Supremum of a concave function

I have a function I want to calculate its supremum. The function is below.

-0.25* (c+A^t-v)^T *(c+A^t-v)/v for all v>0

[Image Analyst]

numPoints = 1000;
v = linspace(0.02, 1, 1000);
% Guesses:
c = 1 * ones(1, numPoints);
A = 2 * ones(1, numPoints);
T = 2 * ones(1, numPoints);
t = 3 * ones(1, numPoints);
% Compute function
y = -0.25 * (c+A.^t-v).^T .* (c+A.^t-v)./v %for all v>0
% Plot it.
plot(v, y, 'b.-', 'LineWidth', 2);
grid on;

What are c, A, t, and T?

[John D'Errico]

What do you know about c, A, t, and T? T is most important, of course. For example, if T is not an integer, then things are, let me say, difficult? That is because noninteger powers of negative numbers will be complex, so that supremem will be a nasty thing.

Answer 1

[Walter Roberson]

This creates a list of supermum for the function, together with the conditions under which the supermum hold. The calculations would have been easier if we had been given more information about the symbols.

syms A t T v c;
f = -0.25* (c+A^t-v)^T *(c+A^t-v)/v;
df = diff(f,v);
sol = solve(df == 0,v,'returnconditions', true);
flavor = simplify(subs(diff(df,v),v,sol.v));
conditional_flavor = arrayfun(@(F,C) simplify(piecewise(C & F>0,symtrue,nan)), flavor, sol.conditions);

bs1 = [T == -1, T==0, T==1, T~=-1 & T~=0 & T~=1 & 1<real(T), T~=-1 & T~=0 & T~=1 & 1>real(T)];
bs2 = [c + A^t~=0, c + A^t==0];
branches = and(bs1, bs2(:));
for bidx = 1 : numel(branches)
    assume(assumptions, 'clear')
    assume(branches(bidx));
    constrained_conditions(:,bidx) = simplify(conditional_flavor);
end
assume(assumptions, 'clear')

supermum= [];
for K = 1: size(constrained_conditions,1)
    for bidx = find(~isnan(constrained_conditions(K,:)))
      temp = arrayfun(@(C) simplify(piecewise(v == sol.v(K) & C, subs(f, v, sol.v(K)))), branches(bidx) & constrained_conditions(K,bidx));
      supermum = [supermum; temp];
    end
end

supermum

supermum = 

There is also a saddle point of f = 0 when v = c + A^t