The L¨ owner-John ellipsoids - Semideﬁnite Optimization

Using Proposition 9.3.1 and Proposition 9.3.2 one can find the ellipsoid of largest volume contained in a polytopeP as well as the ellipsoid of smallest vol-ume containingPby solving determinant maximization problems. In both cases one maximizes the logarithm of the determinant ofA. Because the logarithm of the determinant is astrictlyconcave function both optimization problems have a unique solution. The ellipsoids are called theL¨owner-John ellipsoids¹ ofP. Notation:E_inpPqfor the ellipsoid giving the optimal inner approximation ofP andE_outpPqfor the ellipsoid giving the optimal outer approximation ofP.

The following theorem can be traced back to John (1948). Historically, it is considered to be one of the first theorems involving an optimality condition for nonlinear optimization.

Theorem 9.4.1. (a) LetP be a polytope. The L¨owner-John ellipsoidEoutpPqis equal to the unit ball if and only ifP is contained in the unit ball and there is are positive numbersλ1, . . . , λN and verticesx1, . . . , xN ofP having unit length so that

i“1

λixi“0 and

i“1

λixix^T_i “In

holds.

(b) Let P be a polytope. The L¨owner-John ellipsoidEinpPqis equal to the unit ball if and only ifP is containing the unit ball and if there are unit vectors x1, . . . , xN on the boundary ofP and there are positive numbersλ1, . . . , λN

so that

i“1

λixi“0 and

i“1

λixix^T_i “In

holds.

Before we give the proof we comment on the optimality conditions. The first equality makes sure that not all the vectorsx₁, . . . x_N lie on one side of the sphere. The second equality shows that the vectors behave similar to an

1In the literature the terminology seems to differ from author to author.

orthonormal basis in the sense that we can compute the inner product of two vectorsxandyby

x^Ty“

i“1

λipx^T_ixqpx^T_iyq.

Proof. Statement (a) follows from the optimality conditions of the underlying determinant maximization problem. See Exercise 9.1 (b).

Statement (b) follows from (a) by polarity:

LetCĎRⁿbe a convex body. Itspolar bodyC^˚ is C^˚“ txPRⁿ:x^Tyď1for allyPCu.

The unit ball is self-polar,B^˚“B. Furthermore, for every ellipsoidEwe have volEvolE^˚ ě pvolBq²,

because direct verification yields pEA,cq^˚“EA¹,c¹ with A¹“

ˆ A^TA p1`¹₄c^TpA^TAq^´1cq

˙^´1{2

, c¹“ ´1

2pA^TAq^´1c, and so

detAdetA¹ ě1.

Let P be a polytope and assume that EinpPq “ B. We will now prove by contradiction thatB“E_outpP^˚q. For suppose not. Then the volume ofE_outpP^˚q is strictly smaller than the volume ofB sinceP^˚ ĎB. However, by taking the polar again we have

EoutpP^˚q^˚ĎP,

andvolEoutpP^˚q^˚ ą volpBq a contradiction. So by (a) we have for vertices x1, . . . , xN ofP, which are of unit length, the conditions

i“1

λ_ix_i “0 and

i“1

λ_ix_ix^T_i “I_n,

for positiveλ1, . . . , λN. Then the unit vectorsxi also lie on the boundary of the polytopeP because

P “ pP^˚q^˚“ txPRⁿ:x^T_ixď1, i“1, . . . , Nu.

Now let

EinpPq “ txPRⁿ :px´cq^TA^´2px´cq ď1u

be the optimal inner approximation ofP. We want to derive from the optimality conditions thatdetAď1holds.

First we realize that the second optimality condition implies that the equa-tionřN

i“1λi“nholds; simply take the trace.

The points

yi“c` px^T_iA²xiq^´1{2A²xi

lie inEinpPqand soy_i^Txi ď1 holds because this inequality determines a sup-porting hyperplanes ofP. Then,

ně

i“1

λ_iy^T_ix_i“

i“1

λ_ipx^T_iA²x_iq^1{2 where we used the first equality when simplifyingřN

i“1λic^Txi “0. The trace ofAcan be estimated by using the second equality

xA, Iny “@ A,

i“1

λixix^T_iD

“

i“1

λix^T_iAxiď

i“1

λipx^T_iA²xiq^1{2ďn, where we used in the first inequality the spectral factorization ofA “P^TDP, with orthogonal matrix P and diagonal matrix D, together with the Cauchy-Schwarz inequality

px^T_iP^TDqpP xiq ď px^T_iP^TD²P xiq^1{2ppP xiq^TP xiq^1{2“ px^T_iP^TD²P xiq^1{2. Now we finish the proof by realizing that (lnxďx´1)

ln detAďTrpAq ´nď0, and sodetAď1.

This optimality condition is helpful in surprisingly many situation. For ex-ample one can use them to prove an estimate on the quality of the inner and outer approximation.

Corollary 9.4.2. LetP ĎRⁿbe ann-dimensional polytope, then there are invert-ible affine transformationsTinandToutso that

B“T_inE_inpPq ĎT_inP ĎnT_inE_inpPq “nB

and 1

nB“ 1

nToutEoutpPq ĎToutPĎToutEoutpPq “B holds.

Proof. We only prove the first statement, the second follows again by polarity.

It is clear that we can map E_inpPq to the unit ball by an invertible affine transformation. So we can use the equations

i“1

λixi“0 and

i“1

λixix^T_i “In

to show TinP Ď nEinpPq. By taking the trace on both sides of the second equations we also have

i“1

λi“n.

The supporting hyperplane through the boundary pointxiofPis orthogonal the unit vectorxi(draw a figure). Hence,

BĎPĎQ“ txPRⁿ:x^T_ixď1, i“1, . . . , Nu.

Letxbe inQ, then becausex^TxiP r´}x},1swe have 0ď

i“1

λ_ip1´x^Tx_iqp}x} `x^Tx_iq

“ }x}

i“1

λ_i` p1´ }x}q

i“1

λ_ix^Tx_i´

i“1

λ_ipx^Tx_iq²

“ }x}n`0´ }x}², and so}x} ďn.

IfP is centrally symmetric, i.e.P “ ´P, then in the above inequalities n can be replaced by?

n. See Exercise 9.1 (c).

Another nice mathematical application of the uniqueness L¨owner-John el-lipsoids is the following.

Proposition 9.4.3. Let P be a polytope and consider the group Gof all affine transformations which mapPinto itself. Then there is an affine transformationT so thatT GT^´1is a subgroup of the orthogonal group.

Proof. Since the volume is invariant under affine transformations with determi-nant equal to1or´1(only those affine transformations can be inG) and since the L¨owner-John ellipsoid is the unique maximum volume ellipsoid contained in a polytope we have

AEinpPq “EinpAPq “EinpPq for allAPG.

Let T be the affine transformation which maps the L¨owner-John ellipsoid E_inpPqto the unit ballB. Then for everyAPG

T AT^´1B“T AE_inpPq “TE_inpPq “B.

SoT AT^´1leaves the unit ball invariant, hence it is an orthogonal transforma-tion.

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