## Write Constraints for Problem-Based Cone Programming

To ensure that solve or prob2struct calls coneprog for a second-order cone problem, specify the second-order cone constraints as one of two types:

• norm(linear expression) + constant <= linear expression

• sqrt(sum of squares) + constant <= linear expression

Here, linear expression means a linear expression in the optimization variables. sum of squares means a sum of explicit squares of optimization variables, such as sum(x.^2). The objective function for coneprog must be linear in the optimization variables. For more information on the sum of squares form, see Write Objective Function for Problem-Based Least Squares.

solve and prob2struct also call coneprog when the constraint type has an equivalent form to the two listed:

• linear expression >= sqrt(sum of squares) + constant

• linear expression >= norm(linear expression) + constant

• const*norm(linear expression) + constant <= linear expression provided const > 0

• (sum of squares)^0.5 instead of sqrt(sum of squares)

For example, coneprog is the default solver for each of the following two equivalent problem formulations when you call solve.

x = optimvar('x',3,...
'LowerBound',[-Inf,-Inf,0],...
'UpperBound',[Inf,Inf,2]);
A = diag([1,1/2,0]);
d = [0;0;1];
f = [-1,-2,0];
probnorm = optimproblem('Objective',f*x);
probsumsq = optimproblem('Objective',f*x);

consnorm = norm(A*x) <= d'*x;
probnorm.Constraints.consnorm = consnorm;
conssumsq = sqrt(sum((A*x).^2)) <= dot(d,x);
probsumsq.Constraints.conssumsq = conssumsq;

optnorm = optimoptions(probnorm);
class(optnorm)
ans =

'optim.options.ConeprogOptions
optsumsq = optimoptions(probsumsq);
class(optsumsq)
ans =

'optim.options.ConeprogOptions

If you write the second-order constraints differently, such as the mathematically equivalent sqrt(x'*x), solve calls a different solver, such as fmincon. In this case, you need to supply solve with an initial point, and the solution process can be different (and often is less efficient), as in the following example.

x = optimvar('x',3,...
'LowerBound',[-Inf,-Inf,0],...
'UpperBound',[Inf,Inf,2]);
A = diag([1,1/2,0]);
d = [0;0;1];
f = [-1,-2,0];
prob = optimproblem('Objective',f*x);
cons = sqrt(x'*A'*A*x) <= d'*x;
prob.Constraints.cons = cons;
opt = optimoptions(prob);
class(opt)
ans =

'optim.options.Fmincon'