Polyhedral approximation of spectrahedral shadows via homogenization

This article is concerned with the problem of approximating a not necessarily bounded spectrahedral shadow, a certain convex set, by polyhedra. By identifying the set with its homogenization the problem is reduced to the approximation of a closed convex cone. We introduce the notion of homogeneous δ -approximation of a convex set and show that it defines a meaningful concept in the sense that approximations converge to the original set if the approximation error δ diminishes. Moreover, we show that a homogeneous δ -approximation of the polar of a convex set is immediately available from an approximation of the set itself under mild conditions. Finally, we present an algorithm for the computation of homogeneous δ -approximations of spectrahedral shadows and demonstrate it on examples.


Introduction
Polyhedral approximation of convex sets is a profound method in convex analysis with applications ranging through various branches of mathematics.In the 1903 article [Min03] Minkowski argues that every compact convex set in three-dimensional Euclidean space can be approximated arbitrarily well by polyhedra, i.e. intersections of finitely many closed halfspaces.Although it is not explicitly stated in the article, his line of argumentation also applies to general n-dimensional Euclidean spaces.This is later mentioned by Bonnesen and Fenchel in [BF34].
Since then, polyhedral approximation has been applied in mathematical programming, in particular in multiple objective optimization, e.g.[LRU14;Dör+22;Wag+23], in approximation methods for general convex optimization problems [Vei67; BY11] and for mixed-integer convex programs [WP95;KLW16].Other areas of application include machine learning [Gie+19; YBR08] and large deviation probability theory [NR95].In [Kam92] Kamenev introduces a class of approximation algorithms that compute outer and inner polyhedral approximations of a compact convex set called cutting and augmenting schemes, respectively, by iteratively improving upon a current approximation.The prevalent method of quantifying the approximation error in these algorithms is via the Hausdorff distance.Kamenev proves that under certain conditions the sequence of polyhedral approximations generated by such a scheme converges to the original set in the Hausdorff distance and derives explicit bounds for the Hausdorff distance between an approximation and the set to be approximated.These general schemes subsume many of the polyhedral approximation algorithms in the literature.The literature concerned with polyhedral approximation of not necessarily bounded sets is scarce.The reason appears to be the fact that the Hausdorff distance only defines a metric on the family of nonempty compact subsets of R n but not on the larger family of nonempty closed subsets.In fact, restrictive conditions need to be satisfied for polyhedral approximation of unbounded sets in the Hausdorff distance to be possible.For instance, the recession cones of the approximation and the original set need to coincide, which limits approximation to sets with polyhedral recession cones.A characterization of those sets that are approximable by polyhedra in the Hausdorff distance is due to Ney and Robinson [NR95].Similar results are obtained in [Ulu18] in the context of convex multiple objective optimization.Evidently, the Hausdorff distance is no suitable measure of approximation quality in the unbounded case.One of the authors recently introduced the notion of (ε, δ)-approximation, a concept of polyhedral outer approximation of closed convex sets containing no lines, see [Dör22].One feature is that these approximations bound the so-called truncated Hausdorff distance, a metric on the family of closed cones in R n , between the recession cones, but allow them to differ.Here, we present another notion of polyhedral approximation that is not limited to outer approximation, applicable to a larger class of sets and exhibits elegant behavior under polarity not present for (ε, δ)-approximations.These approximations, denoted homogeneous δ-approximations, are motivated by the fact that every closed convex set in R n can be identified with a closed convex cone in R n+1 , called its homogenization, and that it can also be recovered therefrom.They are defined via the truncated Hausdorff distance between the homogenizations of the approximation and the original set.This article is structured as follows.In the next section we introduce necessary notation and basic definitions.Section 3 presents the notion of homogeneous δ-approximation for the polyhedral approximation of a convex set C. We show that homogeneous δ-approximations converge to cl C in the sense of Painlevé-Kuratowski if the approximation error δ tends to zero.Moreover, we prove that a polyhedron P is a homogeneous δ-approximation of C given that the Hausdorff distance between P and C is finite and compute the approximation error δ in terms of this distance.This shows that the concept of homogeneous δ-approximation is more general than polyhedral approximation with respect to the Hausdorff distance.Finally, we investigate the behaviour of homogeneous δ-approximations under polarity and show that a homogeneous δ-approximation of the polar set C • can be obtained from a homogeneous δ-approximation of C under mild assumptions.In the last section an algorithm for the computation of outer and inner homogeneous δ-approximations for a special class of convex sets called spectrahedral shadows is presented.These sets are the images of spectrahedra, the feasible regions of semidefinite programs, under linear transformations and are special in the sense that their homogenizations can easily be described explicitly as spectrahedral shadows at least up to closure.The algorithm is based on a recently presented algorithm for the computation of polyhedral approximations of the recession cone of a spectrahedral shadow, see [DL22].Two examples illustrating the algorithm are presented.

Preliminaries
Given a set C ⊆ R n we denote by cl C, int C and conv C the closure, interior and convex hull of C, respectively.The Euclidean unit ball of appropriate dimension is written as The smallest, with respect to inclusion, convex cone containing C is called the conical hull of C and written cone C. We use the convention cone ∅ = {0}.Furthermore, a point x ∈ C is called an extreme point of C if C \ {x} is convex.The set of extreme points of C is written ext C. We denote the hyperplane with normal vector ω ∈ R n and offset γ ∈ R by H(ω, γ), i.e.H(ω, γ) = x ∈ R n ω T x = γ , and the lower closed halfspace x ∈ R n ω T x ⩽ γ by H − (ω, γ).A polyhedron P ⊆ R n is the intersection of finitely many such halfspaces.Its extreme points are called vertices and are denoted by vert P. According to the wellknown Weyl-Minkowski theorem, see [Gru07, Theorem 14.3], every polyhedron can alternatively be expressed as the Minkowski sum of the convex hull of finitely many points and the conical hull of finitely many directions, i.e.P = conv {v 1 , . . . ,v m } + cone {d 1 , . . . ,d r } for suitable v 1 , . . . ,v m ∈ R n and d 1 , . . ., d r ∈ R n \ {0}.The polar C • and polar cone C * of C are defined as u ∈ R n ∀x ∈ C : x T u ⩽ γ for γ = 1 and γ = 0, respectively.It is easily verified that (cone C) • = C * .We define d (x, C) = inf {∥x − y∥ | y ∈ C} as the Euclidean distance between x and C. If C is closed and convex, the infimum is uniquely attained, see e.g.[HL01].In this case, the point at which the infimum is attained is called the projection of x onto C and denoted by π C (x).Now, the Hausdorff distance between sets C 1 , C 2 ⊆ R n is then expressed as The function d H defines a metric on the class of nonempty compact subsets of R n .For convex cones K 1 , K 2 ⊆ R n the truncated Hausdorff distance between K 1 and K 2 , denoted d tH (K 1 , K 2 ), is defined as That is, it is the usual Hausdorff distance between K 1 and K 2 restricted to B. Since cones contain the origin, d tH (K 1 , K 2 ) ⩽ 1 always holds.The truncated Hausdorff distance provides a metric on the class of closed convex cones in R n , see [WW67].

Approximation concept for convex sets
In this section we present a concept of polyhedral approximation of a convex set C ⊆ R n that need not be bounded.The truncated Hausdorff distance being a metric between closed convex cones is the motivation for our approach.In order to work in a conic setting, we assign to C a convex cone from R n+1 called its homogenization.
Every closed convex set C ⊆ R n can be identified with its homogenization because C can be recovered from homog C. In particular, i.e. intersecting homog C with the hyperplane Clearly, if P is a homogeneous δ-approximation of C it is also one of cl C because homog C = homog cl C. The notion of homogeneous δ-approximation can be understood as a relative error measure between the involved sets.Consider a convex set C ⊆ R n and a polyhedron P ⊆ R n .If x ∈ P with d (x, C) ⩽ ε, then the distance of the corresponding direction of homog P to homog C obeys the relation In particular, the distance depends inversely on ∥x∥.Hence, if P is a homo- geneous δ-approximation of C, then points of one set that are far from the origin are allowed a larger distance to the other set than points closer to the origin.This observation is explained from the fact that the distinction between points and directions of a closed convex set C collapses when transitioning to its homogenization.One has the identity see [Roc70, Theorem 8.2], i.e. points as well as directions of C correspond to directions of homog C. Thus, a convergent sequence (x k , µ k ) T k∈N of directions of homog C does not necessarily yield a convergent sequence in C itself.For instance, if (x k , µ k ) T k∈N converges to a direction ( x, 0) T and provided all µ k are positive, then the sequence This means that closeness of homogenizations with respect to the truncated Hausdorff distance does not imply closeness of the sets themselves with respect to the Hausdorff distance.In fact, the Hausdorff distance, which measures absolute distances, between a homogeneous δapproximation P of C and C may be infinite for arbitrary δ ∈ (0, 1]. The second equality is evident by taking into account that the projection of (1, 0) T onto homog P δ is attained on the ray generated by Hence, P δ is a homogeneous δ-approximation of C with d H (P δ , C) = ∞.
A converse relation does hold.
Let x be an element from the right hand side set.Then there exist The above proposition and Example 3.3 show that the concept of homogeneous δ-approximation is more general than approximation with respect to the Hausdorff distance.Moreover, homogeneous δ-approximations provide a meaningful notion of approximation in the sense that a sequence of homogeneous δ-approximations of C converge to C if δ tends to 0. The common notion of convergence in this scenario is that of Painlevé-Kuratowski, a theory of convergence for sequences of sets that applies to a broader class than Hausdorff convergence but coincides with it in a compact setting.It is discussed in detail in the book [RW98] by Rockafellar and Wets and we adhere to the notation therein.Let N ∞ and N # ∞ denote the collections {N ⊆ N | N \ N is finite} and {N ⊆ N | N is infinite} of subsets of N, re- spectively.The set N ∞ can be regarded as the set of subsequences of N that contain all natural numbers larger than some n ∈ N and N # ∞ as the set of all subsequences of N. Obviously, N ∞ ⊆ N # ∞ .
Definition 3.5.For a sequence {M k } k∈N of subsets of R n the outer limit is the set denoted by lim sup k→∞ M k , and the inner limit lim inf k→∞ M k is the set The outer limit of a sequence {M k } k∈N can be paraphrased as the set of all points for which every open neighbourhood intersects infinitely many elements M k of the sequence.Similarly, the inner limit is the set for which the same is true but for all elements M k beyond some k ∈ N. In particular, both sets are closed.Definition 3.6.A sequence {M k } k∈N of subsets of R n is said to converge to a set M ⊆ R n in the sense of Painlevé-Kuratowski or PK-converge, written Now, homogeneous δ-approximations define a suitable notion of approximation for closed convex sets because PK-convergence can be characterized in terms of homogenizations and the truncated Hausdorff distance.
Proposition 3.7 (cf.[RW98, Corollary 4.47]).Let {C k } k∈N be a sequence of closed convex subsets of R n and C ⊆ R n be a closed convex set.Then the following are equivalent: Hence, if {P k } k∈N is a sequence of homogeneous δ k -approximations of C and lim k→∞ δ k = 0, then {P k } k∈N PK-converges to C. We will now emphasize another property of homogeneous δ-approximations that is not present for approximations in the Hausdorff distance.Surprisingly, the polar set of a homogeneous δ-approximation of C is a homogeneous δ-approximation of C • provided the origin is contained in the sets.This is different from the compact case.
where r 0 (P) and r 0 (C) are the radius of the largest Euclidean ball centered at the origin that is contained in P and C, respectively, see [Kam02, Lemma 1].In particular, the Hausdorff distance between P • and C • does not only depend on ε but also on constants related to the geometry of C. In order to establish the result, we need the a connection between the polar cone of homog C and C • .The following result is originally presented as [BBW23, Theorem 3.1] for closed convex sets C. Here, we prove a slight generalization.
Proposition 3.8.Let C ⊆ R n be a convex set containing the origin.Then Proof.The first equality is proved in [BBW23, Theorem 3.1] for C being closed and convex.However, it also holds for arbitrary convex sets C because For the other direction consider a point that means we fix µ ⩾ 0 and x ∈ cl C. Thus, there exists a sequence and by taking closures To show the second equation in the original statement we compute The penultimate line follows from [Roc70, Theorem 9.1].
Figure 1 illustrates Proposition 3.8 for the set C = x ∈ R 2 x 2 ⩾ x 2 1 .Theorem 3.9.Let C ⊆ R n be a convex set containing the origin.If P is a homogeneous δ-approximation of C containing the origin, then P • is a homogeneous δ-approximation of C • .Proof.Let M : R n+1 → R n+1 denote the isometry defined by ⩽ δ.The first equation is true by [IS10, Proposition 2.1].The second and third follow from Proposition 3.8 using the fact that 0 ∈ P ∩ C. Theorem 1 of [WW67] yields the equality in line 4. Finally, the inequality holds because P is a homogeneous δ-approximation of C. Thus, 1 and its polar cone.According to Proposition 3.8 the polar cone is the closure of the cone generated by C • × {−1}.

Computing homogeneous δ-approximations of spectrahedral shadows
In this section we present an algorithm for the computation of a homogeneous δ-approximation of a convex sets C. In order to do this, we need to be able to explicitly describe the set homog C. Therefore, we restrict ourselves to a certain class of convex sets for which this is possible called spectrahedral shadows.Let S ℓ denote the linear space of real symmetric ℓ × ℓ-matrices.If X ∈ S ℓ is positive semidefinite and positive definite, it is written as X ≽ 0 and X ≻ 0, respectively.Moreover, for X, Y ∈ S ℓ an inner product is defined by the trace of the matrix product, i.e.X • Y = tr (XY).Now, a spectrahedral shadow S ⊆ R n can be represented as for A 0 ∈ S ℓ and linear functions A : R n → S ℓ , B : R m → S ℓ .Every linear function A between R n and S ℓ can be defined as A(x) = ∑ n i=1 A i x i for suitable A 1 , . . ., A n ∈ S ℓ , see [GM12].Spectrahedral shadows are the images of spectrahedra, the feasible regions of semidefinite programs, under orthogonal projections.They are closed under many set operations, e.g.intersections, Minkowski sums, linear transformations, taking polar or conical hulls, see, for instance, [GR95; HN12; Net11; BPT13; Dör23].The algorithm we present is based on an algorithm for the polyhedral approximation of recession cones of convex sets that was recently and independently developed in [Dör22] and [Wag+23].Given a spectrahedral shadow S and an error tolerance δ > 0, Algorithm 2 of [DL22] computes polyhedral cones It is shown that the algorithm works correctly and terminates after finitely many steps under the following assumptions: (A1) a point p ∈ R n is knwon for which there exists y p ∈ R m such that For ease of reference we recall the algorithm here as Algorithm 1 below.Therefore, the following primal/dual pair of semidefinite programs depending on parameters p, d ∈ R n and a spectrahedral shadow S is needed.
Here, A T and B T denote the adjoints of A and B, respectively.It holds A T (V) = (A 1 • V, . . . ,A n • V) T and the corresponding for B T , see [GM12, Lemma 4.5.3].Typically, we assume that p ∈ S and d is a direction.Then problem (P (p, d, S)) can be motivated as follows.Starting at the point p ∈ S the maximum distance is to be determined that can be moved in direction d without leaving the set.If S is compact, then the maximum distance will be attained in the point where halfline {p Proposition 4.1.Let S ⊆ R n be a spectrahedral shadow and p ∈ S. Then the following hold: )) is strictly feasible with finite optimal value t * , then an optimal solution (V * , w * ) to (D (p, d, S)) exists and the hyperplane H(w * , w * T p + t * ) supports cl S at p + t * d.
After finding initial outer and inner approximations K O and K I or determining that S is the whole space in lines 1-9, Algorithm 1 iteratively shrinks K O or enlarges K I until the truncated Hausdorff distance between them is certifiably not larger than δ.To this end, in every iteration of the main loop in lines 10-25 a new search direction d is chosen in line 14 and the problem (P (p, d, S)) is investigated.If it is unbounded, then d ∈ 0 ∞ cl S according to Proposition 4.1(i) and K I is updated as cone (K I ∪ {d}).Oth- erwise, a solution (V d , w d ) to (D (p, d, S)) exists and H(w d , w T d p + t d ) is a supporting hyperplane of cl S for the optimal value t d of (P (p, d, S)) ac- cording to Proposition 4.1(ii).Therefore, H(w d , 0) supports 0 ∞ cl S and the current outer approximation is updated as K O ∩ H − (w d , 0).After an update of either approximation occurs, a new search direction is computed.For more details about the algorithm we refer the reader to [DL22].In order to apply Algorithm 1 to the problem of computing homogeneous δ-approximations of S, observe that every convex cone is its own recession cone.This suggests we could apply Algorithm 1 to homog S and obtain a homogeneous δ-approximation of S by undoing the homogenization.This approach presupposes that S satisfies suitable assumptions such that the requirements of Algorithm 1 are fulfilled for homog S and, more importantly, that homog S is a spectrahedral shadow for which a representation is available or can be computed from a representation of S. The following proposition shows that the latter is possible.
This set is a spectrahedral shadow because the three inequalities can be expressed as one ALGORITHM 1: Approximation algorithm for recession cones of spectrahedral shadows Input: a spectrahedral shadow S ⊆ R n , point p satisfying (A1), direction d satisfying (A2), error tolerance δ > 0 Output: outer and inner polyhedral approximation According to [Net11, Proposition 4.3.1]cone (S × {1}) is the spectrahedral shadow Using the implicit equality x n+1 = µ we eliminate µ from the description of the set.The resulting spectrahedral shadow differs from the claimed set only through the occurence of the matrix inequalities for i = 1, . . ., n + 1.Now, there exists t ∈ R such that (5) holds for i = 1, . . ., n + 1 if and only if the same is true but for i = 1, . . ., n.Indeed, if x n+1 = 0, then positive definiteness in (5) implies x = 0 and we can choose t = 0 in both cases.Otherwise, if x n+1 > 0, then (5) holds for some i, if and only if t ⩾ x −1 n+1 x 2 i according to the Schur complement, see [Zha05].Thus, we can set t ⩾ x −1 n+1 max x 2 i i = 1, . . ., n + 1 in both cases.The case x n+1 < 0 does not occcur as it violates the positive semidefiniteness in (5).By a similar argument using Schur complement there exists t ∈ R such that (5) holds for all i = 1, . . ., n if and only if there exists t ∈ R such that where I denotes the identity matrix of size n.
Proposition 4.2 and its preceding paragraph motivate the formulation of the following algorithm for the computation of homogeneous δ-approximations of S.
ALGORITHM 2: homogeneous δ-approximation algorithm for spectrahedral shadows Input: a spectrahedral shadow S ⊆ R n , point x ∈ int S satisfying (A1), error tolerance δ > 0 Output: outer and inner homogeneous δ-approximation P O and P I of S 1 K ← cone (S × {1}) 2 compute outer and inner approximation K O and K I of K using Algorithm 1 with input parameters p = xT , 1 T , d = p/∥p∥ and error tolerance δ The function π R n : R n+1 → R n in lines 3 and 4 denotes the projection given by Theorem 4.3.Algorithm 2 works correctly, in particular it terminates with an outer homogeneous δ-approximation P O and an inner homogeneous δ-approximation P I of S.
Proof.Since x satisfies Assumption (A1) for S, the point p = xT , 1 T satisfies analogous conditions for K.This is seen from a direct calculation using the description of K derived in Proposition 4.2.Similarly, x ∈ int S implies xT , 1 From the correctness of Algorithm 1, see [DL22, Theorem 4.4], it follows that line 2 of Algorithm 2 works correctly and returns polyhedral cones K O and K I satisfying Remark 4.4.The inner homogeneous δ-approximation P I of S computed by Algorithm 2 is always compact.Due to the containment of its homogenization K I in the set cone (S × {1}), one has µ > 0 whenever (x, µ) T ∈ K I \ {0}.Thus, every direction of K I corresponds to a point of P I , in particular according to Equation (2).
We conclude this section with two examples.
Example 4.5.Algorithm 2 is applied to the spectrahedral shadow which is the Minkowski sum of the epigraphs of the functions x → x −1 restricted to R ++ and x → x 2 .As input parameters we set δ = 0.01 and x ≈ (1.8846, 1.8846) T .Figure 2 depicts P O as well as the determined inner homogeneous δ-approximation P I .In total, 534 semidefinite programs are solved during the execution of Algorithm 2 for finding P O and P I .
Example 4.6.Consider the spectrahedral shadow S consisting of points x ∈ R 3 that admit the existence of y 1 , y 2 ∈ R 3 and Z ∈ S 3 , Z ≽ 0, such that the system is satisfied.Algorithm 2 with δ = 0.03 and x = 0 requires the solutions to 1989 problems of type (P (p, d, K)), where K = cone (S × {1}).The re- sulting outer approximation P O has 92 vertices and 24 extreme directions, while the compact inner approximation P I is composed of 177 vertices.According to Theorem 3.9 their polar sets P • O and P • I are an inner and outer homogeneous δ-approximation of S • , respectively.They are both compact with 94 and 232 vertices each.Sets P O and P I as well as their polars are shown in Figure 3.

Conclusions
We have introduced the notion of homogeneous δ-approximation for the polyhedral approximation of convex sets, which uses the common concept of homogenization of a convex set.The advantage of this approach compared to approximation with respect to the Hausdorff distance is its compatibility with sets that are unbounded.Moreover, we have shown that homogeneous δ-approximations behave well under polarity in the sense that the polar of a homogeneous δ-approximation of a convex set is a homogeneous δ-approximation of the polar set under mild assumptions.This Figure 3: The top row shows homogeneous δ-approximations P O and P I of the set S from Example 4.6 for δ = 0.03 restricted to balls of radius 7 and 100, respectively.Their polar sets, which are homogeneous δapproximations of S • , are depicted in the bottom row.Both are contained in a ball of radius 1.02.behaviour is not encountered when working with the Hausdorff distance.Finally, we have presented an algorithm for the computation of homogeneous δ-approximations of spectrahedral shadows, a subclass of convex sets.The algorithm is shown to be correct and finite and its practicability is demonstrated on two examples.
intersects the boundary of S. A solution to the dual program, if it exists, gives rise to a supporting hyperplane of cl S. The following proposition is a slight variation of [Dör23, Proposition 3.11].

x − y 1 Figure 2 :
Figure2: Depicted are the outer and inner homogeneous δ-approximation P O and P I of the set from Example 4.5 computed by Algorithm 2 with δ = 0.01 in red and blue, respectively.On the left is a section around the origin with vertices marked by black dots.Note that the distance between vertices of P O and P I increases with their distance to the origin.The right figure shows the scale of P I , which is compact.
and projecting the result onto the first n variables yields the original set C. In the general case, the right hand side set in Equation (1) equals cl C because homog C = homog cl C, cf.Proposition 3.8 below.Thus, we can identify convex sets with their homogenizations up to closures.Using this correspondence, one can work with convex cones entirely.This is a standard tool in convex analysis to study problems in a conic framework, see e.g.[Roc70; RW98; Bri20].Here, we use it to reduce the problem of approximating convex sets to the problem of approximating convex cones.Definition 3.2.For a convex set C Moreover, ∥ x∥ ⩽ ∥x∥ ⩽ 1, because the projection mapping onto homog C is nonexpansive, see [HL01, Proposition 3.1.3].Thus, x = x + (x − x) ∈ homog C ∩ B + εB.Using (3) we obtain the inclusion cone (P × {1}) ∩ B ⊆ homog C ∩ B + εB and, because the right hand side set is closed, also homog P ∩ B ⊆ homog C ∩ B + εB.
6) as well as d tH (K O , homog S) ⩽ δ and d tH (K I , homog S) ⩽ δ.For the sets P O and P I in lines 3 and 4 it holds homog P O = K O ∩ H + (e n+1 , 0) and homog P I = K I .The difference arises from the inclusions in (6) and the fact that homogenizations are contained in the halfspace H + (e n+1 , 0) by definition.However,d tH K O ∩ H + (e n+1 , 0), homog S ⩽ d tH (K O , homog S)⩽ δ holds due to the inclusion homog S ⊆ K O ∩ H + (e n+1 , 0).Therefore, P O and P I are homogeneous δ-approximations of S. The finiteness of Algorithm 2 is implied by the finiteness of Algorithm 1, see [DL22, Theorem 4.4].