszig [42], Theorem 9.1-1; and Rudin [59], Theorem 12.15[b]). SinceV is also compact, the Spectral Theorem states that the Hilbert space Z has an orthonormal basis

14

formed by eigenvectors of V (Knapp [41], Theorem 2.3). Let {ϕθ : θ ∈ Θ} be such an orthonormalbasis, and let λθ be the corresponding real eigenvalues. Since V is also PSD, these eigenvalues mustbe nonnegative, because taking the inner product of both sides of V (ϕθ) = λθϕθ with ϕθ yields

V (ϕθ) • ϕθ = λθϕθ • ϕθ = λθ||ϕθ||2Z.

Thus, λθ = V (ϕθ)•ϕθ||ϕθ||2Z

≥ 0 because the eigenvector ϕθ 6= 0 and V (ϕθ) •ϕθ ≥ 0. Then, computing the

trace of TV using this orthonormal basis yields

Trace(TV ) =∑θ∈Θ

T (ϕθ) • V (ϕθ) =∑θ∈Θ

T (ϕθ) • λθϕθ =∑θ∈Θ

λθT (ϕθ) • ϕθ ≥ 0,

because T is PSD.End Proof of Claim 2.Claims 1 and 2 imply that [S+(Z)]′ = S+(Z), as required.

This discussion implies that the dual of (SDP ) is the conic LP

(DSDP ) supy′

Trace(Gy′) (52)

F ′(y′) + c ∈ {0X} (53)

y′ ∈ −S+(Z) (54)

y′ ∈ S(Z). (55)

Observe that this dual is itself an SDP. The generic inverse optimization problem (Q) formulatedin (17-20) thus specializes in this context to the conic LP

(QSDP ) infu,y′,r

r

(u, r) ∈WTrace(Gy′) + u • x− c • x ∈ {0}

(53− 55)

u ∈ X

r ∈ <,

where W is the second-order cone {(u, r) ∈ (X ×<) :√u • u ≤ r}. This inverse problem itself can

be seen as an SDP if the constraint (u, r) ∈W can be equivalently written in a semi-definite form.This calls for a generalization of the Schur complement property of finite-dimensional matrices(Boyd and Vandenberghe [17], Appendix A.5.5; and Alizadeh and Goldfarb [4], page 7). Lemma 6below achieves this using additional notation that is introduced next.

View < as a Hilbert space with inner product s • t = st, for s, t ∈ <. The product space <×X

is then also a Hilbert space with inner product defined as (s, x)• (t, y)def= s• t+x•y = st+x•y and

the corresponding natural sum norm (Luenberger [44], Example 6 on page 37). Fix any 0 ≤ r ∈ <.View r : < → < as a linear operator defined by r(t) = r • t = rt. It is easy to verify that this r isself-adjoint. This operator is also PSD in the sense that t2r ≥ 0 for all t ∈ <. Fix any u ∈ X. Viewit as a linear operator from X into < defined by u(x) = u • x. Let u′ : < → X be its adjoint. Forany y ∈ <, the adjoint should satisfy u(x)× y = yu •x = x • (yu′). Consequently, u′ = u. Considera linear operator H : (<×X)→ (<×X) defined as

H(t, x) = (rt+ u • x, tu+ rx).

15

This operator is self-adjoint. To see this, fix any (t, x) ∈ < × X and let (s, y) ∈ (<,X). Observethat

(s, y) • (H(t, x)) = (s, y) • (rt+ u • x, tu+ rx) = s(rt+ u • x) + y • (tu+ rx)

= trs+ su • x+ tu • y + ry • x = t(rs+ u • y) + (su+ ry) • x= (rs+ u • y, su+ ry) • (t, x) = (H(s, y)) • (t, x).

This self-adjoint operator is defined to be PSD when (H(t, x)) • (t, x) ≥ 0 for all (t, x) ∈ (< ×X).Note that

(H(t, x)) • (t, x) = (rt+ u • x, tu+ rx) • (t, x) = rt2 + 2tu • x+ rx • x.

Consequently, H is PSD if, and only if, rt2 + 2tu • x+ rx • x ≥ 0, for all (t, x) ∈ < ×X.

Lemma 6. (u, r) ∈ W if, and only if, operator H is PSD; that is, if, and only if, rt2 + 2tu • x +rx • x ≥ 0, for all (t, x) ∈ < ×X.

Proof. Suppose r = 0. If (u, r) = (u, 0) is in W , that is, if√u • u ≤ 0, then u = 0X. Thus,

rt2 + 2tu • x + rx • x = 0 + 0 + 0 = 0, for all (t, x) ∈ < × X. Thus, this direction holds triviallyin this case. Conversely, suppose rt2 + 2tu • x + rx • x = 0 + 2tu • x + 0 ≥ 0 for all (t, x). Thatis, tu • x ≥ 0, for all (t, x) ∈ (< × X). This implies that u = 0X and hence (u, 0) = (u, r) is in W .Thus, this direction also holds trivially in this case.

Now suppose r > 0. Suppose (u, r) is in W . That is,

r ≥√u • u = ||u||X. (56)

Then,

rt2 + 2tu • x+ rx • x ≥ ||u||Xt2 + 2tu • x+ ||u||X||x||2X≥ ||u||Xt2 − 2|t||u • x|+ ||u||X||x||2X≥ ||u||Xt2 − 2|t|||u||X||x||X + ||u||X||x||2X= ||u||X(t2 − 2|t|||x||X + ||x||2X)

= ||u||X(|t| − ||x||X)2 ≥ 0.

Here, the first inequality follows from (56). The second inequality holds because, for any two realnumbers a, b, we have ab ≥ −|a||b|. The third inequality is Cauchy-Schwarz.Conversely, suppose r <

√u • u = ||u||X. Then choose t = −r and x = u to obtain

rt2 + 2tu • x+ rx • x = r(r2 − 2||u||2X + ||u||2X) = r(r2 − ||u||2X) < 0.

This means, by contrapositive, that rt2 + 2tu • x+ rx • x ≥ 0, for all (t, x), implies r ≥ ||u||X. Thiscompletes the proof.

This discussion yields

Proposition 4. The inverse problem (QSDP ) belongs to the same subclass as the forward problem(SDP ) — both are semi-definite programs.

However, the author is not aware of any existing algorithms to solve the forward problem (SDP )and hence the inverse problem (QSDP ) in general.

16

7 Conclusions and future work

This paper established a duality-based framework to formulate inverse problems for a large classof convex programs — conic LPs wherein the costs belong to a normed linear space. The maininitial conclusion was that the inverse problem is also a conic LP. Detailed derivations for threespecial cases were described: continuous conic LPs; normed cone programs in Banach spaces; andsemi-definite programs in Hilbert spaces. In each of these cases, the initial result was strengthened— the inverse problem is in the same class as the forward. That is, the inverse continuous conicLP is a continuous conic LP; the inverse normed cone program is a normed cone program (and asa further special case, the inverse second-order cone program is a second-order cone program); andthe inverse semi-definite program is a semi-definite program. Algorithmic research on solving theforward problems in these classes is limited at this point. Consequently, efficient algorithms maynot be available to solve the inverse problems either. The paper identified some cases where anexisting solution algorithm for the forward problem can be adapted to the inverse problem. Furtheralgorithmic work is needed to solve the forward and the inverse problems in other cases.

Acknowledgement

This research was funded in part by the National Science Foundation [grant #CMMI 1561918]. Theauthor is grateful to an anonymous reviewer of a previous version of this paper, whose suggestionsshortened and improved the presentation; helped merge two longer sections into a single Section4; strengthened the proof of Lemma 2 by including measure-theoretic issues that the author hadoverlooked; and simplified the proof of Lemma 6 via a concrete approach instead of relying on amore general result from [32].

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