smooth non-linear optimization

I am trying to minimize cost function (using Matlab optimization toolbox) with constraints x1 < x2 and 0 < x1, 0 < x2. I've the following issues:

1. When I try using fmincon; it says Error running optimization. Inputs must be a scalar and a square matrix. To compute elementwise POWER, use POWER (.^) instead. I've tried the same and got the error "Not enough input arguments" (I've given start point [0 0] in both cases.)
2. This problem is not convex, so I need to use Global solvers to find global optimum. I've Global optimization tool box installed (I've checked using 'ver' command) but I don't see Global optimization tool box in apps. Please let me know how to start it.

My function defined in m-file is as below:

function [cost] = optim1( x1, x2 )
%   Input arguments
k1=150;
k2=200;
c1=0.1;
c2=0.75;
c3=0.2;
c4=100;
alpha=0.01;
beta=2;
a=80;
b=0.3;
lamda=0.2;
x=0.5;
y=0.9; 
%time=[x1 x2];

% To break coeff functions in smaller parts

A =(lamda*x-lamda^2/2)* x1^2;
B = (alpha*x1^(beta+2))/(beta+1); 
C = (- beta * lamda^(beta+2))/(beta+2)+(x*(lamda^(beta+1))- lamda*        (x^(beta+1)));
D = (x - lamda)^2 * (x1^2)/2;
E = (alpha*(x1^(beta+2)))/(beta+1);
F = C + (beta * x^(beta+2))/(beta+2);
G = ((1 - x)^2 * (x1^2))/2;
H = beta/(beta+2)-(beta*(x^(beta+2)))/(beta+2)- x + (x^(beta+1));
I = ((2 - 3 * x)*((x1)^3)/3);
J =  alpha*((x1)^(beta+3));
K = (2*beta)/((beta+1)*(beta+3)) - ((2*x)/(beta+2)) - x + ((x^(beta+1))/(beta+1));
L = (((y-1)*(x2^2))/2) - x2*x1;
M = (a*((y*x2 - x1)^2))/2;
N = ((x1)^3)+2*(x2^3) - 3*((x2)^2)*x1;

% coeff functions  
coeff1    = (k1-a)*(A + (B*C)) + (k2-a)*(D - E*F) + a*(G + B*H) + (b/2)*(I + J*K);
coeff2    = (k1-a)*(1-y)*L - M - (b/6)*N;
coeff3 =  (k1*lamda*x1 + (k2*(x-lamda)*x1) - a*x1 - (b*((x1)^2))/2)*c3;

% To define objective function to minimize

cost = ((c1*coeff1) + (c2*coeff2) + (c3*coeff3) + c4)/x2;
end

Answer 1

[Walter Roberson]

function [cost] = optim1( x )
  x1 = x(:,1);
  x2 = x(:,2);

and convert pretty much all of your ^ to .^

Note: you might need x(1,:) and x(2,:) instead -- the shape passed in is not always well documented.

[learner]

Thanks Walter, it worked but I couldn't understand the logic. What's wrong with function [cost] = optim1( x1, x2 ) as in command window, if I run it, it works fine. Would you please discuss in detail. Also, I've seen functions defined using ^ in math-work documentations. but if we use optimization solver for such functions, why do we need to change ^ to .^ function from some math-work documentation is c(1) = (x(1)^2)/9 + (x(2)^2)/4 - 1

I am trying to learn MATLAB but find it difficult for such small issues. Please help !