· joseph-louis lagrange (1736-1813) source: wikipedia w.w.r. ball, history of mathematics (3rd...
TRANSCRIPT
![Page 1: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/1.jpg)
Price Signalsin
Energy Optimization
Claudia Sagastizabal(IMECC-UNICAMP, adjunct researcher)
Workshop de Otimizacao
IM-UFRJRio, May 7, 2018
![Page 2: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/2.jpg)
Joseph-Louis Lagrange (1736-1813)
source: Wikipedia
W.W.R. Ball, History of Mathematics (3rd Ed., 1901)
I regard as quite useless the reading of largetreatises of pure analysis: too large a number of
methods pass at once before the eyes. It is in theworks of application that one must study them; one
judges their utility there and appraises the manner ofmaking use of them.
Je considere comme completement inutile la lecture de grostraites d’analyse pure: un trop grand nombre de methodes
passent en meme temps devant les yeux. C’est dans les travauxd’application qu’on doit les etudier; c’est la qu’on juge leurs
capacites et qu’on apprend la maniere de les utiliser.
There are many great quotes by Lagrange . . .
![Page 3: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/3.jpg)
Joseph-Louis Lagrange (1736-1813)
source: Wikipedia
W.W.R. Ball, History of Mathematics (3rd Ed., 1901)
I regard as quite useless the reading of largetreatises of pure analysis: too large a number of
methods pass at once before the eyes. It is in theworks of application that one must study them; one
judges their utility there and appraises the manner ofmaking use of them.
Je considere comme completement inutile la lecture de grostraites d’analyse pure: un trop grand nombre de methodes
passent en meme temps devant les yeux. C’est dans les travauxd’application qu’on doit les etudier; c’est la qu’on juge leurs
capacites et qu’on apprend la maniere de les utiliser.There are many great quotes by Lagrange . . .
![Page 4: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/4.jpg)
Joseph-Louis Lagrange (1736-1813)
source: Wikipedia
Before we take to sea we walk on land.Before we create we must understand.
Following Lagrange’s leadin today’s talkwe shall walk on the landof energy optimizationand make a few interesting observations
![Page 5: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/5.jpg)
Joseph-Louis Lagrange (1736-1813)
source: Wikipedia
Before we take to sea we walk on land.Before we create we must understand.
Following Lagrange’s leadin today’s talkwe shall walk on the landof energy optimizationand make a few interesting observations
![Page 6: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/6.jpg)
Joseph-Louis Lagrange (1736-1813)
source: Wikipedia
Thanks to Lagrangewe know electricity prices
1st obse
rvat
ion
![Page 7: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/7.jpg)
Walking on the application’s landManage at minimum cost the different technologies of aninterconnected electric power system, so that demand is satisfied
source: ONS
![Page 8: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/8.jpg)
Illustration with a simple example
Two power plants
pT ∈PT pH ∈PH
CT (pT ) CH(pH)
pT + pH = d (demand)
![Page 9: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/9.jpg)
Illustration with a simple example
Two power plants
pT ∈PT pH ∈PH
CT (pT ) CH(pH)
pT + pH = d (demand)
![Page 10: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/10.jpg)
Illustration with a simple example
Two power plants
pT ∈PT pH ∈PH
CT (pT ) CH(pH)
pT + pH = d (demand)
![Page 11: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/11.jpg)
Illustration with a simple example
Two power plants
pT ∈PT pH ∈PH
CT (pT ) CH(pH)
pT + pH = d (demand)
![Page 12: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/12.jpg)
Illustration with a simple example min CT (pT ) + CH(pH)s.t. pT ∈PT ,pH ∈PH
pT + pH = dTwo power plants
pT ∈PT pH ∈PH
CT (pT ) CH(pH)
pT + pH = d (demand)
![Page 13: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/13.jpg)
Pricing via marginal costThe perturbation function
p( u ) :=
inf CT (pT ) + CH(pH)s.t. pT ∈PT ,pH ∈PH
pT + pH = d+uis the key to price electricity:what is the cost increaseif demand increases in one unit?
Answer:compute p(·)’s directional derivative at 0
(p(0) is the optimal value of our problem)
![Page 14: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/14.jpg)
Pricing via marginal costThe perturbation function
p( u ) :=
inf CT (pT ) + CH(pH)s.t. pT ∈PT ,pH ∈PH
pT + pH = d+uis the key to price electricity:what is the cost increaseif demand increases in one unit?
Answer:compute p(·)’s directional derivative at 0
(p(0) is the optimal value of our problem)
![Page 15: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/15.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u ← x + ↑
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
s.t. pT + pH = d + u
![Page 16: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/16.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup??
of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u ← x + ↑
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
s.t. pT + pH = d + u
![Page 17: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/17.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u ← x + ↑
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
s.t. pT + pH = d + u
![Page 18: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/18.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u
← x + ↑
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
s.t. pT + pH = d + u
![Page 19: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/19.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u ← x
+ ↑
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
s.t. pT + pH = d + u
![Page 20: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/20.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u ← x + ↑
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
s.t. pT + pH = d + u
![Page 21: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/21.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u ← x + ↑
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
s.t. pT + pH = d + u
![Page 22: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/22.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u ← x + ↑
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
s.t. pT + pH = d + u
![Page 23: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/23.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
supx
infp∈P
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
supx{F(x ,u) := 〈x ,u〉+ constant terms in u}
![Page 24: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/24.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
supx
infp∈P
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
supx{F(x ,u) := 〈x ,u〉+ constant terms in u}
![Page 25: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/25.jpg)
Pricing via marginal cost: Danskin’s theoremFor max-functions, differentiating the argument provides the
directional derivative:
ϕ(u) = supx
F(x ,u) =⇒ ϕ′(u, ·) = 〈∇uF(x ,u), ·〉
p(u) = sup?? of the Lagrangian!
p(u) =
{inf
p∈PCT (pT ) + CH(pH)
s.t. pT + pH = d + u
= infp∈P
supx
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
supx
infp∈P
CT (pT ) + CH(pH) + 〈x ,d + u−pT −pH〉
supx{F(x ,u) := 〈x ,u〉+ constant terms in u}
![Page 26: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/26.jpg)
Pricing via marginal cost: Danskin’s theoremAt u = 0 Lagrange multiplier x gives directional derivative
p(0) =
min CT (pT ) + CH(pH)s.t. pT ∈PT ,pH ∈PH
pT + pH = d+0
= minp∈P
maxx
CT (pT ) + CH(pH) + 〈x ,d−pT −pH〉
L(p,x) = CT (pT ) + CH(pH) + 〈x ,d−pT −pH〉is the Lagrangian
p(0) = minp∈P
maxx
L(p,x)
maxx
minp∈P
L(p,x)
thanks to Lagrangewe know electricity prices
![Page 27: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/27.jpg)
Pricing via marginal cost: Danskin’s theoremAt u = 0 Lagrange multiplier x gives directional derivative
p(0) =
min CT (pT ) + CH(pH)s.t. pT ∈PT ,pH ∈PH
pT + pH = d+0= min
p∈Pmax
xCT (pT ) + CH(pH) + 〈x ,d−pT −pH〉
L(p,x) = CT (pT ) + CH(pH) + 〈x ,d−pT −pH〉is the Lagrangian
p(0) = minp∈P
maxx
L(p,x)
maxx
minp∈P
L(p,x)
thanks to Lagrangewe know electricity prices
![Page 28: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/28.jpg)
Pricing via marginal cost: Danskin’s theoremAt u = 0 Lagrange multiplier x gives directional derivative
p(0) =
min CT (pT ) + CH(pH)s.t. pT ∈PT ,pH ∈PH
pT + pH = d+0= min
p∈Pmax
xCT (pT ) + CH(pH) + 〈x ,d−pT −pH〉
L(p,x) = CT (pT ) + CH(pH) + 〈x ,d−pT −pH〉is the Lagrangian
p(0) = minp∈P
maxx
L(p,x)
maxx
minp∈P
L(p,x)
thanks to Lagrangewe know electricity prices
![Page 29: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/29.jpg)
Pricing via marginal cost: Danskin’s theoremAt u = 0 Lagrange multiplier x gives directional derivative
p(0) =
min CT (pT ) + CH(pH)s.t. pT ∈PT ,pH ∈PH
pT + pH = d+0= min
p∈Pmax
xCT (pT ) + CH(pH) + 〈x ,d−pT −pH〉
L(p,x) = CT (pT ) + CH(pH) + 〈x ,d−pT −pH〉is the Lagrangian
p(0) = minp∈P
maxx
L(p,x)
maxx
minp∈P
L(p,x)
thanks to Lagrangewe know electricity prices
![Page 30: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/30.jpg)
Pricing via marginal cost: Danskin’s theoremAt u = 0 Lagrange multiplier x gives directional derivative
p(0) =
min CT (pT ) + CH(pH)s.t. pT ∈PT ,pH ∈PH
pT + pH = d+0= min
p∈Pmax
xCT (pT ) + CH(pH) + 〈x ,d−pT −pH〉
L(p,x) = CT (pT ) + CH(pH) + 〈x ,d−pT −pH〉is the Lagrangian
p(0) = minp∈P
maxx
L(p,x)
maxx
minp∈P
L(p,x)
thanks to Lagrangewe know electricity prices
![Page 31: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/31.jpg)
Pricing via marginal cost
At u = 0 Lagrange multiplier x gives directional derivative
p(0) = minp∈P
maxx
L(p,x)
maxx
minp∈P
L(p,x)
without convexity
![Page 32: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/32.jpg)
Pricing via marginal cost: there is a catch!
At u = 0 Lagrange multiplier x gives directional derivative
p(0) = minp∈P
maxx
L(p,x)
p∗∗(0) = maxx
minp∈P
L(p,x)
without convexity p(0)≥ p∗∗(0)= dualPros and cons+ Separability ( 6= technologies )
+ Accurate marginal prices in short CPU timeif using a bundle method with on-demand accuracy
(subgradient/Uzawa-like methods do not get enough precision)– Loss of primal feasibility
![Page 33: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/33.jpg)
Pricing via marginal cost: there is a catch!
At u = 0 Lagrange multiplier x gives directional derivative
p(0) = minp∈P
maxx
L(p,x)
p∗∗(0) = maxx
minp∈P
L(p,x)
without convexity p(0)≥ p∗∗(0)= dualPros and cons+ Separability ( 6= technologies )
+ Accurate marginal prices in short CPU timeif using a bundle method with on-demand accuracy
(subgradient/Uzawa-like methods do not get enough precision)– Loss of primal feasibility
![Page 34: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/34.jpg)
Pricing via marginal cost: there is a catch!
At u = 0 Lagrange multiplier x gives directional derivative
p(0) = minp∈P
maxx
L(p,x)
p∗∗(0) = maxx
minp∈P
L(p,x)
without convexity p(0)≥ p∗∗(0)= dual
Pros and cons+ Separability ( 6= technologies )
+ Accurate marginal prices in short CPU timeif using a bundle method with on-demand accuracy
(subgradient/Uzawa-like methods do not get enough precision)– Loss of primal feasibility
![Page 35: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/35.jpg)
Pricing via marginal cost: there is a catch!
At u = 0 Lagrange multiplier x gives directional derivative
p(0) = minp∈P
maxx
L(p,x)
p∗∗(0) = maxx
minp∈P
L(p,x)
without convexity p(0)≥ p∗∗(0)= dualPros and cons+ Separability ( 6= technologies )
+ Accurate marginal prices in short CPU timeif using a bundle method with on-demand accuracy
(subgradient/Uzawa-like methods do not get enough precision)– Loss of primal feasibility
![Page 36: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/36.jpg)
Illustration with a simple examplePower plants have same capacity (P) but different technology
0-1 variables continuous variablesgenerates either 0 or P any power in [0,P]
PT = {0,P} PH = [0,P]expensive cheap
Suppose demand is d = 1.5P so that both power plants aredispatched: pT = P and pH = 1
2PThe dual approach, that finds x via p∗∗(0), amounts to solvingan LP with PT = {0,P} replaced by conv(PT ) = [0,P]Because p∗∗ = conv(p) < p, the associated dispatch isinfeasible: pdual
T = 12P and pdual
H = P
![Page 37: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/37.jpg)
Illustration with a simple examplePower plants have same capacity (P) but different technology
0-1 variables continuous variables
generates either 0 or P any power in [0,P]PT = {0,P} PH = [0,P]
expensive cheapSuppose demand is d = 1.5P so that both power plants aredispatched: pT = P and pH = 1
2PThe dual approach, that finds x via p∗∗(0), amounts to solvingan LP with PT = {0,P} replaced by conv(PT ) = [0,P]Because p∗∗ = conv(p) < p, the associated dispatch isinfeasible: pdual
T = 12P and pdual
H = P
![Page 38: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/38.jpg)
Illustration with a simple examplePower plants have same capacity (P) but different technology
0-1 variables continuous variablesgenerates either 0 or P any power in [0,P]
PT = {0,P} PH = [0,P]
expensive cheapSuppose demand is d = 1.5P so that both power plants aredispatched: pT = P and pH = 1
2PThe dual approach, that finds x via p∗∗(0), amounts to solvingan LP with PT = {0,P} replaced by conv(PT ) = [0,P]Because p∗∗ = conv(p) < p, the associated dispatch isinfeasible: pdual
T = 12P and pdual
H = P
![Page 39: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/39.jpg)
Illustration with a simple examplePower plants have same capacity (P) but different technology
0-1 variables continuous variablesgenerates either 0 or P any power in [0,P]
PT = {0,P} PH = [0,P]expensive cheap
Suppose demand is d = 1.5P so that both power plants aredispatched:
pT = P and pH = 12P
The dual approach, that finds x via p∗∗(0), amounts to solvingan LP with PT = {0,P} replaced by conv(PT ) = [0,P]Because p∗∗ = conv(p) < p, the associated dispatch isinfeasible: pdual
T = 12P and pdual
H = P
![Page 40: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/40.jpg)
Illustration with a simple examplePower plants have same capacity (P) but different technology
0-1 variables continuous variablesgenerates either 0 or P any power in [0,P]
PT = {0,P} PH = [0,P]expensive cheap
Suppose demand is d = 1.5P so that both power plants aredispatched: pT = P and pH = 1
2P
The dual approach, that finds x via p∗∗(0), amounts to solvingan LP with PT = {0,P} replaced by conv(PT ) = [0,P]Because p∗∗ = conv(p) < p, the associated dispatch isinfeasible: pdual
T = 12P and pdual
H = P
![Page 41: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/41.jpg)
Illustration with a simple examplePower plants have same capacity (P) but different technology
0-1 variables continuous variablesgenerates either 0 or P any power in [0,P]
PT = {0,P} PH = [0,P]expensive cheap
Suppose demand is d = 1.5P so that both power plants aredispatched: pT = P and pH = 1
2PThe dual approach, that finds x via p∗∗(0), amounts to solvingan LP with PT = {0,P} replaced by conv(PT ) = [0,P]
Because p∗∗ = conv(p) < p, the associated dispatch isinfeasible: pdual
T = 12P and pdual
H = P
![Page 42: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/42.jpg)
Illustration with a simple examplePower plants have same capacity (P) but different technology
0-1 variables continuous variablesgenerates either 0 or P any power in [0,P]
PT = {0,P} PH = [0,P]expensive cheap
Suppose demand is d = 1.5P so that both power plants aredispatched: pT = P and pH = 1
2PThe dual approach, that finds x via p∗∗(0), amounts to solvingan LP with PT = {0,P} replaced by conv(PT ) = [0,P]Because p∗∗ = conv(p) < p, the associated dispatch isinfeasible: pdual
T = 12P and pdual
H = P
![Page 43: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/43.jpg)
Illustration with a simple example
For our simple problem p(u) ={
inf CT (pT )+CH (pH )s.t. pT ∈PT ,pH ∈PH
pT +pH = d +u
is the minimum of two quadratic functions
There is a duality gap
when p is not convex(p(0) > p∗∗(0))
p(0)→← p∗∗(0)
![Page 44: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/44.jpg)
Illustration with a simple example
For our simple problem p(u) ={
inf CT (pT )+CH (pH )s.t. pT ∈PT ,pH ∈PH
pT +pH = d +u
is the minimum of two quadratic functions
There is a duality gap
when p is not convex
(p(0) > p∗∗(0))
p(0)→← p∗∗(0)
![Page 45: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/45.jpg)
Illustration with a simple example
For our simple problem p(u) ={
inf CT (pT )+CH (pH )s.t. pT ∈PT ,pH ∈PH
pT +pH = d +u
is the minimum of two quadratic functions
There is a duality gap
when p is not convex(p(0) > p∗∗(0))
p(0)→← p∗∗(0)
![Page 46: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/46.jpg)
Remarks:
Price signals in Energy Optimization
I are always related to dual variables
I nonconvexity complicates calculations
I direct 0-1 solution not possible:there are no dual variables!
![Page 47: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/47.jpg)
Remarks:
Price signals in Energy Optimization
I are always related to dual variables
I nonconvexity complicates calculations
I direct 0-1 solution not possible:there are no dual variables!
![Page 48: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/48.jpg)
Remarks:
Price signals in Energy Optimization
I are always related to dual variables
I nonconvexity complicates calculations
I direct 0-1 solution not possible:there are no dual variables!
![Page 49: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/49.jpg)
Remarks:
Price signals in Energy Optimization
I are always related to dual variables
I nonconvexity complicates calculations
I direct 0-1 solution not possible:
there are no dual variables!
![Page 50: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/50.jpg)
Remarks:
Price signals in Energy Optimization
I are always related to dual variables
I nonconvexity complicates calculations
I direct 0-1 solution not possible:there are no dual variables!
![Page 51: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/51.jpg)
Another star? in decomposition methods
source: Twitter
![Page 52: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/52.jpg)
A less simple example
Two power plants
pT ∈PT pH ∈PH
PT = [0,P] PH = [0,P]
x ∈ {0,1} and pT ≤ x P〈c,x〉+ CT (pT ) CH(pH)
pT + pH = d(demand)
![Page 53: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/53.jpg)
A less simple example
Two power plants
pT ∈PT pH ∈PH
PT = [0,P] PH = [0,P]x ∈ {0,1} and pT ≤ x P
〈c,x〉+ CT (pT ) CH(pH)
pT + pH = d(demand)
![Page 54: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/54.jpg)
A less simple example
Two power plants
pT ∈PT pH ∈PH
PT = [0,P] PH = [0,P]x ∈ {0,1} and pT ≤ x P〈c,x〉+ CT (pT ) CH(pH)
pT + pH = d(demand)
![Page 55: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/55.jpg)
A less simple example
Two power plants
pT ∈PT pH ∈PH
PT = [0,P] PH = [0,P]x ∈ {0,1} and pT ≤ x P〈c,x〉+ CT (pT ) CH(pH)
pT + pH = d(demand)
![Page 56: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/56.jpg)
Optimization problemmin 〈c,x〉+CT (pT )+CH(pH)
s.t. x ∈ {0,1} ,pT ,pH ∈ [0,P]pT ≤ x PpT +pH = d
![Page 57: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/57.jpg)
Optimization problemmin 〈c,x〉+CT (pT )+CH(pH)
s.t. x ∈ {0,1} ,pT ,pH ∈ [0,P]pT ≤ x PpT +pH = d
![Page 58: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/58.jpg)
Optimization problem
min 〈c,x〉+ CT (pT ) + CH(pH)s.t. x ∈ {0,1} ,pT ,pH ∈ [0,P]
pT ≤ x PpT + pH = d
min 〈c,x〉+ Q(x)s.t. x ∈ {0,1}
Ax = bfor Q(x) :=
min CT (pT ) + CH(pH) ↔ C(p)s.t. pT ,pH ∈ [0,P] ↔ p ∈ [0,P]
pT ≤ x PpT + pH = d
}↔ Wp = h−Tx
Bender’s decomposition separates the optimizationproblem in two decision levels.
![Page 59: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/59.jpg)
Optimization problem
min 〈c,x〉+ CT (pT ) + CH(pH)s.t. x ∈ {0,1} ,pT ,pH ∈ [0,P]
pT ≤ x PpT + pH = d
min 〈c,x〉+ Q(x)s.t. x ∈ {0,1}
Ax = bfor Q(x) :=
min CT (pT ) + CH(pH) ↔ C(p)s.t. pT ,pH ∈ [0,P] ↔ p ∈ [0,P]
pT ≤ x PpT + pH = d
}↔ Wp = h−Tx
Bender’s decomposition separates the optimizationproblem in two decision levels.
![Page 60: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/60.jpg)
Optimization problem
min 〈c,x〉+ CT (pT ) + CH(pH)s.t. x ∈ {0,1} ,pT ,pH ∈ [0,P]
pT ≤ x PpT + pH = d
min 〈c,x〉+ Q(x)s.t. x ∈ {0,1}
Ax = bfor Q(x) :=
min CT (pT ) + CH(pH) ↔ C(p)s.t. pT ,pH ∈ [0,P] ↔ p ∈ [0,P]
pT ≤ x PpT + pH = d
}↔ Wp = h−Tx
Bender’s decomposition separates the optimizationproblem in two decision levels.
![Page 61: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/61.jpg)
Optimization problem
min 〈c,x〉+ CT (pT ) + CH(pH)s.t. x ∈ {0,1} ,pT ,pH ∈ [0,P]
pT ≤ x PpT + pH = d
min 〈c,x〉+ Q(x)s.t. x ∈ {0,1}
Ax = bfor Q(x) :=
min CT (pT ) + CH(pH) ↔ C(p)s.t. pT ,pH ∈ [0,P] ↔ p ∈ [0,P]
pT ≤ x PpT + pH = d
}↔ Wp = h−Tx
Bender’s decomposition separates the optimizationproblem in two decision levels.
![Page 62: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/62.jpg)
Optimization problem
min 〈c,x〉+ CT (pT ) + CH(pH)s.t. x ∈ {0,1} ,pT ,pH ∈ [0,P]
pT ≤ x PpT + pH = d
min 〈c,x〉+ Q(x)s.t. x ∈ {0,1}
Ax = bfor Q(x) :=
min CT (pT ) + CH(pH) ↔ C(p)s.t. pT ,pH ∈ [0,P] ↔ p ∈ [0,P]
pT ≤ x PpT + pH = d
}↔ Wp = h−Tx
Bender’s decomposition separates the optimizationproblem in two decision levels.
If d or C uncertain,this is a 2-stage stochastic programming problem
![Page 63: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/63.jpg)
Optimization problem
min 〈c,x〉+ CT (pT ) + CH(pH)s.t. x ∈ {0,1} ,pT ,pH ∈ [0,P]
pT ≤ x PpT + pH = d
min 〈c,x〉+ Q(x)s.t. x ∈ {0,1}
Ax = bfor Q(x) :=
min CT (pT ) + CH(pH) ↔ C(p)s.t. pT ,pH ∈ [0,P] ↔ p ∈ [0,P]
pT ≤ x PpT + pH = d
}↔ Wp = h−Tx
Bender’s decomposition separates the optimizationproblem in two decision levels. If d or C uncertain,this is a 2-stage stochastic programming problem
![Page 64: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/64.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx
← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x , ωs)
![Page 65: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/65.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x , ωs)
![Page 66: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/66.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}
⇒ E [Q(x ,ω)]≈ 1S ∑
sQ(x ,ωs) and E [π(x ,ω)]≈ 1
S ∑s
π(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x , ωs)
![Page 67: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/67.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs)
and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x , ωs)
![Page 68: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/68.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x , ωs)
![Page 69: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/69.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}
⇒ E [Q(x ,ω)]≈ 1S ∑
sQ(x , ωs) and E [π(x ,ω)]≈ 1
S ∑s
π(x , ωs)
![Page 70: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/70.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x , ωs)
![Page 71: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/71.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x,ω)]≈ 1S ∑
sπ(x, ωs)
source: Clara Lage
How much prices change? 1
[1]R&D contract with , joint with C. Lage and M. Solodov
![Page 72: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/72.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x,ω)]≈ 1S ∑
sπ(x, ωs)
source: Clara Lage
How much prices change? 1
[1]R&D contract with , joint with C. Lage and M. Solodov
![Page 73: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/73.jpg)
Price Signals in 2-SLPmin 〈c,x〉+E [Q(x ,ω)]s.t. x ∈ {0,1}
Ax = bQ(x ,ω) =
min C(p)s.t. p ∈ [0,P]
Wp = h(ω)−Tx ← π(x ,ω)
SAA: ω ∈ {ω1, . . . ,ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x ,ωs) and E [π(x ,ω)]≈ 1S ∑
sπ(x ,ωs)
A different sample: ω ∈ {ω1, . . . , ωS}⇒ E [Q(x ,ω)]≈ 1
S ∑s
Q(x , ωs) and E [π(x,ω)]≈ 1S ∑
sπ(x, ωs)
source: Clara Lage
How much prices change? 1
[1]R&D contract with , joint with C. Lage and M. Solodov
![Page 74: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/74.jpg)
Multiplier sensitivity to uncertainty representationA simple exercise
I Take a 2-SLP with uncertain right-hand side(h(ω))
I Given the sample ω ∈ {ω1m, . . . ,ω
Sm}:
I The SAAm has first-stage solution xmI The associated optimal mean price is
πm :=1S ∑
sπ(xm,ω
sm)
I Repeat for m = 1, . . . ,M and compute thevariance of π1, . . . , πM
Var [π] can be very large!
![Page 75: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/75.jpg)
Multiplier sensitivity to uncertainty representationA simple exercise
I Take a 2-SLP with uncertain right-hand side(h(ω))
I Given the sample ω ∈ {ω1m, . . . ,ω
Sm}:
I The SAAm has first-stage solution xmI The associated optimal mean price is
πm :=1S ∑
sπ(xm,ω
sm)
I Repeat for m = 1, . . . ,M and compute thevariance of π1, . . . , πM
Var [π] can be very large!
![Page 76: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/76.jpg)
Stabilizing dual variablesTo mitigate wild oscillations, for β > 0 consider
Qβ (x ,ω) :=
min C(p) + 1
2β‖Tx + Wp−h(ω)‖2
s.t. p ∈ [0,P]
Wp = h(ω)−Tx ↗
I corresponds to a quadratic term β
2‖π‖2 in the
dualI if pβ = pβ (x ,ω) solves the regularized problem,
then
πβ (x ,ω) :=
1β
(Tx + Wpβ −h(ω)
)I xβ now solves
{min 〈c,x〉+E
[Qβ (x ,ω)
]s.t. Ax = b ,x ≥ 0
![Page 77: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/77.jpg)
Stabilizing dual variablesTo mitigate wild oscillations, for β > 0 consider
Qβ (x ,ω) :=
min C(p) + 1
2β‖Tx + Wp−h(ω)‖2
s.t. p ∈ [0,P]
Wp = h(ω)−Tx ↗
I corresponds to a quadratic term β
2‖π‖2 in the
dual
I if pβ = pβ (x ,ω) solves the regularized problem,then
πβ (x ,ω) :=
1β
(Tx + Wpβ −h(ω)
)I xβ now solves
{min 〈c,x〉+E
[Qβ (x ,ω)
]s.t. Ax = b ,x ≥ 0
![Page 78: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/78.jpg)
Stabilizing dual variablesTo mitigate wild oscillations, for β > 0 consider
Qβ (x ,ω) :=
min C(p) + 1
2β‖Tx + Wp−h(ω)‖2
s.t. p ∈ [0,P]
Wp = h(ω)−Tx ↗
I corresponds to a quadratic term β
2‖π‖2 in the
dualI if pβ = pβ (x ,ω) solves the regularized problem,
then
πβ (x ,ω) :=
1β
(Tx + Wpβ −h(ω)
)
I xβ now solves
{min 〈c,x〉+E
[Qβ (x ,ω)
]s.t. Ax = b ,x ≥ 0
![Page 79: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/79.jpg)
Stabilizing dual variablesTo mitigate wild oscillations, for β > 0 consider
Qβ (x ,ω) :=
min C(p) + 1
2β‖Tx + Wp−h(ω)‖2
s.t. p ∈ [0,P]
Wp = h(ω)−Tx ↗
I corresponds to a quadratic term β
2‖π‖2 in the
dualI if pβ = pβ (x ,ω) solves the regularized problem,
then
πβ (x ,ω) :=
1β
(Tx + Wpβ −h(ω)
)I xβ now solves
{min 〈c,x〉+E
[Qβ (x ,ω)
]s.t. Ax = b ,x ≥ 0
![Page 80: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/80.jpg)
Stabilizing dual variablesTo mitigate wild oscillations, for βk → 0 consider
Qβk (x ,ω) :=
{min C(p) + 1
2βk‖Tx + Wp−h(ω)‖2
s.t. p ∈ [0,P]
I corresponds to a quadratic term βk2 ‖π‖
2 in thedual
I if pk = pβk (x ,ω) solves the regularized problem,then
πk(x ,ω) :=
1βk
(Tx + Wpk −h(ω)
)I xk now solves
{min 〈c,x〉+E
[Qβk (x ,ω)
]s.t. Ax = b ,x ≥ 0
![Page 81: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/81.jpg)
Stabilizing dual variables: theoremUnder suitable CQ and 2nd order conditions,as βk → 0 ,
I the sequence {xk} is bounded,I each accumulation point x∞ solves the original
problemI the sequence of mean regularized prices{
πk :=
1S ∑
sπ
k(xk ,ωs)
}converges to the
multiplier π with minimum norm .
![Page 82: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/82.jpg)
Stabilizing dual variablesAs βk decreases the variance Var
[πk(xk ,ω)
]increases
=⇒ a suitable value for βk must be found
![Page 83: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/83.jpg)
Remarks (suite):
Price signals in Energy Optimization
I are always related to dual variables
I uncertainty complicates calculations
I quadratic regularization in the dual stabilizes theoutcome
I nonconvexity complicates calculations
I regarding nonconvexity . . .
![Page 84: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/84.jpg)
Remarks (suite):
Price signals in Energy Optimization
I are always related to dual variables
I uncertainty complicates calculations
I quadratic regularization in the dual stabilizes theoutcome
I nonconvexity complicates calculations
I regarding nonconvexity . . .
![Page 85: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/85.jpg)
Remarks (suite):
Price signals in Energy Optimization
I are always related to dual variables
I uncertainty complicates calculations
I quadratic regularization in the dual stabilizes theoutcome
I nonconvexity complicates calculations
I regarding nonconvexity . . .
![Page 86: · Joseph-Louis Lagrange (1736-1813) source: Wikipedia W.W.R. Ball, History of Mathematics (3rd Ed., 1901) I regard as quite useless the reading of large treatises of pure analysis:](https://reader033.vdocument.in/reader033/viewer/2022043010/5fa0e5dce6bded6cc9149a07/html5/thumbnails/86.jpg)
Solucoes Matematicas para Problemas Industriais
I Modelling schoolI WorkshopI 2018 topic:
Formacao de precosno Despacho Hidrotermico
de Curto PrazoI July 9 to 20th, 2018I Sao Carlos, SP
http://www.cemeai.icmc.usp.br/3WSMPI/