secondordercone

Create second-order cone constraint

Since R2020b

Syntax

socConstraint = secondordercone(A,b,d,gamma)

Description

The secondordercone function creates a second-order cone constraint representing the inequality

$$\Vert A\cdot x-b\Vert \le d^T\cdot x-\gamma$$

from the input matrices A, b, d, and gamma.

socConstraint = secondordercone(A,b,d,gamma) creates a second-order cone constraint object socConstraint.

Solve problems with second-order cone constraints by using the coneprog function. To represent multiple cone constraints, pass an array of these constraints to coneprog as shown in the example Several Cone Constraints.

Examples

Single Cone Constraint

To set up a problem with a second-order cone constraint, create a second-order cone constraint object.

A = diag([1,1/2,0]);
b = zeros(3,1);
d = [0;0;1];
gamma = 0;
socConstraints = secondordercone(A,b,d,gamma);

Create an objective function vector.

f = [-1,-2,0];

The problem has no linear constraints. Create empty matrices for these constraints.

Aineq = [];
bineq = [];
Aeq = [];
beq = [];

Set upper and lower bounds on x(3).

lb = [-Inf,-Inf,0];
ub = [Inf,Inf,2];

Solve the problem by using the coneprog function.

[x,fval] = coneprog(f,socConstraints,Aineq,bineq,Aeq,beq,lb,ub)

Optimal solution found.

x = 3×1

    0.4851
    3.8806
    2.0000

fval = 
-8.2462

The solution component x(3) is at its upper bound. The cone constraint is active at the solution:

norm(A*x-b) - d'*x % Near 0 when the constraint is active

ans = 
-2.5677e-08

Several Cone Constraints

To set up a problem with several second-order cone constraints, create an array of constraint objects. To save time and memory, create the highest-index constraint first.

A = diag([1,2,0]);
b = zeros(3,1);
d = [0;0;1];
gamma = -1;
socConstraints(3) = secondordercone(A,b,d,gamma);

A = diag([3,0,1]);
d = [0;1;0];
socConstraints(2) = secondordercone(A,b,d,gamma);

A = diag([0;1/2;1/2]);
d = [1;0;0];
socConstraints(1) = secondordercone(A,b,d,gamma);

Create the linear objective function vector.

f = [-1;-2;-4];

Solve the problem by using the coneprog function.

[x,fval] = coneprog(f,socConstraints)

Optimal solution found.

x = 3×1

    0.4238
    1.6477
    2.3225

fval = 
-13.0089

Input Arguments

A — Linear factor of cone

real matrix

Linear factor of the cone, specified as a real matrix. The number of columns in A must equal the number of elements in d, and the number of rows in A must equal the number of elements in b.

Example: diag([1,1/2,0])

Data Types: double

b — Center of cone

real vector

Center of the cone, specified as a real vector. The number of elements in b must equal the number of rows in A.

Example: zeros(3,1)

Data Types: double

d — Linear bound

real vector

Linear bound, specified as a real vector. The number of elements in d must equal the number of columns in A.

Example: [0;0;1]

Data Types: double

gamma — Bound

real scalar

Bound, specified as a real scalar. Smaller values of gamma correspond to looser constraints.

Example: -1

Data Types: double

Output Arguments

socConstraint — Second-order cone constraint

SecondOrderConeConstraint object

Second-order cone constraint, returned as a SecondOrderConeConstraint object. Use this object as a constraint for the coneprog solver. If you have multiple cone constraints, pass a vector of constraints to coneprog; see Several Cone Constraints.

Version History

Introduced in R2020b