# 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 Compatibility Considerations.

## Syntax

``````[x,status] = mpcqpsolver(Linv,f,A,b,Aeq,beq,iA0,options)``````
``````[x,status,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:$J=\frac{1}{2}{x}^{⊺}Hx+{f}^{⊺}x$subject to inequality constraints $Ax\ge b$, and equality constraints ${A}_{eq}x={b}_{eq}$. `status` indicates the validity of `x`.```

example

``````[x,status,iA,lambda] = mpcqpsolver(Linv,f,A,b,Aeq,beq,iA0,options)``` also returns the active inequalities, `iA`, at the solution, and the Lagrange multipliers, `lambda`, for the solution.```

## Examples

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Find the values of x that minimize

`$f\left(x\right)=0.5{x}_{1}^{2}+{x}_{2}^{2}-{x}_{1}{x}_{2}-2{x}_{1}-6{x}_{2},$`

subject to the constraints

`$\begin{array}{l}{x}_{1}\ge 0\\ {x}_{2}\ge 0\\ {x}_{1}+{x}_{2}\le 2\\ -{x}_{1}+2{x}_{2}\le 2\\ 2{x}_{1}+{x}_{2}\le 3.\end{array}$`

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];```

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 ```

Find the values of x that minimize

`$f\left(x\right)=3{x}_{1}^{2}+0.5{x}_{2}^{2}-2{x}_{1}{x}_{2}-3{x}_{1}+4{x}_{2},$`

subject to the constraints

`$\begin{array}{l}{x}_{1}\ge 0\\ {x}_{1}+{x}_{2}\le 5\\ {x}_{1}+2{x}_{2}\le 7.\end{array}$`

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];```

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

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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.

Multiplier of objective function linear term, specified as a column vector of length n.

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 `[]`.

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)`.

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 `[]`.

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)`.

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 argument `iA` from a previous solution to specify `iA0` in this way. If both `iA0(i)` and `iA0(j)` are `true`, then rows i and j of `A` should be linearly independent. Otherwise, the solution can fail with `status = -2`.

Option set for `mpcqpsolver`, specified as a structure created using `mpcqpsolverOptions`.

## Output Arguments

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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`.

Solution validity indicator, returned as an integer according to the following:

ValueDescription
`> 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 $Ax\ge b$ cannot be satisfied.
`-2`An unrecoverable numerical error occurred.

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.

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

FieldDescription
`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.

## Compatibility Considerations

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Warns starting in R2020a

## 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.