maximization with fmincon which of the two codes provided do the correct job

I have the following maximization problem , solve for a_j:

s.t

k_j = 1x6, R_t= 6x1 , u = @(a) 1/(1-gamma)*a.^(1-gamma); gamma=2 ,

in the code W is 6x6 matrix , where each row correspond to k_j

code 1 : I have implemented as

clear all
W = csvread('optw.csv'); % Assuming W.csv contains a 6x6 matrix , where each row correspond to k_j
R = csvread('rt.csv'); % Assuming R.csv contains a 6x249 matrix

% Define the objective function
gamma = 2;
n = 6; % Number of variables (a_j)
u = @(x) 1/(1-gamma) * x.^(1-gamma);
obj = @(a) -sum(u(sum(a.* (W' * R), 1)));

% Set up optimization problem
Aeq = ones(1, n); % Equality constraint: sum_{j=1}^{j=6} a_j = 1
beq = 1; % Equality constraint value
lb = zeros(n, 1); % Lower bounds for variables a_j
ub = ones(n, 1); % Upper bounds for variables a_j

% Solve the optimization problem
options = optimoptions('fmincon', 'Display', 'iter'); % Set additional options if needed
[a_opt, fval] = fmincon(obj, ones(n, 1) / n, [], [], Aeq, beq, lb, ub, [], options);

% Display the optimal solution and objective value
disp("Optimal a_j:");
disp(a_opt');
disp("Objective value:");
disp(-fval);

however another objective function I implemented provide different results :

W = csvread('optw.csv'); % Assuming W.csv contains a 6x6 matrix
R = csvread('rt.csv'); % Assuming R.csv contains a 6x249 matrix

% Define the objective function
gamma = 2;
n = 6; % Number of variables (a_j)
u = @(x) 1/(1-gamma) * x.^(1-gamma);
obj = @(a) -sum(u(sum(a .* (W* R), 1)));

% Set up optimization problem
Aeq = ones(1, n); % Equality constraint: sum_{j=1}^{j=6} a_j = 1
beq = 1; % Equality constraint value
lb = zeros(n, 1); % Lower bounds for variables a_j
ub = ones(n, 1); % Upper bounds for variables a_j

% Solve the optimization problem
options = optimoptions('fmincon', 'Display', 'iter'); % Set additional options if needed
[a_opt, fval] = fmincon(obj, ones(n, 1) / n, [], [], Aeq, beq, lb, ub, [], options);

% Display the optimal solution and objective value
disp("Optimal a_j:");
disp(a_opt');
disp("Objective value:");
disp(-fval);

Can you please help me which objective function is doing the correct calculation based on the the problem I described former.

Thanks in advance

[Matt J]

Your objective function is not clear. What is 'E'? Also, R_t is presumably independent of j, so why does it appear inside the summation?

[Az.Sa]

E is the sum_t=1 to 249

[Torsten]

If R_t is 6x1, the k_j must be 6x1, too, for that k_j^T*R_t can be evaluated.

[Az.Sa]

R at observation t : is 6x1 , R does not depend on j

the dimention of k_j: is 1x6 ( you are correct , I editted the question by removing the tranpose sympole on k_j and make the k_j: is 1x6 )

we sum the objective function evaluated at each t and maximize the sum over the vector of m_j

Answer 1

[Torsten]

I suggest using a function instead of a function handle.

I compared it with your implementation and random matrices for W and R. The second code gives the same result.

function value = obj(a,W,R)
  value_gamma = 2;
  u = @(x) 1/(1-value_gamma) * x.^(1-value_gamma);
  vec = a.'*W;
  value = 0;
  for i = 1:size(R,2)
    value = value + u(vec*R(:,i));
  end
  value = -value;
end

[Az.Sa]

thank you very much for your answer! I notice you wrote vec = a.'*W;

and this is the main dfference between the two objective functions of the two codes provided previously. May I ask what can we interpret from the calculation : sum(a.* (W' * R)) and sum(a.'* (W * R)) how can this affect the final results

[Torsten]

Play a little:

syms a [2,1] real
syms W [2,2] real
syms R [2,4] real
s1 = sum(a.* (W' * R))

s1 = 

size(s1)

ans = 1×2

1 4

s2 = sum(a.'* (W * R))

s2 = 

size(s2)

ans = 1×2

1 1

[Az.Sa]

that is so great! Thank you very much for your help!