mpcqpsolver
(To be removed) Solve a quadratic programming problem using the KWIK algorithm
mpcqpsolver
will be removed in a future release. Use
mpcActiveSetSolver
instead. For more information, see Version History.
Syntax
Description
Examples
Solve Quadratic Programming Problem Using Active-Set Solver
Find the values of x that minimize
subject to the constraints
Specify the Hessian and linear multiplier vector for the objective function.
H = [1 -1; -1 2]; f = [-2; -6];
Specify the inequality constraint parameters.
A = [1 0; 0 1; -1 -1; 1 -2; -2 -1]; b = [0; 0; -2; -2; -3];
Define Aeq
and beq
to indicate that
there are no equality constraints.
Aeq = []; beq = zeros(0,1);
Find the lower-triangular Cholesky decomposition of
H
.
[L,p] = chol(H,'lower');
Linv = inv(L);
It is good practice to verify that H
is positive
definite by checking if p = 0
.
p
p = 0
Create a default option set for
mpcActiveSetSolver
.
opt = mpcqpsolverOptions;
To cold start the solver, define all inequality constraints as inactive.
iA0 = false(size(b));
Solve the QP problem.
[x,status] = mpcqpsolver(Linv,f,A,b,Aeq,beq,iA0,opt);
Examine the solution, x
.
x
x = 2×1
0.6667
1.3333
Check Active Inequality Constraints for QP Solution
Find the values of x that minimize
subject to the constraints
Specify the Hessian and linear multiplier vector for the objective function.
H = [6 -2; -2 1]; f = [-3; 4];
Specify the inequality constraint parameters.
A = [1 0; -1 -1; -1 -2]; b = [0; -5; -7];
Define Aeq
and beq
to indicate that
there are no equality constraints.
Aeq = []; beq = zeros(0,1);
Find the lower-triangular Cholesky decomposition of
H
.
[L,p] = chol(H,'lower');
Linv = inv(L);
Verify that H
is positive definite by checking if
p = 0
.
p
p = 0
Create a default option set for mpcqpsolver
.
opt = mpcqpsolverOptions;
To cold start the solver, define all inequality constraints as inactive.
iA0 = false(size(b));
Solve the QP problem.
[x,status,iA,lambda] = mpcqpsolver(Linv,f,A,b,Aeq,beq,iA0,opt);
Check the active inequality constraints. An active inequality constraint is at equality for the optimal solution.
iA
iA = 3x1 logical array
1
0
0
There is a single active inequality constraint.
View the Lagrange multiplier for this constraint.
lambda.ineqlin(1)
ans = 5.0000
Input Arguments
Linv
— Inverse of lower-triangular Cholesky decomposition of Hessian matrix
n-by-n matrix
Inverse of lower-triangular Cholesky decomposition of Hessian matrix,
specified as an n-by-n matrix, where n > 0 is the number of optimization variables. For a given
Hessian matrix, H, Linv
can be
computed as follows:
[L,p] = chol(H,'lower');
Linv = inv(L);
H is an n-by-n matrix, which must be symmetric and positive definite. If p = 0, then H is positive definite.
Note
The KWIK algorithm requires the computation of
Linv
instead of using H
directly, as in the quadprog
(Optimization Toolbox) command.
f
— Multiplier of objective function linear term
column vector
Multiplier of objective function linear term, specified as a column vector of length n.
A
— Linear inequality constraint coefficients
m-by-n matrix | []
Linear inequality constraint coefficients, specified as an m-by-n matrix, where m is the number of inequality constraints.
If your problem has no inequality constraints, use
[]
.
b
— Right-hand side of inequality constraints
column vector of length m
Right-hand side of inequality constraints, specified as a column vector of length m.
If your problem has no inequality constraints, use
zeros(0,1)
.
Aeq
— Linear equality constraint coefficients
q-by-n matrix | []
Linear equality constraint coefficients, specified as a
q-by-n matrix, where
q is the number of equality constraints, and q <= n. Equality constraints must be linearly independent with
rank(Aeq) =
q.
If your problem has no equality constraints, use
[]
.
beq
— Right-hand side of equality constraints
column vector of length q
Right-hand side of equality constraints, specified as a column vector of length q.
If your problem has no equality constraints, use
zeros(0,1)
.
iA0
— Initial active inequalities
logical vector of length m
Initial active inequalities, where the equal portion of the inequality is true, specified as a logical vector of length m according to the following:
If your problem has no inequality constraints, use
false(0,1)
.For a cold start,
false(m,1)
.For a warm start, set
iA0(i) == true
to start the algorithm with the ith inequality constraint active. Use the optional output argumentiA
from a previous solution to specifyiA0
in this way. If bothiA0(i)
andiA0(j)
aretrue
, then rows i and j ofA
should be linearly independent. Otherwise, the solution can fail withstatus = -2
.
options
— Option set for mpcqpsolver
structure
Option set for mpcqpsolver
, specified as a structure
created using mpcqpsolverOptions
.
Output Arguments
x
— Optimal solution to the QP problem
column vector
Optimal solution to the QP problem, returned as a column vector of length
n. mpcqpsolver
always returns a
value for x
. To determine whether the solution is
optimal or feasible, check the solution status
.
status
— Solution validity indicator
positive integer | 0
| -1
| -2
Solution validity indicator, returned as an integer according to the following:
Value | Description |
---|---|
> 0 | x is optimal.
status represents the number of
iterations performed during optimization. |
0 | The maximum number of iterations was reached. The
solution, x , may be suboptimal or
infeasible. |
-1 | The problem appears to be infeasible, that is, the constraint cannot be satisfied. |
-2 | An unrecoverable numerical error occurred. |
iA
— Active inequalities
logical vector of length m
Active inequalities, where the equal portion of the inequality is true,
returned as a logical vector of length m. If
iA(i) == true
, then the
ith inequality is active for the solution
x
.
Use iA
to warm start a
subsequent mpcqpsolver
solution.
lambda
— Lagrange multipliers
structure
Lagrange multipliers, returned as a structure with the following fields:
Field | Description |
---|---|
ineqlin | Multipliers of the inequality constraints, returned as a
vector of length n. When the solution is
optimal, the elements of ineqlin are
nonnegative. |
eqlin | Multipliers of the equality constraints, returned as a vector of length q. There are no sign restrictions in the optimal solution. |
Tips
The KWIK algorithm requires that the Hessian matrix, H, be positive definite. When calculating
Linv
, use:[L, p] = chol(H,'lower');
If p = 0, then H is positive definite. Otherwise, p is a positive integer.
mpcqpsolver
provides access to the QP solver used by Model Predictive Control Toolbox™ software. Use this command to solve QP problems in your own custom MPC applications.
Algorithms
mpcqpsolver
solves the QP problem using an active-set method, the
KWIK algorithm, based on [1]. For more information, see
QP Solvers.
References
[1] Schmid, C., and L.T. Biegler. ‘Quadratic Programming Methods for Reduced Hessian SQP’. Computers & Chemical Engineering 18, no. 9 (September 1994): 817–32. https://doi.org/10.1016/0098-1354(94)E0001-4.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
You can use
mpcqpsolver
as a general-purpose QP solver that supports code generation. Create the functionmyCode
that usesmpcqpsolver
.function [out1,out2] = myCode(in1,in2) %#codegen ... [x,status] = mpcqpsolver(Linv,f,A,b,Aeq,Beq,iA0,options); ...
Generate C code with MATLAB® Coder™.
func = 'myCode'; cfg = coder.config('mex'); % or 'lib', 'dll' codegen('-config',cfg,func,'-o',func);
For code generation, use the same precision for all real inputs, including options. Configure the precision as
'double'
or'single'
usingmpcqpsolverOptions
.
Version History
Introduced in R2015bR2020a: mpcqpsolver
will be removed
mpcqpsolver
will be removed in a future release. Use
mpcActiveSetSolver
instead. There are differences between
these functions that require updates to your code.
The following differences require updates to your code:
For
mpcActiveSetSolver
, you define inequality constraints in the form Ax≤b. Previously, formpcqpsolver
, you defined inequality constraints in the form Ax≥bFor
mpcActiveSetSolver
, you specify solver options with a structure created using thempcActiveSetOptions
function. Previously, formpcqpsolver
, you created an option structure using thempcqpsolverOptions
function. These option structures contain the same options, though some option names have changed.By default, you pass the Hessian matrix to
mpcActiveSetSolver
. Previously, you passed the inverse of lower-triangular Cholesky decomposition (Linv
) of the Hessian matrix tompcqpsolver
. To continue to useLinv
, set theUseHessianAsInput
field of the structure returned bympcActiveSetSolver
tofalse
.When your QP problem has either no inequality constraints or no equality constraints, the corresponding
A
orAeq
input argument tompcActiveSetSolver
must bezeros(0,n)
, wheren
is the number of decision variables. Previously, formpcqpsolver
, you specified these input arguments as[]
.
This table shows some typical usages of mpcqpsolver
and
how to update your code to use mpcActiveSetSolver
instead.
Not Recommended | Recommended |
---|---|
opt = mpcqpsolverOptions; [x,status] = mpcqpsolver(Linv,f,A,b,... Aeq,beq,iA0,opt); | opt = mpcActiveSetOptions; opt.UseHessianAsInput = false; [x,status] = mpcActiveSetSolver(Linv,f,... -A,-b,Aeq,beq,iA0,opt); Alternatively,
you can use the Hessian matrix,
opt = mpcActiveSetOptions; [x,status] = mpcActiveSetSolver(H,f,... -A,-b,Aeq,beq,iA0,opt); |
opt = mpcqpsolverOptions('single'); [x,status] = mpcqpsolver(Linv,f,A,b,... Aeq,beq,iA0,opt); |
opt = mpcActiveSetOptions('single'); opt.UseHessianAsInput = false; [x,status] = mpcActiveSetSolver(Linv,f,... -A,-b,Aeq,beq,iA0,opt); |
opt = mpcqpsolverOptions; opt.MaxIter = 300; opt.FeasibilityTol = 1e-5; [x,status] = mpcqpsolver(Linv,f,A,b,... Aeq,beq,iA0,opt); |
opt = mpcActiveSetOptions; opt.UseHessianAsInput = false; opt.MaxIterations = 300; opt.ContraintTolerance = 1e-5; [x,status] = mpcActiveSetSolver(Linv,f,... -A,-b,Aeq,beq,iA0,opt); |
[x,status] = mpcqpsolver(Linv,f,[],... zeros(0,1),Aeq,beq,iA0,opt); |
n = length(f); opt.UseHessianAsInput = false; [x,status] = mpcActiveSetSolver(Linv,f,... zeros(0,n),zeros(0,1),Aeq,beq,iA0,opt); |
[x,status] = mpcqpsolver(Linv,f,A,b,... [],zeros(0,1),iA0,opt); |
n = length(f); opt.UseHessianAsInput = false; [x,status] = mpcActiveSetSolver(Linv,f,... -A,-b,zeros(0,n),zeros(0,1),iA0,opt); |
See Also
Functions
mpcqpsolverOptions
|mpcActiveSetSolver
|mpcActiveSetOptions
|mpcInteriorPointSolver
|mpcInteriorPointOptions
|setCustomSolver
|quadprog
(Optimization Toolbox)
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