Find the optimal value to maximize a function

3 min readSep 13, 2023

Hi everyone…..

could anyone help……

I need to find th optimal value of x0 and x1 to maximize R, where x1=d-x0……….the function MATLAB code:

close all; clear all; clc;

% System parameters
F = 10000;    % Number of files
S = 100;     % SBS cache capacity (set to 100)
M = 50;      % Cash capacity fraction
alpha = 2; % Path loss exponent for LOS link
c = 3e8;     % Light speed
fr = 1e12;  % Operating frequency
B = 10e6;    % System bandwidth
epsilon = 0.8; % Skewness factor
K = 0.0016;  % Molecular absorption coefficient
P_M = 10^(64/10); % Transmit power MBS
P_S = 10^(30/10); % Transmit power SBS
sigma = 10^(-90/10); % Noise power
N_L = 512;
N_M = 16;
eta = 1;
x2 =30;      % RIS-MBS distance
d_range = 1:1:40;R = zeros(size(d_range));for i = 1:length(d_range)
    d = d_range(i);
    
    sum1 = 0;
    for f = 1:F
        sum1 = sum1 + f^(-epsilon);
    end    sum2 = 0;  
    for f = 1:M
        sum2 = sum2 + f^(-epsilon)/sum1;
    end    sum3 = 0;
    for f = (M + 1):(M + (S - M) / eta)
        sum3 = sum3 + (f^(-epsilon)) / sum1;
    end    sum3 = sum3 * eta;
    
    beta = (c/(4*pi*fr))^2;  % Spreading loss index
    Rs = B * log2(1 + (P_S * beta * (d-x1)^(-alpha) * exp(-K * (d-x1))) / sigma);
    Rm = B * log2(1 + (P_M * (N_L^2) * N_M * (beta * (d-x0)^(-alpha) * exp(-K * (d-x0))) * (beta * x2^(-alpha) * exp(-K * x2))) / sigma);
    Rt = Rs * (sum2 + sum3) + Rm * (1 - (sum2 + sum3));
    R(i) = Rt;
    
  
endfigure
plot(d_range, R, 'b^-')
xlabel('SBS-RIS Distance, d')
ylabel('Achievable Rate')

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seems to me that R curves goes up as x0 goes closer to zero (but not rqual to zero otherwise you get Inf values)

also all the small for loops to create the sum1, sum2,sum3 variables can be replaced with sum function directly

close all; clear all; clc;

x0_range  = [0.01 0.25 0.5 1];% System parameters
F = 10000;    % Number of files
S = 100;     % SBS cache capacity (set to 100)
M = 50;      % Cash capacity fraction
alpha = 2; % Path loss exponent for LOS link
c = 3e8;     % Light speed
fr = 1e12;  % Operating frequency
B = 10e6;    % System bandwidth
epsilon = 0.8; % Skewness factor
K = 0.0016;  % Molecular absorption coefficient
P_M = 10^(64/10); % Transmit power MBS
P_S = 10^(30/10); % Transmit power SBS
sigma = 10^(-90/10); % Noise power
N_L = 512;
N_M = 16;
eta = 1;
x2 =30;      % RIS-MBS distance
d_range = 1:1:40;R = zeros(numel(d_range),numel(x0_range));for k = 1:numel(x0_range)
    x0 = x0_range(k);
    
    for i = 1:length(d_range)
        d = d_range(i);
        x1 = d-x0; % added line %         sum1 = 0;
%         for f = 1:F
%             sum1 = sum1 + f^(-epsilon);
%         end
        f = 1:F;
        sum1 = sum(f.^(-ep

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