mpcqpsolver

（要删除）使用KWIK算法解决二次编程问题

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mpcqpsolverwill be removed in a future release. UsempcActiveSetSolverinstead. For more information, seeCompatibility Considerations。

Syntax

[x，status] = mpcqpsolver（linv，f，a，b，aeq，beq，ia0，options）

[X，状态，IA，lambda] = mpcqpsolver（linv，f，a，b，aeq，beq，ia0，options）

Description

example

[x,status] = mpcqpsolver(Linv,f,A,b,Aeq,beq,iA0,options)finds an optimal solution,x, to a quadratic programming problem by minimizing the objective function:

$J = \frac{1}{2} x^{⊺} H x + f^{⊺} x$

subject to inequality constraints $A x \geq b$ , and equality constraints $A_{e q} x = b_{e q}$ 。statusindicates the validity ofx。

example

[x,status,iA,lambda] = mpcqpsolver(Linv,f,A,b,Aeq,beq,iA0,options)also returns the active inequalities,iA，在解决方案和拉格朗日乘数，lambda, for the solution.

Examples

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Solve Quadratic Programming Problem Using Active-Set Solver

找到值x最小化

$f (x) = 0 。 5 x_{1}^{2} + x_{2}^{2} - x_{1} x_{2} - 2 x_{1} - 6 x_{2},$

subject to the constraints

$\begin{array}{l} x_{1} \geq 0 \\ x_{2} \geq 0 \\ x_{1} + x_{2} \leq 2 \\ - x_{1} + 2 x_{2} \leq 2 \\ 2 x_{1} + x_{2} \leq 3 。 \end{array}$

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];

DefineAeqandbeqto indicate that there are no equality constraints.

Aeq = []; beq = zeros(0,1);

Find the lower-triangular Cholesky decomposition ofH。

[L,p] = chol(H,'lower'); Linv = inv(L);

It is good practice to verify thatHis positive definite by checking ifp = 0。

p = 0

Create a default option set formpcActiveSetSolver。

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 =2×10.6667 1.3333

Check Active Inequality Constraints for QP Solution

找到值x最小化

$f (x) = 3 x_{1}^{2} + 0 。 5 x_{2}^{2} - 2 x_{1} x_{2} - 3 x_{1} + 4 x_{2},$

subject to the constraints

$\begin{array}{l} x_{1} \geq 0 \\ x_{1} + x_{2} \leq 5 \\ x_{1} + 2 x_{2} \leq 7 。 \end{array}$

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];

DefineAeqandbeqto indicate that there are no equality constraints.

Aeq = []; beq = zeros(0,1);

Find the lower-triangular Cholesky decomposition ofH。

[L,p] = chol(H,'lower'); Linv = inv(L);

Verify thatHis positive definite by checking ifp = 0。

p = 0

Create a default option set formpcqpsolver。

opt = mpcqpsolverOptions;

To cold start the solver, define all inequality constraints as inactive.

iA0 = false(size(b));

Solve the QP problem.

[X，状态，IA，lambda] = mpcqpsolver（linv，f，a，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 array1 0 0

There is a single active inequality constraint.

View the Lagrange multiplier for this constraint.

lambda.ineqlin（1）

ans = 5.0000

Input Arguments

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`Linv`—逆矩阵的下三角Cholesky decomposition of Hessian matrix
n-经过-nmatrix

逆矩阵的下三角Cholesky decomposition of Hessian matrix, specified as ann-经过-n矩阵，哪里n> 0is the number of optimization variables. For a given Hessian matrix,H,Linvcan be computed as follows:

[L,p] = chol(H,'lower'); Linv = inv(L);

His ann-经过-n矩阵，必须是对称和积极的。如果p= 0, thenHis positive definite.

Note

The KWIK algorithm requires the computation ofLinv而不是使用Hdirectly, as in theQuadprog（优化工具箱）command.

`f`—Multiplier of objective function linear term
column vector

Multiplier of objective function linear term, specified as a column vector of lengthn。

`A`—Linear inequality constraint coefficients
m-经过-nmatrix|`[]`

Linear inequality constraint coefficients, specified as anm-经过-n矩阵，哪里mis the number of inequality constraints.

如果your problem has no inequality constraints, use[]。

`b`—Right-hand side of inequality constraints
column vector of lengthm

Right-hand side of inequality constraints, specified as a column vector of lengthm。

如果your problem has no inequality constraints, use零（0,1）。

`Aeq`—Linear equality constraint coefficients
q-经过-nmatrix|`[]`

Linear equality constraint coefficients, specified as aq-经过-n矩阵，哪里qis the number of equality constraints, andq<=n。Equality constraints must be linearly independent withrank(Aeq) =q。

如果your problem has no equality constraints, use[]。

`beq`—Right-hand side of equality constraints
column vector of lengthq

Right-hand side of equality constraints, specified as a column vector of lengthq。

如果your problem has no equality constraints, use零（0,1）。

`iA0`—Initial active inequalities
logical vector of lengthm

Initial active inequalities, where the equal portion of the inequality is true, specified as a logical vector of lengthmaccording to the following:

如果your problem has no inequality constraints, usefalse(0,1)。
For a冷启动,false(m,1)。
For awarm start, setiA0(i) == trueto start the algorithm with theith inequality constraint active. Use the optional output argumentiAfrom a previous solution to specifyiA0in this way. If bothiA0(i)andiA0(j)aretrue, then rowsiandjofAshould be linearly independent. Otherwise, the solution can fail withstatus = -2。

`options`—Option set for`mpcqpsolver`
structure

Option set formpcqpsolver, specified as a structure created usingmpcqpsolverOptions。

Output Arguments

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`x`— Optimal solution to the QP problem
column vector

Optimal solution to the QP problem, returned as a column vector of lengthn。mpcqpsolveralways returns a value forx。To determine whether the solution is optimal or feasible, check the solutionstatus。

`status`— Solution validity indicator
positive integer |`0`|`-1`|`-2`

解决方案有效性指标，根据以下内容返回为整数：

Value	Description
`> 0`	`x`是最佳的。`status`表示优化过程中执行的迭代次数。
`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 $A x \geq b$ cannot be satisfied.
`-2`	An unrecoverable numerical error occurred.

`iA`— Active inequalities
logical vector of lengthm

主动不平等现象是不平等的相等部分，作为长度的逻辑向量返回m。如果iA(i) == true, then thei对于解决方案，不平等是有效的x。

UseiAtowarm starta subsequentmpcqpsolversolution.

`lambda`— Lagrange multipliers
structure

Lagrange multipliers, returned as a structure with the following fields:

Field	Description
`Ineqlin`	不平等约束的乘数，作为长度向量返回n。当解决方案是最优的,the elements of`Ineqlin`are nonnegative.
`eqlin`	Multipliers of the equality constraints, returned as a vector of lengthq。最佳解决方案中没有符号限制。

尖端

The KWIK algorithm requires that the Hessian matrix,H, be positive definite. When calculatingLinv, use:
```
[L, p] = chol(H,'lower');
```
如果p= 0, thenHis positive definite. Otherwise,pis a positive integer.
mpcqpsolver提供对模型预测控制Toolbox™软件使用的QP求解器的访问。使用此命令在您自己的自定义MPC应用程序中解决QP问题。

Algorithms

mpcqpsolversolves the QP problem using an active-set method, the KWIK algorithm, based on[1]。For more information, seeQP Solvers。

参考

[1]Schmid, C., and L.T. Biegler. ‘Quadratic Programming Methods for Reduced Hessian SQP’.Computers & Chemical Engineering18, 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 usempcqpsolveras a general-purpose QP solver that supports code generation. Create the functionmyCodethat usesmpcqpsolver。

function[out1,out2] = myCode(in1,in2)%#codegen。。。[x,status] = mpcqpsolver(Linv,f,A,b,Aeq,Beq,iA0,options);。。。

Generate C code withMATLAB^®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 R2015b

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R2020a:`mpcqpsolver`will be removed

从R2020a开始警告

mpcqpsolverwill be removed in a future release. UsempcActiveSetSolverinstead. There are differences between these functions that require updates to your code.

Update Code

The following differences require updates to your code:

FormpcActiveSetSolver, you define inequality constraints in the formAx≤b。以前，为mpcqpsolver, you defined inequality constraints in the formAx≥b
FormpcActiveSetSolver, you specify solver options with a structure created using thempcActiveSetOptions功能。以前，为mpcqpsolver, you created an option structure using thempcqpsolverOptions功能。These option structures contain the same options, though some option names have changed.
By default, you pass the Hessian matrix tompcActiveSetSolver。Previously, you passed the inverse of lower-triangular Cholesky decomposition (Linv）Hessian Matrixmpcqpsolver。To continue to useLinv, set theUseHessianAsInputfield of the structure returned bympcActiveSetSolvertofalse。
When your QP problem has either no inequality constraints or no equality constraints, the correspondingAorAeqinput argument tompcActiveSetSolvermust bezeros(0,n), wherenis the number of decision variables. Previously, formpcqpsolver, you specified these input arguments as[]。

This table shows some typical usages ofmpcqpsolverand how to update your code to usempcActiveSetSolverinstead.

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,`H`。 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 =长度（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 =长度（f）;opt.usehessianasinput = false;[x，status] = mpcactivesetsolver（linv，f，... -a，-b，zeros（0，n），零（0,1），ia0，opt）;

mpcqpsolver

Syntax

Description

Examples

Solve Quadratic Programming Problem Using Active-Set Solver

Check Active Inequality Constraints for QP Solution

Input Arguments

`Linv`—逆矩阵的下三角Cholesky decomposition of Hessian matrix
n-经过-nmatrix

`f`—Multiplier of objective function linear term
column vector

`A`—Linear inequality constraint coefficients
m-经过-nmatrix|`[]`

`b`—Right-hand side of inequality constraints
column vector of lengthm

`Aeq`—Linear equality constraint coefficients
q-经过-nmatrix|`[]`

`beq`—Right-hand side of equality constraints
column vector of lengthq

`iA0`—Initial active inequalities
logical vector of lengthm

`options`—Option set for`mpcqpsolver`
structure

Output Arguments

`x`— Optimal solution to the QP problem
column vector

`status`— Solution validity indicator
positive integer |`0`|`-1`|`-2`

`iA`— Active inequalities
logical vector of lengthm

`lambda`— Lagrange multipliers
structure

尖端

Algorithms

参考

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

R2020a:`mpcqpsolver`will be removed

See Also

Topics

mpcqpsolver

Syntax

Description

Examples

Solve Quadratic Programming Problem Using Active-Set Solver

Check Active Inequality Constraints for QP Solution

Input Arguments

Linv—逆矩阵的下三角Cholesky decomposition of Hessian matrixn-经过-nmatrix

f—Multiplier of objective function linear termcolumn vector

A—Linear inequality constraint coefficientsm-经过-nmatrix|[]

b—Right-hand side of inequality constraintscolumn vector of lengthm

Aeq—Linear equality constraint coefficientsq-经过-nmatrix|[]

beq—Right-hand side of equality constraintscolumn vector of lengthq

iA0—Initial active inequalitieslogical vector of lengthm

options—Option set formpcqpsolverstructure

Output Arguments

x— Optimal solution to the QP problemcolumn vector

status— Solution validity indicatorpositive integer |0|-1|-2

iA— Active inequalitieslogical vector of lengthm

lambda— Lagrange multipliersstructure

尖端

Algorithms

参考

Extended Capabilities

C/C++ Code GenerationGenerate C and C++ code using MATLAB® Coder™.

Version History

R2020a:mpcqpsolverwill be removed

See Also

Topics

`Linv`—逆矩阵的下三角Cholesky decomposition of Hessian matrix
n-经过-nmatrix

`f`—Multiplier of objective function linear term
column vector

`A`—Linear inequality constraint coefficients
m-经过-nmatrix|`[]`

`b`—Right-hand side of inequality constraints
column vector of lengthm

`Aeq`—Linear equality constraint coefficients
q-经过-nmatrix|`[]`

`beq`—Right-hand side of equality constraints
column vector of lengthq

`iA0`—Initial active inequalities
logical vector of lengthm

`options`—Option set for`mpcqpsolver`
structure

`x`— Optimal solution to the QP problem
column vector

`status`— Solution validity indicator
positive integer |`0`|`-1`|`-2`

`iA`— Active inequalities
logical vector of lengthm

`lambda`— Lagrange multipliers
structure

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

R2020a:`mpcqpsolver`will be removed