Strong Lagrangian Duality using Norm-like Penalties

The following results demonstrate that ψN

8 and ψ1N satisfy the conditions given in Assumption 6.3,

and therefore than when they are employed as penalty functions Theorem 6.2 may be applied. Lemma 6.15 If two functions ψA : Rm Ñ R and ψB : Rm Ñ R are positive homogeneous, contin-

uous, and strictly positive for all u 0 then there exists a finite γ ¡ 0 such that ψApuq ¥ γψBpuq for all uP Rm.

Proof. Since they are positive homogeneous, ψA and ψB vanish at zero and so the required

property trivially holds with equality at u 0. To obtain the required inequality for nonzero u, set V  tu : ||u||  1u (where ||  || is any norm) and take α  minuPV ψApuq and β  maxuPV ψBpuq.

Since ψAand ψB are continuous and defined on the closed and bounded set V , by the Extreme Value

Theorem these extrema exist and are attained by their respective functions. Since these functions are strictly positive and finite valued on V , and they attain their extrema, α and β both strictly positive and finite.

For any point uP Rm_{zt0u, ||u|| is strictly positive and the point} u

||u|| is in V . Therefore, by the

positive homogeneity of ψApuq and ψBpuq we have

ψApuq  ||u||ψA  u ||u|| ¥ α||u|| and β||u|| ¥ ||u||ψB  u ||u||  ψBpuq.

Let γ  α{β. Since α and β are strictly positive and finite, γ is also strictly positive and finite. The required inequality follows:

ψApuq ¥ α||u|| 

α

ββ||u||  γβ||u|| ¥ γψBpuq.

Proposition 6.16 For any positive basis N , the augmenting functions ψN

8 and ψ1N given in (6.11)

137 Proof. Let V  B_ε8p0q be an open ball in the infinity norm with radius ε ¡ 0 centred at the origin. This is an appropriate open neighbourhood of 0 for the purposes of Conditions 2 and 3 of Assumption 6.3.

Condition 1: ψp0q  0.

If u 0 then nJ_i u 0 and therefore ψN

8puq  0 and ψ1Npuq  0, as required.

Condition 2: ψpuq ¥ δ ¡ 0, @u R V for some positive scalar δ.

Using Theorem 6.8, for any u 0 we have some i such that nJ_i u ¡ 0 and hence ψ₈Npuq ¡ 0. Now define

δ : min

u t maxi1,...,ltn J

i uu | }u}8  εu ¡ 0, (6.13)

where δ ¡ 0 follows from the compactness of the ε- ball, the continuity of u ÞÑ maxi1,...,ltnJi uu,

and Theorem 6.8. For any uR V , the point v : ε_}u}u

8 is in V and hence ψ

N

8pvq ¥ δ ¡ 0. Using the

positive homogeneity property we have

ε }u}8ψ N 8puq ¥ δ ¡ 0 and so ψN 8puq ¥ δ}u}ε8 ¥ δ ¡ 0,

using the fact that uR V means }u}₈ ¥ ε. This is the required inequality for ψN 8.

Apply Lemma 6.15 to deduce that there exists a η ¡ 0 such that: ψN₁ puq ¥ ηψ₈Npuq ¥ ηδ ¡ 0 for all u R V. ηδ is also a positive scalar and so this is the required inequality for ψN

1 .

Condition 3: ψpuq ¥ γ||u||₈,@u P V for some positive scalar γ.

The property holds trivially for u 0. For any u P V zt0u, the point v : ε_}u}u

8 is in V and using

the same δ as defined in (6.13) we have

ε }u}8ψ

N

8puq ¥ δ ¡ 0

and so ψ₈Npuq ¥ δ}u}_ε8 ¥ δ_ε}u}₈ ¡ 0, and so we may place γ : δ_ε ¡ 0. This is the required inequality for ψN

8.

As above, apply Lemma 6.15 to deduce that there exists a η ¡ 0 such that: ψ₁Npuq ¥ ηψ₈Npuq ¥ ηγ}u}₈ ¡ 0.

ηγ is also a positive scalar and so this is the required inequality for ψN 1 .

Corollary 6.17 Suppose that ζSIP _{(as defined in (6.1)) satisfies Assumption 6.1, and that the}

penalty function ψ appearing in zLR

ρ pωq and ζρLD (as defined in (6.6) and (6.7) respectively) has

the form of (6.11) or (6.12). Then, for each ω P Ω, there is a finite ρ ¡ 0 for which we have

ζ_ρLD  zLR_ρ pωq  ζSIP (6.14)

Proof. Equalities (6.14) follow directly from Theorem 6.2, Remark 6.3, and Proposition 6.16. This result may be generalised to an even wider class of penalty functions.

Remark 6.18 Consider a positive basis N  tn1, . . . , nlu and the functions

gipuq : maxtnJi u, 0u.

Each function gipuq is non-negative, positive homogeneous and finite valued, and these properties

are preserved if multiple gis are summed, or their maximum is taken. By Theorem 6.8, for any

non-zero u there exists an index iP t1, . . . , lu such that gipuq is strictly positive. Therefore, if every

one of the gi functions is combined using a combination of summation and/or maximisation, the

resulting function gpuq will be strictly positive for all non-zero u. Applying Lemma 6.15 to bound g below by a positive multiple of ψ₈N (as ψ₁N was treated in Proposition 6.16) shows that this function gpuq satisfies the conditions of Assumption 6.3, and as such will close the duality gap if used as an augmenting function (as per Corollary 6.17).

Remark 6.19 By using the positive basist eiumi1Yt

°m

i1
eiu or similar to define an augmenting

function, we can obtain penalty terms analogous to the Lagrangian terms obtained through surrogate semi-Lagrangian relaxation (see Section 2.2.2).