Special Cases - Mathematical Optimization for the Discrete Problem

Chapter IV: Mathematical Optimization for the Discrete Problem

4.4 Special Cases

Subtracting ¹₂𝑋^𝑇

1𝑋₁+ ¹

2𝑋^𝑇

2𝑋₂from both sides, we find 0=−1

4𝑋^𝑇

1𝑋₁+ 1 4𝑋^𝑇

1𝑋₂+ 1 4𝑋^𝑇

2𝑋₁− 1 4𝑋^𝑇

2𝑋₂

=−1

4(𝑋₁−𝑋₂)^𝑇(𝑋₁− 𝑋₂).

This implies 𝑋₁=𝑋₂. Hence, the solution of (4.5) is unique.

Due to exactness, any solution𝑝₁, . . . , 𝑝_𝑘of (4.4) corresponds to an optimal solution

𝑍 = 𝐼_𝑑 𝑋

𝑋^𝑇 𝑋^𝑇𝑋

of (4.5) via 𝑋 =

𝑝₁ . . . 𝑝_𝑘

. Consequently, the solution of

(4.4) is also unique. □

We have shown that 𝑝_𝑖 ∈H^𝑟 ∩𝐵_𝑖 for𝑖 > 𝑟.

minimize |𝜁_𝑖−𝑧|² s.t.

𝑧− 𝑝_𝑟 +𝜃_𝑖 2

≤

𝑝_𝑟 −𝜃_𝑖 2

, (𝜃_𝑟 − 𝑝_𝑟) · (𝑧− 𝑝_𝑟) ≤ 0.

The Lagrangian is 𝐿(𝑧, 𝜆₁, 𝜆₂)

= |𝜁_𝑖−𝑧|²+𝜆₁

𝑧− 𝑝_𝑟 +𝜃_𝑖 2

−

𝑝_𝑟 −𝜃_𝑖 2

+𝜆₂(𝜃_𝑟 −𝑝_𝑟) · (𝑧−𝑝_𝑟)

= |𝜁_𝑖−𝑧|²+𝜆₁(𝑝_𝑟 −𝑧) (𝜃_𝑖−𝑧) +𝜆₂(𝜃_𝑟 − 𝑝_𝑟) · (𝑧− 𝑝_𝑟)

= (1+𝜆₁) |𝑧|²+

−2𝜁₁−𝜆₁(𝑝_𝑟 +𝜃_𝑖) +𝜆₂(𝜃_𝑟 −𝑝_𝑟)

·𝑧 + |𝜁_𝑖|²+𝜆₁𝑝_𝑟 ·𝜃_𝑖−𝜆₂𝜃_𝑟 · 𝑝_𝑟,

where𝜆₁, 𝜆₂ ≥ 0.

∇𝑧𝐿(𝑧, 𝜆₁, 𝜆₂) =𝜆₁(−𝑝_𝑟 −𝜃_𝑖+2𝑧) +𝜆₂(−𝑝_𝑟 +𝜃_𝑟) +2𝑧−2𝜁_𝑖. It’s easy to verify that𝑧 = 𝑝_𝑟,𝜆₁= ²^(𝜃^𝑟^−𝜁^𝑖⁾

𝜃𝑖−𝜃𝑟 ,𝜆₂= ²^(𝜃^𝑖^−𝜁^𝑖⁾

𝜃𝑖−𝜃𝑟 satisfy∇𝑧𝐿(𝑧, 𝜆₁, 𝜆₂) =0.

Note that 𝑧 = 𝑝_𝑟 is primal feasible. Also, since 𝜁_𝑖 ≤ 𝜃_𝑟 < 𝜃_𝑖, 𝜆₁, 𝜆₂ ≥ 0 must be dual feasible. We found a primal-dual pair satisfying the Karush-Kuhn-Tucker condition. Hence, 𝑝_𝑟 is the minimizer.

Define e𝑝₁, . . . ,

e𝑝_𝑘 by e𝑝_𝑖 = 𝑝_𝑖 for𝑖 =1, . . . , 𝑟 −1, and e𝑝_𝑖 = 𝑝_𝑟 for𝑖 =𝑟 , . . . , 𝑘. We claim that they satisfy

(𝜃_𝑖− e𝑝_𝑖) · (

𝑝e_𝑗 −

𝑝e_𝑖) ≤0 for all𝑖 ≠ 𝑗 , (𝜃_𝑖−

e𝑝_𝑖) · (−

𝑝e_𝑖) ≤0 for all𝑖 . By construction, it suffices to check

(𝜃_𝑗− 𝑝_𝑟) · (𝑝_𝑖− 𝑝_𝑟) ≤ 0 for𝑖 < 𝑟 < 𝑗 , (𝜃_𝑗 − 𝑝_𝑟) · (−𝑝_𝑟) ≤ 0 for 𝑗 > 𝑟 .

To prove the first inequality, fix𝑖 < 𝑟 < 𝑗. First note that 𝑝_𝑖 ≤ 𝜃_𝑖. This follows directly from (𝜃_𝑖 − 𝑝_𝑖) · (−𝑝_𝑖) ≤ 0. Next, we have 𝑝_𝑖 ≤ 𝑝^′

𝑟 (recall that 𝑝^′

𝑟 is the

first coordinate of 𝑝_𝑟 ∈ R²). Indeed, suppose, to the contrary, 𝑝_𝑖 > 𝑝^′

𝑟. Then 𝑝^′

𝑟 < 𝑝_𝑖 ≤ 𝜃_𝑖 ≤ 𝜃_𝑟, and|𝜃_𝑟 − 𝑝_𝑟|² = |𝜃_𝑟 − 𝑝^′

𝑟|²+ |𝑞_𝑟|² > |𝜃_𝑟 − 𝑝_𝑖|². 𝑝_𝑖 would be a closer point in Γto 𝜃_𝑟 than 𝑝_𝑟, contradiction. Since𝑖 < 𝑟 < 𝑗, 𝑝_𝑖 ≤ 𝑝^′

𝑟, 𝜃_𝑟 ≤ 𝜃_𝑗, and

(𝜃_𝑗−𝜃_𝑟) · (𝑝_𝑖− 𝑝_𝑟) =(𝜃_𝑗−𝜃_𝑟) (𝑝_𝑖− 𝑝^′

𝑟) ≤ 0. We already know

(𝜃_𝑟 −𝑝_𝑟) · (𝑝_𝑖−𝑝_𝑟) ≤0, so adding the two inequalities gives

(𝜃_𝑗 −𝑝_𝑟) · (𝑝_𝑖−𝑝_𝑟) ≤0.

For the second inequality, fix 𝑗 > 𝑟. We know that (𝜃_𝑟 − 𝑝_𝑟) · (−𝑝_𝑟) = (𝜃_𝑟 − 𝑝^′

𝑟) (−𝑝^′

𝑟) + |𝑞_𝑟|² ≤ 0. From this, we deduce that 𝑝^′

𝑟 ≥ 0 (if 𝑝^′

𝑟 < 0, then from (𝜃_𝑟 −𝑝^′

𝑟) (−𝑝^′

𝑟) ≤0 we find𝜃_𝑟− 𝑝^′

𝑟 ≤0, so 0 ≤ 𝜃_𝑟 ≤ 𝑝^′

𝑟 < 0, contradiction). Since 𝜃_𝑗 ≥ 𝜃_𝑟 and𝑝^′

𝑟 ≥ 0,

(𝜃_𝑗 −𝜃_𝑟) · (−𝑝_𝑟) ≤0. We already know

(𝜃_𝑟− 𝑝_𝑟) · (−𝑝_𝑟) ≤0, so adding the two inequalities gives

(𝜃_𝑗 −𝑝_𝑟) · (−𝑝_𝑟) ≤0.

Now define 𝑝

1, . . . , 𝑝

𝑘 as follows: 𝑝

𝑖 =

𝑝e_𝑖 = 𝑝_𝑖 for𝑖 < 𝑟, 𝑝

𝑖 = 𝑝^′

𝑟

for𝑖 ≥ 𝑟. It’s easy to verify that they satisfy

(𝜃_𝑖− 𝑝

𝑖) · (𝑝

𝑗 −𝑝

𝑖) ≤0 for all𝑖 ≠ 𝑗 , (𝜃_𝑖− 𝑝

𝑖) · (−𝑝

𝑖) ≤0 for all𝑖 .

By Proposition 8, there exists a convex region Γ ⊆ R^𝑑 containing 0 such that 𝑝𝑖 = 𝑝

Γ(𝜃_𝑖) for𝑖 =1, . . . , 𝑘. Moreover,

𝑘

∑︁

𝑖=1

𝑤_𝑖|𝜁_𝑖− 𝑝_𝑖|²≥

𝑟−1

∑︁

𝑖=1

𝑤_𝑖|𝜁_𝑖−𝑝_𝑖|²+

𝑘

∑︁

𝑖=𝑟

𝑤_𝑖|𝜁_𝑖−𝑝_𝑟|²

𝑘

∑︁

𝑖=1

𝑤_𝑖|𝜁_𝑖− 𝑝

𝑖|².

It follows that the convex setΓwe started with cannot be optimal. □

The situation covered by Proposition 17 is one-dimensional. We next study a special case that goes beyond one dimension. We show that when 𝜃₁, . . . , 𝜃_𝑘 are linearly independent, (4.5) is exact. It results from the convenient structure of the dual problem (4.12). We will use the following lemma:

Lemma 18. Suppose 𝑦^∗

𝑖 𝑗, 𝑧^∗

𝑖, 𝑉^∗ are optimal for the dual problem (4.12). Let𝑆^∗ = 𝐶−Í

𝑖≠𝑗𝑦^∗

𝑖 𝑗𝐴_{𝑖 𝑗} −Í^𝑘

𝑖=1𝑧^∗

𝑖𝐵_𝑖− 𝑉 0

0 0

. If rank𝑆^∗ ≥ 𝑘, then (4.5) is exact.

Proof. Suppose 𝑍^∗ is optimal for (4.5). Then 𝑍^∗ • 𝑆^∗ = 0. Since 𝑍^∗, 𝑆^∗ are both positive semidefinite, we have the matrix product 𝑍^∗𝑆^∗ = 0. Since𝑍^∗, 𝑆^∗ are (𝑑+ 𝑘) × (𝑑 +𝑘), rank 𝑍^∗+rank𝑆^∗ ≤ 𝑑+ 𝑘. Since rank𝑆^∗ ≥ 𝑘, it must be that rank𝑍^∗ ≤ 𝑑. By Proposition 10, Problem (4.5) is exact. □ With Lemma 18, it suffices to show that rank 𝑆^∗ ≥ 𝑘 whenever we wish to show that the SDP relaxation is exact. We now present the special case of interest.

Proposition 19.Suppose𝜃₁, . . . , 𝜃_𝑘are linearly independent. Then for any𝑤₁, . . . , 𝑤_𝑘 >

0, problem (4.5) with parameters (𝜃 , 𝑤)is exact.

Proof. Suppose 𝑦^∗

𝑖 𝑗, 𝑧^∗

𝑖, 𝑉^∗ are optimal for (4.12). Let 𝑆^∗ = 𝐶 − Í

𝑖≠𝑗 𝑦^∗

𝑖 𝑗𝐴_{𝑖 𝑗} − Í^𝑘

𝑖=1𝑧^∗

𝑖𝐵_𝑖− 𝑉 0

0 0

. Then𝑆^∗takes the form

𝑆^∗ =

« Í^𝑘

𝑖=1𝑤_𝑖

𝜃𝑖𝜃^𝑇

𝑖

4 −𝑉^∗ 𝑓₁ 𝑓₂ . . . 𝑓_𝑘 𝑓^𝑇

1 𝑒₁₁ 𝑒₁₂ . . . 𝑒₁_𝑘 𝑓^𝑇

2 𝑒₂₁ 𝑒₂₂ . . . 𝑒₂_𝑘 ..

.. .

... .. . 𝑓^𝑇

𝑘 𝑒_𝑘₁ 𝑒_𝑘₂ . . . 𝑒_{𝑘 𝑘} ª

¬ ,

where

𝑓_𝑖 =−𝑤_𝑖 𝜃_𝑖

2 +∑︁

𝑗≠𝑖

𝑦^∗

𝑖 𝑗

𝜃_𝑖 2 +𝑧^∗

𝑖

𝜃_𝑖 2 −∑︁

𝑗≠𝑖

𝑦^∗

𝑗 𝑖

𝜃_𝑗

2, 𝑖=1, . . . , 𝑘 , 𝑒_𝑖𝑖 =𝑤_𝑖−∑︁

𝑗≠𝑖

𝑦^∗

𝑖 𝑗 −𝑧^∗

𝑖, 𝑖=1, . . . , 𝑘 , 𝑒_{𝑖 𝑗} =−1

2𝑦^∗

𝑖 𝑗 − 1 2𝑦^∗

𝑗 𝑖, 𝑖 ≠ 𝑗 .

We show that 𝑓₁, 𝑓₂, . . . , 𝑓_𝑘 are linearly independent. Write 𝑓_𝑖 =

−𝑤_𝑖+∑︁

𝑗≠𝑖

𝑦^∗

𝑖 𝑗 +𝑧^∗

𝑖

𝜃_𝑖 2 −∑︁

𝑗≠𝑖

𝑦^∗

𝑗 𝑖

𝜃_𝑗 2 . Since𝜃₁, . . . , 𝜃_𝑘 are linearly independent, it suffices to show that

−𝑤₁+Í

𝑗≠1𝑦^∗

1𝑗 +𝑧^∗

1 −𝑦^∗

12 . . . −𝑦^∗

1𝑘

−𝑦^∗

21 −𝑤₂+Í

𝑗≠2𝑦^∗

2𝑗 +𝑧^∗

2 . . . −𝑦^∗

2𝑘

.. .

−𝑦^∗

𝑘1 −𝑦^∗

𝑘2 . . . −𝑤_𝑘 +Í

𝑗≠𝑘𝑦^∗

𝑘 𝑗 +𝑧^∗

𝑘

¬ is nonsingular. Since 𝑦^∗

𝑖 𝑗 ≤ 0, 𝑧^∗

𝑖 ≤ 0, this matrix is strictly diagonally dominant, hence indeed nonsingular.

Having established linear independence of 𝑓₁, . . . , 𝑓_𝑘, we have shown that rank𝑆^∗ ≥

𝑘. By Lemma 18, Problem (4.5) is exact. □

This result can be partially extended to the case when𝑑 ≥ 𝑘.

Lemma 20. Let 𝑣(𝜃 , 𝑤) be the optimal value of problem (4.5) with parameters (𝜃 , 𝑤). Then for any pair𝜃 , 𝜃^′∈R^𝑑×𝑘,

|𝑣(𝜃 , 𝑤) −𝑣(𝜃^′, 𝑤) | ≤ 9 4

𝑘

∑︁

𝑖=1

𝑤_𝑖[2 max{|𝜃_𝑖|,|𝜃^′

𝑖|}|𝜃_𝑖−𝜃^′

𝑖| + |𝜃_𝑖−𝜃^′

𝑖|²]. (4.14) In particular, if𝜃^𝑙 ∈R^𝑑×𝑘 converges to𝜃, thenlim𝑙→∞𝑣(𝜃^𝑙, 𝑤) =𝑣(𝜃 , 𝑤).

Proof. Put 𝜁_𝑖 = ^𝜃₂^𝑖. By Theorem 15, there exists 𝑟 ≥ 0 and a closed convex set Γ ⊆ R^𝑑⁺^𝑟 such that 0 ∈ Γ and 𝑣(𝜃 , 𝑤) = Í^𝑘

𝑖=1𝑤_𝑖|𝜁_𝑖 − 𝑝_Γ(𝜃_𝑖) |². The map 𝑝_Γ : R^𝑑 → R^𝑑 is contractive, that is, |𝑝_Γ(𝑥) − 𝑝_Γ(𝑦) | ≤ |𝑥 − 𝑦| (see [13]).

Consequently,

|𝜁^′

𝑖 − 𝑝_Γ(𝜃^′

𝑖) | =|𝜁_𝑖−𝑝_Γ(𝜃_𝑖) − [𝜁_𝑖− 𝑝_Γ(𝜃_𝑖)] + [𝜁^′

𝑖 − 𝑝_Γ(𝜃^′

𝑖)] |

≤ |𝜁_𝑖− 𝑝_Γ(𝜃_𝑖) | + |𝜁_𝑖−𝜁^′

𝑖| + |𝑝_Γ(𝜃_𝑖) − 𝑝_Γ(𝜃^′

𝑖) |

≤ |𝜁_𝑖− 𝑝_Γ(𝜃_𝑖) | + 1

2|𝜃_𝑖−𝜃^′

𝑖| + |𝜃_𝑖−𝜃^′

𝑖|, and

|𝜁^′

𝑖 − 𝑝_Γ(𝜃^′

𝑖) |²

≤ |𝜁_𝑖− 𝑝_Γ(𝜃_𝑖) |²+3|𝜁_𝑖−𝑝_Γ(𝜃_𝑖) ||𝜃_𝑖−𝜃^′

𝑖| + 9

4|𝜃_𝑖−𝜃^′

𝑖|²

≤ |𝜁_𝑖− 𝑝_Γ(𝜃_𝑖) |²+ 9

2|𝜃_𝑖||𝜃_𝑖−𝜃^′

𝑖| + 9

4|𝜃_𝑖−𝜃^′

𝑖|².

Upon summation over𝑖, we find 𝑣(𝜃^′, 𝑤) ≤

𝑘

∑︁

𝑖=1

𝑤_𝑖|𝜁^′

𝑖 − 𝑝_Γ(𝜃^′

𝑖) |²

≤ 𝑣(𝜃 , 𝑤) + 9 4

𝑘

∑︁

𝑖=1

𝑤_𝑖[2|𝜃_𝑖||𝜃_𝑖−𝜃^′

𝑖| + |𝜃_𝑖−𝜃^′

𝑖|²].

Inequality (4.14) follows from reversing the roles of𝜃 and𝜃^′. □ Corollary 21. When𝑑 ≥ 𝑘, Problem (4.5) admits a solution of rank𝑑.

Proof. Since𝑑 ≥ 𝑘, there exists a sequence (𝜃^𝑙

1, . . . , 𝜃^𝑙

𝑘) → (𝜃₁, . . . , 𝜃_𝑘) such that for each 𝑙, 𝜃^𝑙

1, . . . , 𝜃^𝑙

𝑘 are linearly independent. Let 𝐶^𝑙, 𝐴^𝑙

𝑖 𝑗, 𝐵^𝑙

𝑖 be the coefficient matrices for Problem (4.5) with input (𝜃^𝑙

1, . . . , 𝜃^𝑙

𝑘).

Let 𝑍^𝑙 =

𝐼_𝑑 𝑝^𝑙

1 . . . 𝑝^𝑙

𝑘

(𝑝^𝑙

1)^𝑇 𝑦^𝑙

11 . . . 𝑦^𝑙

1𝑘

.. .

... .. . (𝑝^𝑙

𝑘)^𝑇 𝑦^𝑙

𝑘1 . . . 𝑦^𝑙

𝑘 𝑘

be the optimal solution for Problem (4.5) with

input 𝜃^𝑙. By Proposition 19, rank 𝑍^𝑙 = 𝑑, and there exists a closed convex set Γ^𝑙 ⊆ R^𝑑 such that 0 ∈ Γ^𝑙, and 𝑝^𝑙

𝑖 = 𝑝_Γ𝑙(𝜃^𝑙

𝑖). We know that |𝑝^𝑙

𝑖| ≤ |𝜃^𝑙

𝑖|. Moreover, since rank 𝑍^𝑙 = 𝑑, we must have |𝑦^𝑙

𝑖 𝑗| = |𝑝^𝑙

𝑖 · 𝑝^𝑙

𝑗| ≤ |𝜃^𝑙

𝑖||𝜃^𝑙

𝑗|. This analysis shows that the sequence{𝑍^𝑙}^∞

𝑙=1is bounded. After passing to a subsequence,𝑍^𝑙converges to some matrix 𝑍. Note that 𝑍 is feasible for Problem (4.5) with input 𝜃, and that rank𝑍 =𝑑. Also, lim𝑙→∞𝑍^𝑙 •𝐶^𝑙 = 𝑍•𝐶. By Lemma 20,𝑍 is optimal.

□ Remark 8. This result is weaker than Proposition 19 because it does not imply uniqueness of solution for (4.5). It guarantees that an optimal solution of rank 𝑑 exists, but there may be additional solutions with higher rank.

Stated in terms of the fund menu problem, if there are at least as many assets as the number of types, then the manager cannot (strictly) increase aggregate fee by introducing additional independent assets with zero mean return.

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A p p e n d i x A

GAMMA CONVERGENCE

We provide justification for focusing on the finite case of minimizing Í^𝑘

𝑖=1𝑤_𝑖

𝜃𝑖

2 − 𝑝_Γ(𝜃_𝑖)

Consider a bounded regionΩ ⊆ R^𝑑and a finite measure𝜇. We may discretize𝜇by partitioning R^𝑑 into dyadic cubes and forming a weighted sum of point-masses at the centers. Formally, letD𝑛be the collection of dyadic cubes inR^𝑑 of length 2⁻^𝑛. For each𝑄_𝑖 ∈ D𝑛, let𝜃_𝑖be the center of𝑄_𝑖. Define

𝜇_𝑛:= ∑︁

𝑄𝑖∈D𝑛

𝜇(Ω∩𝑄_𝑖)𝛿_𝜃

𝑖 (A.1)

Let C be the collection of closed convex subsets of R^𝑑 containing the origin, and let F𝑛(Γ) := ∫

𝜃

2 − 𝑝_Γ(𝜃)

𝑑𝜇_𝑛. Loosely speaking, as the dyadic cube partition gets finer, the discretized problem minimizeΓ∈CF𝑛(Γ) “approximates” the target problem minimizeΓ∈CF (Γ)better. The precise formulation of such approximation (often known as gamma-convergence in calculus of variations) is given by the following result:

Proposition 22. Suppose Ω ⊆ R^𝑑 is bounded, and 𝜇 is a finite measure on Ω.

Define 𝜇_𝑛 as in (A.1). Then there exists a sequence of minimizers {Γ𝑛} for prob- lems minimizeΓ∈CF𝑛(Γ) having an accumulation point inΔ𝐻. Moreover, any such accumulation point is an optimal solution of minimizeΓ∈CF (Γ).

Lemma 23. Suppose {Γ𝑛} is a sequence in C such that Γ𝑛 → Γ in Δ𝐻. Then F𝑛(Γ𝑛) → F (Γ).

Now consider the first term. On each dyadic cube 𝑄_𝑖, ∫

𝑄𝑖

𝜃 2 − 𝑝_Γ

𝑛(𝜃)

𝑑𝜇_𝑛 =

∫

𝑄𝑖

𝜃𝑖

2 −𝑝_Γ

𝑛(𝜃_𝑖)

2𝑑𝜇. Since|𝑝_Γ

𝑛(𝜃) | ≤ |𝜃|, and|𝑝_Γ

𝑛(𝜃_𝑖) − 𝑝_Γ

𝑛(𝜃) | ≤ |𝜃_𝑖−𝜃|,

∫

𝑄𝑖

𝜃_𝑖 2 −𝑝_Γ

𝑛(𝜃_𝑖)

𝑑𝜇−

∫

𝑄𝑖

𝜃 2 −𝑝_Γ

𝑛(𝜃)

𝑑𝜇 ≲

∫

𝑄𝑖

(|𝜃_𝑖| + |𝜃|) |𝜃_𝑖−𝜃|𝑑𝜇.

By the dominated convergence theorem, as 𝑛 → ∞, Í

𝑄_𝑖∈D𝑛

∫

𝑄𝑖

(|𝜃_𝑖| + |𝜃|) |𝜃_𝑖 −

𝜃|𝑑𝜇→0. □

Proof of Proposition 22. SinceΩis bounded, all𝜇_𝑛are supported in a large enough ball, and we may assume thatΓ𝑛are contained in this ball. By the Blaschke selection theorem, we may assume that {Γ𝑛} contains a subsequence that converges in the Hausdorff distance. SupposeΓis any accumulation point of {Γ𝑛}. By Lemma 23, F𝑛(Γ𝑛) → F (Γ). Moreover, ifΓ^′∈ C, then

F (Γ^′) =limF𝑛(Γ^′) ≥limF𝑛(Γ𝑛) =F (Γ).

Consequently,Γis a minimizer ofF (·). □

A p p e n d i x B

NUMERICAL RESULTS

We examine the numerical solutions of Problem (4.5) in several cases of interest.

Each case consists of points𝜃₁, . . . , 𝜃_𝑘inR²with uniform weight𝑤₁=· · ·=𝑤_𝑘 =1.

We plot the types 𝜃₁, . . . , 𝜃_𝑘 (blue), the optimal convex set Γ (shaded region), as well as the projection points𝑝_Γ(𝜃₁), . . . , 𝑝_Γ(𝜃_𝑘).

In addition to the plots, we will be interested in the numerical value of𝜆_𝑑+1(𝑆^∗)—the (𝑑+1)-st smallest eigenvalue of the optimal dual slack matrix

𝑆^∗ =𝐶−∑︁

𝑖≠𝑗

𝑦^∗

𝑖 𝑗𝐴_{𝑖 𝑗} −

𝑘

∑︁

𝑖=1

𝑧^∗

𝑖𝐵_𝑖− 𝑉^∗ 0

0 0

for Problem (4.12). A strictly positive value implies rank𝑆^∗ ≥ 𝑘. By Lemma (18), it allows us to verify, a posteriori, that the SDP numerical solution is exact for the complete problem (4.4).

The numerical solution is implemented with Python packageCVXPY.

Dalam dokumen TABLE OF CONTENTS (Halaman 36-47)