Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
Robust Optimization
John E. Mitchell
Department of Mathematical SciencesRPI, Troy, NY 12180 USA
March 2020
Mitchell Robust Optimization 1 / 18
Robust LPs
Outline
1 Robust LPs
2 Box-constrained entries
3 Ellipsoid-constrained data
Mitchell Robust Optimization 2 / 18
Robust LPs
Robust LPs
We want to solve an LP with constraints Ax � b,but the data is uncertain.
minx cT xsubject to Ax � b 8 A 2 S
x � 0
for some uncertainty set S, typically a box or ellipsoid.
We have uncountably many scenarios: all A in an ellipsoid or a box.
We set up a subproblem: for a given x , what is the worst possible A?
With these structured uncertainty sets, can find the value Ax of theworst scenario explicitly, as a function of x .
Mitchell Robust Optimization 3 / 18
Box-constrained entries
Outline
1 Robust LPs
2 Box-constrained entries
3 Ellipsoid-constrained data
Mitchell Robust Optimization 4 / 18
Box-constrained entries
Box-constrained entries
Each entry falls in the range [aij � aij , aij + aij ], with aij � 0.
This can be modelled mathematically, where we replace each elementof A by its lower bound.
minx cT xsubject to
Pnj=1 (aij � aij)xj � bi 8 i
x � 0
Mitchell Robust Optimization 5 / 18
ai;ia¥aii→ai;tai;
o
¥2,aiixi.§,aijxi3 b i t i-
Box-constrained entries
Example
Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2].
x1
x2
0
2
6
3 6
Mitchell Robust Optimization 6 / 18
24.1¥.. -Lathan 3 6Enaipi
Box-constrained entries
Example
Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2].
x1
x2
0
2
6
3 6
Robust constraint:2x1 + 3x2 � x1 � 2x2 = x1 + x2 � 6
Mitchell Robust Optimization 6 / 18
-
[ai;x ; Edits
Box-constrained entries
What if only a proportion of the entries can differ?
A more complicated model is when only a proportion of the data differsfrom an expected value [3].
At first glance, this becomes a binary integer program.
By careful modeling, can still formulate the problem asa linear program with a polynomial number of constraints.
Mitchell Robust Optimization 7 / 18
:
Box-constrained entries
The Bertsimas-Sim model
For simplicity, we assume the variables are nonnegative. The modelcan be extended to the general case where l x u.
We allow at most �i entries in row i of the constraint matrix to differfrom their nominal values aij .
Si is the set of entries in row i that differ from their nominal values.
So |Si | �i for each row i .
Model:
minx cT xsubject to
Pnj=1 aijxj �
Pj2Si
aij xj � bi 8 |Si | �i , 8 i
x � 0
Mitchell Robust Optimization 8 / 18
O
-
Earlier: JE,a i ; x ;-§iaiixi?b i
Box-constrained entries
Rewrite the constraint using a max function
Our robust model is
minx cT xsubject to
Pnj=1 aijxj �
Pj2Si
aij xj � bi 8 |Si | �i , 8 i
x � 0
which is linear, but has an exponential number of constraints.
Can write the problem equivalently as a nonlinear program:
minx cT xsubject to
Pnj=1 aijxj � max|Si |�i
nPj2Si
aij xj
o� bi 8 i
x � 0
Mitchell Robust Optimization 9 / 18
- o f -
¥ ¥
Box-constrained entries
Solving the inner max problem parametrically
Our nonlinear formulation:
minx cT xsubject to
Pnj=1 aijxj � max|Si |�i
nPj2Si
aij xj
o� bi 8 i
x � 0
Given x , the max value in the constraint for row i can be found bysolving a linear program:
�i(x) := maxzPn
j=1 aij xj zjsubject to
Pnj=1 zj �i
0 zj 1 8 j
Note: zj = 1 () j 2 Si
Mitchell Robust Optimization 10 / 18
0
Box-constrained entries
Example
Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2], � = 1.
x1
x2
0
2
6
3 6
�(x) := maxz x1 z1 + 2x2 z2subject to z1 + z2 1
0 zj 1 j = 1, 2
Mitchell Robust Optimization 11 / 18
pin: :#' {iii.' I I . 2xitxi-sicxlz.coaerial 2x.tk.
-;Esiaiixi36HSik1.Lx,-13*-43624+3×-2×16
Box-constrained entries
Example
Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2], � = 1.
x1
x2
0
2
6
3 6
(3, 1)
�(x) := maxz x1 z1 + 2x2 z2subject to z1 + z2 1
0 zj 1 j = 1, 2
x = (3, 1) : �(x) = 3, 2x1 +3x2 ��(x) = 6+3�3 = 6 � 6X feasible
Mitchell Robust Optimization 11 / 18
•
P (x)= m a x 32,- 122 ,s t . Z ,t r , E l
° E-jet,;=1,2)2¥44
-
Box-constrained entries
Example
Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2], � = 1.
x1
x2
0
2
6
3 6
(1, 3)�(x) := maxz x1 z1 + 2x2 z2
subject to z1 + z2 10 zj 1 j = 1, 2
x = (3, 1) : �(x) = 3, 2x1 +3x2 ��(x) = 6+3�3 = 6 � 6X feasible
x = (1, 3) : �(x) = 6, 2x1+3x2��(x) = 2+9�6 = 5<6 X infeasible
Mitchell Robust Optimization 11 / 18
ACH= I I ? !762L}sola:z - c o ,l )
B (x)= G
-
Box-constrained entries
Example
Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2], � = 1.
x1
x2
0
2
6
3 6
Robust constraints:2x1 + 3x2 � x1 = x1 + 3x2 � 62x1 + 3x2 � 2x2 = 2x1 + x2 � 6
x = (3, 1) : �(x) = 3, 2x1 +3x2 ��(x) = 6+3�3 = 6 � 6X feasible
x = (1, 3) : �(x) = 6, 2x1+3x2��(x) = 2+9�6 = 5<6 X infeasible
Mitchell Robust Optimization 11 / 18
Lx,tha-j.Esidiixi3 6
Si:{Rc
si:{2}
Box-constrained entries
Use LP duality to remove the nonlinearity
From LP duality,
�i(x) := maxzPn
j=1 aij xj zjsubject to
Pnj=1 zj �i
0 zj 1 8 j
= minyi ,wi. �i yi +P
j wijsubject to yi + wij � aij xj 8 j
yi � 0, wij � 0 8 j
Here, yi is a scalar variable.
The nonlinearity in the objective disappears with duality.Now the RHS is parametrized by x .
Mitchell Robust Optimization 12 / 18
§,aiixi-Bilxbbi V i( l - d a dvyq.int[ d u e l variablewij.it,..-in
Box-constrained entries
Formulating the robust problem as an LPThe robust nonlinear program is
minx cT xsubject to
Pnj=1 aijxj � �i(x) � bi 8 i
x � 0
Using the duality representation for �i(x), this is equivalent to thelinear program
minx ,y ,w cT xsubject to
Pnj=1 aijxj � �i yi �
Pj wij � bi 8 i
aij xj � yi � wij 0 8 i , 8 jx � 0, y � 0, w � 0
Have O(mn) variables and constraints if A is m ⇥ n.
�(x) is a protection function for the constraints, protecting againstuncertainty.
Mitchell Robust Optimization 13 / 18
p i(x) :m i n Figitfwijs e .yituij?hi,-xiY i
- f i3 0 ,Wii?O
l d o .-
Box-constrained entries
Example
Nominal constraint: 2x1 + 3x2 � 6, a = [1, 2], � = 1.
One constraint, so one component y1. Two variables, so get w11, w12.
minx ,y ,w cT xsubject to 2x1 + 3x2 � y1 � w11 � w12 � 6
x1 � y1 � w11 02x2 � y1 � w12 0x � 0, y � 0, w � 0
x = (3, 1) : feasible with y1 = 2,w11 = 1,w12 = 0.
x = (1, 3) : infeasible for any y1,w11,w12.
Mitchell Robust Optimization 14 / 18
t.EE#. -
←mine.fix-iigi-fwipt.in. }hip;-Ji-oiif.fi,. . . -
Need: 2+9-y,- w "-wn76 5 - w, ,7 66 - y. - n o , ,g o }
a d d :
- 6 t o , t o , , 3 0impossible.
Ellipsoid-constrained data
Outline
1 Robust LPs
2 Box-constrained entries
3 Ellipsoid-constrained data
Mitchell Robust Optimization 15 / 18
Ellipsoid-constrained data
Second order cone formulationAnother popular model is to use ellipsoids to constrain possiblechoices of aij [2, 1]. With just one constraint:
minx cT xsubject to (a + a)T x � b 8 a satisfying aT M�1a 1
x � 0
For any given x , we have a subproblem to determine if the constraintholds:
mina
{xT a : aT M�1a 1}.
The solution is a = �1pxT Mx
Mx , so the constraint requires
aT x + mina
{xT a : aT M�1a 1} = aT x �p
xT Mx � b.
This is a second order cone constraint.Mitchell Robust Optimization 16 / 18
atxtatxzb.VE
E t .H i spositivedefinite.
-
O- 0
min I k c a , sulu. using KUTcondition..s i t . I ' M -'a 'E l
I +21M-' 2 = 0
0¥, ⇒ i t .# " I
↳ - IAlso,constraint i s active,
s o I'm-'2=1.
So:fj.INT?yfx=l,so2x=Viir#Soa=-¥r# " "
¥⇒=-TIMEOptimal value:
- I T M I
Ellipsoid-constrained data
Notes
Can extend to multiple constraints.Uncertainty can be drawn from a lower dimensional space.Here, each ai , ai , yi is a vector:
minx cT xs.t. (ai + ai)
T x � bi 8 ai = Hiyi with yTi M�1
i yi 1, 8 i
x � 0
Equivalent to the second order cone program
minx cT xs.t. aT
i x �q
xT HiMiHTi x � bi 8 i
x � 0
Mitchell Robust Optimization 17 / 18
Ellipsoid-constrained data
Notes
Can extend to multiple constraints.Uncertainty can be drawn from a lower dimensional space.Here, each ai , ai , yi is a vector:
minx cT xs.t. (ai + ai)
T x � bi 8 ai = Hiyi with yTi M�1
i yi 1, 8 i
x � 0
Equivalent to the second order cone program
minx cT xs.t. aT
i x �q
xT HiMiHTi x � bi 8 i
x � 0
Mitchell Robust Optimization 17 / 18
I = II,I ' I I I
A-' IE T
Ellipsoid-constrained data
A. Ben-Tal, L. El Ghaoui, and A. Nemirovski.Robust Optimization.Princeton University Press, Princeton, NJ, 2009.
A. Ben-Tal and A. Nemirovski.Robust solutions of uncertain linear programs.Operations Research Letters, 25(1):1–13, 1999.
D. Bertsimas and M. Sim.The price of robustness.Operations Research, 52(1):35–53, 2004.
Mitchell Robust Optimization 18 / 18
Ellipsoid-constrained data
A. Ben-Tal, L. El Ghaoui, and A. Nemirovski.Robust Optimization.Princeton University Press, Princeton, NJ, 2009.
A. Ben-Tal and A. Nemirovski.Robust solutions of uncertain linear programs.Operations Research Letters, 25(1):1–13, 1999.
D. Bertsimas and M. Sim.The price of robustness.Operations Research, 52(1):35–53, 2004.
Mitchell Robust Optimization 18 / 18
Ellipsoid-constrained data
A. Ben-Tal, L. El Ghaoui, and A. Nemirovski.Robust Optimization.Princeton University Press, Princeton, NJ, 2009.
A. Ben-Tal and A. Nemirovski.Robust solutions of uncertain linear programs.Operations Research Letters, 25(1):1–13, 1999.
D. Bertsimas and M. Sim.The price of robustness.Operations Research, 52(1):35–53, 2004.
Mitchell Robust Optimization 18 / 18