Approximative Methods for Monotone Systems
of min-max-Polynomial Equations
Javier Esparza Thomas Gawlitza
Stefan Kiefer Helmut Seidl
Technische Universität München
LAA, Edinburgh, July 23, 2008
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Monotone Systems of Polynomial min-max-Equations
(min-max-MSPEs)
We study equation systems of the form
X = f (X ) ,
Example (n = 2)
X
Y
with
= max{1.2X + 0.3Y 2 , 0.9Y }
= min{0.7X 2 + 0.2, 0.4X }
n variables in X ,
n positive min-max-polynomials in f (X ).
A positive min-max-polynomial is a term composed of
variables, positive reals,
and the operations +, ·, max, min
We are interested in the least nonnegative solution of X = f (X ),
i.e., the least fixed point µf .
(if it exists)
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Kleene and the Least Fixed Point
The “Kleene sequence” 0, f (0), f 2 (0), . . . converges to µf ,
if µf exists.
Example
X
Y
= max{Y , 2}
= min{0.5X 2 + 0.5,
3}
0
2
2
2.5
2
3
0=
, f (0) =
, f (0) =
, f (0) =
,
0
0.5
2.5
2.5
2.5
3
4
5
f (0) =
, f (0) =
= f 6 (0) = µf
3
3
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Example: The Flu
A patient has the flu. The doctor has two options:
Do not treat him with medicine.
Treat him with Muniflu.
U = probability to cure an (initially Untreated) patient
and all people he infects
T = probability to cure a Treated patient
and all people he infects
not treat
treat
z
}|
{ z
}|
{
U = max{0.3 + 0.7UU, 0.9T + 0.1U}
0.35
+{z
0.65TU}
T =
|
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Example: The Flu
A patient has the flu. The doctor has two options:
Do not treat him with medicine.
Treat him with Muniflu.
U = probability to cure an (initially Untreated) patient
and all people he infects
T = probability to cure a Treated patient
and all people he infects
not treat
treat
z
}|
{ z
}|
{
U = max{0.3 + 0.7UU, 0.9T + 0.1U}
0.35
+{z
0.65TU}
T =
|
The probabilities are the least fixed point.
(Not necessarily T = U = 1!)
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Example: The Flu
A patient has the flu. The doctor has two options:
Do not treat him with medicine.
Treat him with Muniflu.
U = probability to cure an (initially Untreated) patient
and all people he infects
T = probability to cure a Treated patient
and all people he infects
not treat
treat
z
}|
{ z
}|
{
U = max{0.3 + 0.7UU, 0.9T + 0.1U}
0.35
+{z
0.65TU} , 0.5
T =
|
| + 0.2TU
{z + 0.3TUU}
Influenza A
Influenza B
The probabilities are the least fixed point.
(Not necessarily T = U = 1!)
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Example: The Flu
A patient has the flu. The doctor has two options:
Do not treat him with medicine.
Treat him with Muniflu.
U = probability to cure an (initially Untreated) patient
and all people he infects
T = probability to cure a Treated patient
and all people he infects
not treat
treat
z
}|
{ z
}|
{
U = max{0.3 + 0.7UU, 0.9T + 0.1U}
+{z
0.65TU} , 0.5
T = min { 0.35
|
| + 0.2TU
{z + 0.3TUU} }
Influenza A
Influenza B
The probabilities are the least fixed point.
(Not necessarily T = U = 1!)
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Need for Approximation
In general µf cannot be computed exactly,
even without min/max.
Example
Let f (X ) = 16 X 6 + 12 X 5 + 13 .
Then µf is not expressible by roots.
Need for approximation
(0.3357037075 < µf < 0.3357037076)
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration may be slow.
Kleene iteration may converge slowly, even without min/max.
Example
The least fixed point of f (X ) = 0.5X 2 + 0.5 is µf = 1.
Kleene sequence:
0, f (0) = 0.5, f 2 (0) = 0.62, f 3 (0) = 0.69, f 4 (0) = 0.74, . . .
f 20 (0) = 0.920, . . . , f 200 (0) = 0.990, . . . , f 2000 (0) = 0.9990
“logarithmic convergence”: k iterations for log k bits
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Kleene Iteration (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
Slow!
0.6
f (X )
(k iterations for about log k bits)
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
0.4
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method (univariate case)
Consider f (X ) = 38 X 2 + 14 X +
3
8
1.2
1
µf
0.8
0.6
f (X )
Fast!
0.4
(k iterations for k bits)
0.2
0
0.2
0.4
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
0.6
0.8
1
1.2
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method
Let X = f (X ) be an MSPE (without min/max for now).
Let ν be some approximate of µf . (We start with ν = 0.)
Improve it as follows to get a better approximate ν 0 :
1
Compute the “tangent” of f at ν:
T (f , ν)(X ) := f (ν) + f 0 (ν) · (X − ν)
2
Find the least fixed point of T (f , ν):
ν 0 := µT (f , ν)
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Systems without Min/Max
For systems without min/max we know well about the
convergence of Newton’s method [EY05, KLE07, EKL08].
In particular (important for this talk):
It converges to µf (from below, started from 0).
It converges (at least) linearly, i.e.,
number of bits depends linearly on number of iterations.
In the example: k bits after k iterations
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Systems without Min/Max
For systems without min/max we know well about the
convergence of Newton’s method [EY05, KLE07, EKL08].
In particular (important for this talk):
It converges to µf (from below, started from 0).
It converges (at least) linearly, i.e.,
number of bits depends linearly on number of iterations.
In the example: k bits after k iterations
What about systems with min/max?
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Min-Systems Does Not Work.
Consider Newton’s Method on f (X ) = min{g(X ), h(X )}.
2.5
h(X )
2
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Min-Systems Does Not Work.
Consider Newton’s Method on f (X ) = min{g(X ), h(X )}.
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Min-Systems Does Not Work.
Consider Newton’s Method on f (X ) = min{g(X ), h(X )}.
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Min-Systems Does Not Work.
Consider Newton’s Method on f (X ) = min{g(X ), h(X )}.
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Min-Systems Does Not Work.
Consider Newton’s Method on f (X ) = min{g(X ), h(X )}.
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Min-Systems Does Not Work.
Consider Newton’s Method on f (X ) = min{g(X ), h(X )}.
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Min-Systems Does Not Work.
Consider Newton’s Method on f (X ) = min{g(X ), h(X )}.
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
1.5
2
2.5
Newton’s method doesn’t work!
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Newton’s Method for Min-Systems Does Not Work.
Consider Newton’s Method on f (X ) = min{g(X ), h(X )}.
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
1.5
2
2.5
Let’s modify it!
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
1.5
2
2.5
This method works!
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Our Method for Systems with Min
Again f (X ) = min{g(X ), h(X )}. But linearize both parts!
2.5
h(X )
2
µf
1.5
1
0.5
0
g(X )
0.5
1
1.5
2
2.5
Asymptotically, it converges as fast to µf
as Newton’s method without min/max.
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
How Our Min-Method Works
Let ν be an approximate of µf . Improve it as follows:
Each component fi (X ) is linearized separately:
If fi (X ) does not have a min, then as before:
Li (X ) := T (fi , ν)(X )
= fi (ν) + fi0 (ν) · (X − ν)
If fi (X ) = min{gi (X ), hi (X )}, linearize both gi and hi :
Li (X ) := min{T (gi , ν)(X ), T (hi , ν)(X )}
The resulting system L(X ) is “min-linear”.
Its least fixed point can be found by solving one linear program:
ν 0 := maximize X1 + · · · + Xn subject to 0 ≤ X ≤ L(X )
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
How Our Min-Method Works
Let ν be an approximate of µf . Improve it as follows:
Each component fi (X ) is linearized separately:
If fi (X ) does not have a min, then as before:
Li (X ) := T (fi , ν)(X )
= fi (ν) + fi0 (ν) · (X − ν)
If fi (X ) = min{gi (X ), hi (X )}, linearize both gi and hi :
Li (X ) := min{T (gi , ν)(X ), T (hi , ν)(X )}
The resulting system L(X ) is “min-linear”.
Its least fixed point can be found by solving one linear program:
ν 0 := maximize X1 + · · · + Xn subject to 0 ≤ X ≤ L(X )
What about max?
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
1.5
g2 (X )
1
g1 (X )
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
For “max” we try again Newton’s method,
i.e., linearize only max{g1 (X ), g2 (X )}
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
For “max” we try again Newton’s method,
i.e., linearize only max{g1 (X ), g2 (X )}
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
For “max” we try again Newton’s method,
i.e., linearize only max{g1 (X ), g2 (X )}
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
For “max” we try again Newton’s method,
i.e., linearize only max{g1 (X ), g2 (X )}
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
This method works!
0.5
(i.e., converges to µf )
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
Let’s treat “max” and “min” the same,
i.e., linearize both g1 (X ) and g2 (X ).
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
Let’s treat “max” and “min” the same,
i.e., linearize both g1 (X ) and g2 (X ).
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
Let’s treat “max” and “min” the same,
i.e., linearize both g1 (X ) and g2 (X ).
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
Let’s treat “max” and “min” the same,
i.e., linearize both g1 (X ) and g2 (X ).
0.5
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
Systems with Min and Max
Let f (X ) = min{max{g1 (X ), g2 (X )}, h(X )}.
2.5
2
h(X )
µf
1.5
g2 (X )
1
g1 (X )
This method also works!
0.5
(i.e., converges to µf )
0
0.5
1
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
1.5
2
2.5
Approximative Methods for Solving min-max-Polynomial Equations
How our Min-Max-Methods Work:
Let ν be an approximate of µf . Improve it as follows:
Each component fi (X ) is linearized separately:
If fi (X ) does not have min or max, then as before:
Li (X ) := T (fi , ν)(X )
If fi (X ) = min{gi (X ), hi (X )}, linearize both gi and hi :
Li (X ) := min{T (gi , ν)(X ), T (hi , ν)(X )}
If fi (X ) = max{gi (X ), hi (X )},
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
How our Min-Max-Methods Work:
Let ν be an approximate of µf . Improve it as follows:
Each component fi (X ) is linearized separately:
If fi (X ) does not have min or max, then as before:
Li (X ) := T (fi , ν)(X )
If fi (X ) = min{gi (X ), hi (X )}, linearize both gi and hi :
Li (X ) := min{T (gi , ν)(X ), T (hi , ν)(X )}
If fi (X ) = max{gi (X ), hi (X )}, let w.l.o.g. gi (ν) ≥ hi (ν) and take:
Li (X ) := T (gi , ν)(X )
The resulting system L(X ) is “min-linear”.
Its least fixed point can be found by solving one linear program:
ν 0 := maximize X1 + · · · + Xn subject to 0 ≤ X ≤ L(X )
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
How our Min-Max-Methods Work:
Let ν be an approximate of µf . Improve it as follows:
Each component fi (X ) is linearized separately:
If fi (X ) does not have min or max, then as before:
Li (X ) := T (fi , ν)(X )
If fi (X ) = min{gi (X ), hi (X )}, linearize both gi and hi :
Li (X ) := min{T (gi , ν)(X ), T (hi , ν)(X )}
If fi (X ) = max{gi (X ), hi (X )},
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
How our Min-Max-Methods Work:
Let ν be an approximate of µf . Improve it as follows:
Each component fi (X ) is linearized separately:
If fi (X ) does not have min or max, then as before:
Li (X ) := T (fi , ν)(X )
If fi (X ) = min{gi (X ), hi (X )}, linearize both gi and hi :
Li (X ) := min{T (gi , ν)(X ), T (hi , ν)(X )}
If fi (X ) = max{gi (X ), hi (X )}, linearize both gi and hi :
Li (X ) := max{T (gi , ν)(X ), T (hi , ν)(X )}
The resulting system L(X ) is “min-max-linear”.
Its least fixed point can be found by a method of [Gawlitza,Seidl,07]
based on strategy improvement.
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Convergence Results
Theorem (Convergence of our methods for min-max-MSPEs)
Both of our methods converge linearly to µf .
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Convergence Results
Theorem (Convergence of our methods for min-max-MSPEs)
Both of our methods converge linearly to µf .
Which method is better?
The asymptotic convergence rates are similar.
Much better than Kleene!
The method producing min-max-linear systems
converges faster, but is more expensive per iteration
(strategy improvement!).
The method producing min-linear systems
computes ε-optimal strategies for the max-player.
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Back to the Flu Example
U = probability to cure an (initially Untreated) patient
and all people he infects
T = probability to cure a Treated patient
and all people he infects
not treat
treat
}|
{ z
}|
{
z
U = max{0.3 + 0.7UU, 0.9T + 0.1U}
T = min { 0.35
+{z
0.65TU} , 0.5
|
| + 0.2TU
{z + 0.3TUU} }
Influenza A
Influenza B
Using the method that produces min-linear systems:
U
0 0.300 0.409 0.524 0.538 0.538
···
T
0 0.350 0.465 0.524 0.538 0.538
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Back to the Flu Example
U = probability to cure an (initially Untreated) patient
and all people he infects
T = probability to cure a Treated patient
and all people he infects
not treat
treat
}|
{ z
}|
{
z
U = max{0.3 + 0.7UU, 0.9T + 0.1U}
T = min { 0.35
+{z
0.65TU} , 0.5
|
| + 0.2TU
{z + 0.3TUU} }
Influenza A
Influenza B
Using the method that produces min-linear systems:
U
0 0.300 0.409 0.524 0.538 0.538
···
T
0 0.350 0.465 0.524 0.538 0.538
Fast convergence!
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Back to the Flu Example
U = probability to cure an (initially Untreated) patient
and all people he infects
T = probability to cure a Treated patient
and all people he infects
not treat
treat
}|
{ z
}|
{
z
U = max{0.3 + 0.7UU, 0.9T + 0.1U}
T = min { 0.35
+{z
0.65TU} , 0.5
|
| + 0.2TU
{z + 0.3TUU} }
Influenza A
Influenza B
Using the method that produces min-linear systems:
U
0 0.300 0.409 0.524 0.538 0.538
···
T
0 0.350 0.465 0.524 0.538 0.538
doctor’s not
not
treat
treat
treat
treat · · ·
action treat treat
Fast convergence!
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Conclusions
Finding the least fixed point of min-max-MSPEs
is a natural problem and appears in
extinction games (like the flu example)
recursive Markov decision processes / stochastic games
Our methods extend Newton’s method and
use linear programming and/or strategy improvement.
Our methods converge linearly.
One of them computes ε-optimal strategies
for the max-player.
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
End of Talk
Thank you!
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations
Notion of Valid Bits
y
µf
µf y
2 bits
3
4 µf y
1 bit
1
2 µf y
0 bits
x
0
1
2 µf x
3
4 µf x
µf x
Notion of Valid Bits
Let x = f (x) be an MSPE. A vector ν has i valid bits of µf if
|µf m − νm |
max
≤ 2−i .
|µf m |
m=1...n
Javier Esparza, Thomas Gawlitza, Stefan Kiefer, Helmut Seidl
Approximative Methods for Solving min-max-Polynomial Equations