# Multivariable Optimization with the FMINCON Function

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Alexander on 12 Dec 2012
I'm using a comprehensive MATLAB code to create a stiffened pressure vessel (that fulfills a known set of structural design criteria). The code to design the pressure vessel is written as a function, with four input variables that define the scantlings of the stiffeners for the design. The objective function (using FMINCON) calls up this design code function, and, in theory, the FMINCON algorithm is expected to modify the four design variables until an ideal solution is found (i.e. the pressure vessel design that represents a minimal weight solution, but satisfies all structural design criteria).
Unfortunately, as written, the current code appears to iterate through a single design variable, rather than all four design variables. As a result, no feasible solution is found. How do I modify the FMINCON function to ensure that the full multivariable design space is considered? Why does the default FMINCON function focus on a single design variable at a time, rather than iterating through all four design variables concurrently?
Any insights would be tremendously appreciated.

Kye Taylor on 12 Dec 2012
Edited: Kye Taylor on 12 Dec 2012
The key is to modify your objective function to be one that can be utilized by the fmincon interface.
For example, if your objective function has a signature that looks like
function f = myfun(x1,x2,x3,x4)
% assuming x1,...,x4 are scalars
% computations with x1,...,x4 like
f = x1*x2*x3*x4
rewrite it like
function f = myfun(x)
% input to objectives passed to fmincon can
% be a vector. So treat its scalar components
x1 = x(1);
x2 = x(2);
x3 = x(3);
x4 = x(4);
% computations with x1,...,x4 like
f = x1*x2*x3*x4
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Alexander on 12 Dec 2012
Thanks for the quick response, Kye!
My objective function is already structured in the manner that you suggested ...
function f = PV_objfun(x)
x7 = x(1); x8 = x(2); x9 = x(3); x10 = x(4);
f = PV_design_code(x7,x8,x9,x10);
The variable "f" represents the value of the objective function (i.e. the weight of the pressure vessel). To calculate the value of f, FMINCON is expected to call up the robust design alogrithm ("PV_design_code"), with the four variables determined by the FMINCON algorithm.
Unfortunately, FMINCON seems to focus on a single variable, such as x(1), without altering any of the other variables.

Matt J on 12 Dec 2012
Edited: Matt J on 12 Dec 2012
Have you checked that small changes in the remaining variables x8, x9, x10 affect the output value of the objective function? If they do not, what you're seeing makes perfect sense.
A common mistake is to have non-differentiable quantizer operations in the objective function like ROUND, FLOOR, CEIL, etc... These make the function locally flat in some or all variables.
Matt J on 12 Dec 2012
It's still questionable whether DiffMinChange would make a difference for FMINCON algorithms where line searches are involved. The finite difference calculations normally control the direction of the search for the next point, but not the step size. If the step size isn't large enough to skip over the locally flat region, then you'll still get stuck.
On the other hand, it would make some sense if the FMINCON code did choose the initial step size of the line search at least as large as DiffMinChange, but I've seen no documentation of that being the case.