How can I use gamma function in optimization problem

epsilon_c = 10e-9; % unitless

lambda = 1e-3; % seconds

P_t = -30; % dB

d = 10; % m

L_H = 40; % bits

L_U = 16; %bits

M_1 = 10; %dB

N_o = -204; %dB

A_o = 30; %dB

m = 1;

alpha = 3;

%%%Variables for Solution

K = optimvar('K', 1, 1, 'Type', 'integer', 'LowerBound', 0, 'UpperBound', 10);

z = optimvar('z', 1, 1, 'Type', 'integer', 'LowerBound', 0,  'UpperBound',10);

%%%Linear Constraints

L = L_H+(K*L_U); % Packet length

R_b = L/lambda; % minimum bit rate

decisioncons = (K/2) - z <= 0;

beta = ((epsilon_c*gamma(m*z+1))^(1/(m*z))*P_t)\(m*N_o*M_1*A_o*d^alpha);

bandwidthcons = B(2^(R_b/B)-1) <= beta;

%%%Solve the Problem

commm = optimproblem('ObjectiveSense','minimize');

commm.Objective = B;

commm.Constraints.decisioncons = decisioncons;

commm.Constraints.bandwidthcons = bandwidthcons;

options = optimoptions('intlinprog','Display','final');

[commmsol,fval,exitflag,output] = solve(commm,'options',options);

sol = commmsol.B

Error:

Undefined function
'gamma' for input
arguments of type
'optim.problemdef.OptimizationExpression'.

[Michele Carone]

I think that the only way to use gamma function in optimization problem is to rewrite the problem in symbolic expression. The function "gamma(X)" cannot take an optimization variable as an input but can take syms variable.

[Matt J]

@Michele Carone No, you can use the gamma function in the solver-based framework, e.g.,

xoptimal=fmincon(@(x) gamma(x).^2, 1,[],[],[],[],0,5)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

xoptimal = 1.4616

Also, in reecent matlab, you can make it work with the problem-based framework using fcn2optimexpr,

x=optimvar('x','Lower',0,'Upper',5);

GamSquare=fcn2optimexpr(@(z) gamma(z).^2, x);

sol0.x=1;
xoptimal = solve(optimproblem('Objective',GamSquare),sol0).x

Solving problem using fmincon.

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

xoptimal = 1.4616

[Michele Carone]

thank you @Matt J, I didn't know this new possibility of recent matlab.

Answer 1

[Matt J]

Your bandwidthcons are not linear, so optimproblem is not applicable here. Since there are only 100 combinations of K and z that satisfy the bounds, you should probably just use exhaustive search.

[Daud]

Oh yes got it, apart of nonlinear constraint... just to clarify. I was supposed to use ceil() or gamma() function in one of my constraints. But it stated error that your function is not defined. What is the reason? Can't we use built-in functions in optimization expressions??

Thanks for your recommendations.

Regards

[Walter Roberson]

gamma() is not defined for datatype optim.problemdef.OptimizationVariable

The defined arithmetic operations for the datatype are:

 .\ (ldivide) -- only when the variable is on the right side, nonscalar constant left permitted
 -  (minus)
 \  (mldivide) -- only when the variable is on the right side and left side is scalar
 ^  (mpower) -- only variable^2
 /  (mrdivide) -- only when variable is on left side and right side is scalar
 *  (mtimes) -- nonscalar left and right permitted and variable^2 terms permitted as long as total degree of any term does not exceed 2
 +  (plus)
 .^ (power) -- only variable.^2
 ./ (rdivide) -- only when variable is on left side; right side can be non-scalar
 .* (times) -- nonscalar left and right permitted and variable^2 terms permitted as long as total degree of any term does not exceed 2
 -  (uminus, unary minus)
 +  (uplus, unary plus)

[Matt J]

And the reason gamma() is not defined for datatype optim.problemdef.OptimizationVariable is because you are not supposed to be doing any nonlinear operations on it, since optimproblem is only intended for linear programming (prior to R2018b).