How to maximize determinant of a matrix?

I want to apply ABC OPTIMIZATION algorithm to maximize determinant of a matrix.

I have my matrix in the form

X=A(nxn) +k.*B(nxn)

where A and B are nxn matrices, and k is any number between 0 to 10.

I want to optimize k with ABC algorithm such that det(X) is maximum, in other words I have to find such k for which det (X) is maximum.

The code for ABC is attached I need help regarding to define objective function for this.

%

clc;
clear;
close all;

%% Problem Definition

CostFunction=@(x) ObjectiveFunction(x);        % Cost Function

nVar=1;             % Number of Decision Variables

VarSize=[1 nVar];   % Decision Variables Matrix Size

VarMin=0;         % Decision Variables Lower Bound
VarMax= 10;         % Decision Variables Upper Bound

%% ABC Settings

MaxIt=200;              % Maximum Number of Iterations

nPop=100;               % Population Size (Colony Size)

nOnlooker=nPop;         % Number of Onlooker Bees

L=round(0.6*nVar*nPop); % Abandonment Limit Parameter (Trial Limit)

a=1;                    % Acceleration Coefficient Upper Bound

%% Initialization

% Empty Bee Structure
empty_bee.Position=[];
empty_bee.Cost=[];

% Initialize Population Array
pop=repmat(empty_bee,nPop,1);

% Initialize Best Solution Ever Found
BestSol.Cost=inf;

% Create Initial Population
for i=1:nPop
    pop(i).Position=unifrnd(VarMin,VarMax,VarSize);
    pop(i).Cost=CostFunction(pop(i).Position);
    if pop(i).Cost<=BestSol.Cost
        BestSol=pop(i);
    end
end

% Abandonment Counter
C=zeros(nPop,1);

% Array to Hold Best Cost Values
BestCost=zeros(MaxIt,1);

%% ABC Main Loop

for it=1:MaxIt
    
    % Recruited Bees
    for i=1:nPop
        
        % Choose k randomly, not equal to i
        K=[1:i-1 i+1:nPop];
        k=K(randi([1 numel(K)]));
        
        % Define Acceleration Coeff.
        phi=a*unifrnd(-1,+1,VarSize);
        
        % New Bee Position
        newbee.Position=pop(i).Position+phi.*(pop(i).Position-pop(k).Position);
        
        % Evaluation
        newbee.Cost=CostFunction(newbee.Position);
        
        % Comparision
        if newbee.Cost<=pop(i).Cost
            pop(i)=newbee;
        else
            C(i)=C(i)+1;
        end
        
    end
    
    % Calculate Fitness Values and Selection Probabilities
    F=zeros(nPop,1);
    MeanCost = mean([pop.Cost]);
    for i=1:nPop
        F(i) = exp(-pop(i).Cost/MeanCost); % Convert Cost to Fitness
    end
    P=F/sum(F);
    
    % Onlooker Bees
    for m=1:nOnlooker
        
        % Select Source Site
        i=RouletteWheelSelection(P);
        
        % Choose k randomly, not equal to i
        K=[1:i-1 i+1:nPop];
        k=K(randi([1 numel(K)]));
        
        % Define Acceleration Coeff.
        phi=a*unifrnd(-1,+1,VarSize);
        
        % New Bee Position
        newbee.Position=pop(i).Position+phi.*(pop(i).Position-pop(k).Position);
        
        % Evaluation
        newbee.Cost=CostFunction(newbee.Position);
        
        % Comparision
        if newbee.Cost<=pop(i).Cost
            pop(i)=newbee;
        else
            C(i)=C(i)+1;
        end
        
    end
    
    % Scout Bees
    for i=1:nPop
        if C(i)>=L
            pop(i).Position=unifrnd(VarMin,VarMax,VarSize);
            pop(i).Cost=CostFunction(pop(i).Position);
            C(i)=0;
        end
    end
    
    % Update Best Solution Ever Found
    for i=1:nPop
        if pop(i).Cost<=BestSol.Cost
            BestSol=pop(i);
        end
    end
    
    % Store Best Cost Ever Found
    BestCost(it)=BestSol.Cost;
    
    % Display Iteration Information
    disp(['Iteration ' num2str(it) ': Best Cost = ' num2str(BestCost(it))]);
    
end
    
%% Results

figure;
%plot(BestCost,'LineWidth',2);
semilogy(BestCost,'LineWidth',2);
xlabel('Iteration');
ylabel('Best Cost');
grid on;

Answer 1

[Walter Roberson]

CostFunction = @(x) -det(x);        % Cost Function

[Ammy]

But how it is linked with k

For example if I have a sphere function then I have

function z=Sphere(x)

z=sum(x.^2);

end

Now this x is randomly generated with ABC ALGORITHM

But I want to use randomly generated k and maximize det (X)

as defined above

[Walter Roberson]

CostFunction = @(k) -det(A+k.*B)

[Ammy]

Thank you very much.

[Walter Roberson]

Note that for the system A+k*B that the optimal K can be calculated without searching, if you have the symbolic toolbox

syms k
C = A * k.*B;
dC = diff(det(C),k);
k_candidates = solve(dC, k);
LB = 0; UB = 10;
mask = imag(k_candidates) == 0 & real(k_candidates) >= LB & real(k_candidates) <= UB;
selected_k = [k_candidates(mask), LB, UB]
selected_det = arrayfun(@(K) det(subs(C, k, K))), selected_k)
[best_det, idx] = max(selected_det);
best_k = selected_k(idx)

vpa(best_det)
vpa(best_k)