Re: Non-linear least squares using piece-wise approximation....


if i understand you right, your "n" is 40 ,
you have variables x(1),...x(40) and

low(i)<= x(i) <= up(i) i=1,...,40

additional constraints like

sum_{ j in J_k} y(j)*x(j) =1 , y(j) is 0 or 1

for several index subsets J_k

and as optimization criterion

sum_{i in I} ( p_i_Data - p_i_computed }^2 = minimum

where

p_i_computed = sum _{k=1,...,40} f(v_i,x(k)) (?)

this is the point there your description becomes vague for me:
there are 40 such functions (n?) but how enters the "i" these functions?
then you write you want to model these functions (as functions of one variable
"x" by a piecewise linear approximation, using finally SOS2 variables
for this? why? you could equally model them by a cubic spline:
this is C^2 with respect to the "free" variable (this is the "x"?) and
linear with respect to unknown function values (the "i" position ?)
and : how large ids the data set (I) ?.
and if "x(k)" is the value of the optimization variable where the k-th
function should be evaluated for _al_ the "i" (the data set),
shouldn't finally your functions be functions of two variables at least?
and you get 40 arrays?

I've not explained this very well. It is similar to a problem I had
posted sometime ago:

http://groups.google.com/group/sci.math.num-analysis/browse_thread/thread/5c3046a643bcd710/8160bd6b5804bb0b?lnk=gst&q=svlad#8160bd6b5804bb0b

In the original problem:

"I have a set of m points controlled linearly by n variables. The
variables move the points along line segments.

For a single point I have:

p` = p0 + sum_(i=1 to n) {( pi1 - p0) * t }

where pi1 is the position of the point when the ith variable has a
value of 1. p0 is the rest position of the point when the ith
variable
is 0. In this case p0i is the same so I am using the notation p0. The
simple diagram below illustrates how a single point might be affected
by three different variables.

x p21
p31 x |
\ |
\ |
\ |
\o---------------------x p11

My goal is to find values for the variables t that minimizes the
distance between the point and its target position. By defining di as
the delta of (pi1 - p0) the problem can be stated in matrix-vector
form:

| d11 d12 . . . d1n | z = [p10 p20 . . . pm0]'
| d21 d22 . . . d2n |
X = | . . . | y = [tp1 tp2 . . . tpm]'
| . . . |
| dm1 dm2 . . . dmn |

X is a mxn matrix containing the n delta values for the m points.
z is a m-vector containing the original position of each point
y is a m-vector containing the target positions for each point.
t is a n-vector containing the variables to be optimized.

The problem can now be stated in matrix-vector terms as:

y = z + (X * t)

and solved as a least-squares problem."


What's different now, is that each of the n controls no longer moves
the m points along line segments, but rather along some non-linear
path by some unknown function. For a single point, I have:

p'_i = sum{k=1,...,n} f_ik(x_k)

low(k)<= x(k) <= up(k) k=1,...,n

Additionally there are mutual exclusivity constraints e.g. x_0 and x_7
cannot both be non-zero.

My initial idea was that I could approximate the nonlinear paths with
piece-wise linear ones. I'm thinking this could be solved using a
constrained, nonlinear least-squares solver, but I'm not clear as to
how I would specify the mutex constraints?

Thanks!


.

Relevant Pages

  • Re: Non-linear least squares using piece-wise approximation....
    ... "x" by a piecewise linear approximation, ... and if "x" is the value of the optimization variable where the k-th ... Additionally there are mutual exclusivity constraints e.g. x_0 and x_7 ... constrained, nonlinear least-squares solver, but I'm not clear as to ...
    (sci.math.num-analysis)
  • Re: Quadratic Optimization Problem
    ... Since my problem is a nonlinear convex ... The problem where constraints are linear and the objective function is ... algorithm for nonlinearly constrained optimization. ...
    (comp.lang.python)
  • Re: Behaviour of FMINCON - question.
    ... The large-scale algorithm is a subspace trust region method and is ... Medium-Scale Optimization ... Use equality constraints and the medium-scale method instead. ... Methods for Large-Scale Nonlinear Minimization Subject to Bounds," ...
    (comp.soft-sys.matlab)
  • Re: Step Control for Gradient Descent ?
    ... I have a constrained global optimization problem to solve. ... constraints are sometimes based on equality, e.g., a = f. ... starting each local search near the estimated bottom of that canyon. ...
    (sci.math)
  • Re: Optimization and Positive Definite Matrices
    ... where my objective function is a quadratic form and the ... Is this still true in the case when a linear contraint is added? ... I have always hated bordered Hessians. ... subspace spanned by the tangent vectors to the binding constraints set ...
    (sci.math)
Loading