QP_OASES – Quadratic programming using active set method

Block SymbolLicensing group: ADVANCED

Function Description
The QP_OASES block solves a quadratic programming problem using active set method [11]

\begin{array}{rcl} min_{x \in ℝ^{nV}} & \frac{1}{2} x^{T} H x + x^{T} G, \\ s. t. & lb A \leq A x \leq ub A, \\ lb \leq x \leq ub, \end{array}

where $nV$ is number of variables, $nC$ is number of constraints, the Hessian matrix $H \in ℝ^{nV \times nV}$ is symmetric and positive (semi-)definite, the gradient vector $G \in ℝ^{nV}$ , the constraint matrix $A \in ℝ^{nC \times nV}$ , bound vectors $lb, ub \in ℝ^{nV}$ and constraint vectors $lb A, ub A \in R^{nC}$ .

The block wraps the qpOASES library (qpOASES is distributed under the GNU Lesser General Public License, see Appendix A of [12].), the use of which is described in the manual [12].

The output references yH, yG, yA, yLB, yUB, yLBA, yUBA, yXopt and yYopt are always set to the corresponding input uH, uG, uA, uLB, uUB, uLBA, uUBA, uXopt and uYopt. If the input uQP is not connected, the particular quadratic problem (QP) is allocated in the first execution of the function block (see below) and the output yQP is set to the reference of the allocated QP. If uQP is connected (to the yQP output of the previous QP_OASES block), the yQP output is set to uQP and the block works with an already allocated QP.

The block uses internal variables nV and nC. The value of nV is set to the number of rows of the vector $G$ referenced by uG, the value of nV is set to the number of rows of the matrix $A$ referenced by uA. If the reference uA is not defined (the matrix $A$ is not connected), the value nC = 0.

To solve the QP problem, a QProblem object is created in the generic case (see Chapter 3 of [12]). However, the block can also solve the following special cases depending on the input references and the hessianType parameter:

uH not connected.: In this case, it is assumed that Hessian matrix has a trivial value of the identity or zero matrix. The hessianType parameter must be equal to HST_ZERO or HST_IDENTITY, see Section 4.5 of the manual [12].
uA not connected.: In this case, the constraint matrix $A$ is not used (nC = 0, the QProblemB object is created, see Section 4.3 of the manual [12]. The hessianType parameter can be any allowed value.
VAR = on.: If the input VAR = on during the first time the block is executed, an object of class SQProblem is created, see Section 4.2 of the manual [12]. In this case, all input matrices and vectors can change in each execution step in which VAR = on.

To obtain the solution of the QP problem, at least one of the input references uXopt and uYopt must be defined (connected to a vector). If connected to uXopt, the yXopt output will refer to the primal solution $Xopt \in ℝ^{nV}$ , if connected to uYopt, the yYopt output will refer to the dual solution $Yopt \in ℝ^{nV + nC}$ of the QP problem. If both inputs are connected, both solutions will be obtained in each step. The optimal objective function value is indicated on the output objval.

The integer input unWSR specifies the maximum number of working set recalculations to be performed during the initial homotopy, see Section 3.2 of the manual [12]. Output ynWSR contains the number of working set recalculations actually performed. If the double input utime is connected and has a positive value, it contains the maximum allowed CPU time in seconds for the whole initialisation. The actually required CPU time for the initialization is indicated on the output ytime.

At least one vector must be connected from the uXopt and uYopt pair must be connected to obtain the solution of the QP problem. If uXopt is connected, the yXopt output will refer to the primary Xopt solution, if uYopt is connected, the yYopt output will refer to the dual Yopt solution of the QP task. If both inputs are connected, both solutions will be obtained in each step.

If the input INIT = on then the particular allocated QP problem is re-initialized. The INIT should be on for only a single period (edge) because no solution is computed during the QP initialisation. If HLD = on then nothing is computed.

The error flag E is set to on and the error code iE is set to zero if:

the reference uG or uLB or uUB is not defined (i.e. input uG or uLB or uUB is not connected),
the reference uA is defined and uLBA or uUBA is not defined (i.e. input uA is connected and uLBA or uUBA is not connected),
both references uXopt and uYopt are not defined (i.e. neither of the inputs uXopt and uYopt is connected),
the Hessian matrix $H$ referenced by uH has a different number of rows and columns than nV,
the number of rows of vectors referenced by uLB and uUB is not equal to nV (or the number of their columns is not equal to 1),
the number of rows of vectors referenced by uLBA and uUBA is not equal to nC (or the number of their columns is not equal to 1) if the matrix $A$ referenced by uA is connected,
the number of rows of the vector referenced by uXopt is not equal to nV or the number of rows of the vector referenced by yOpt is not equal to nV+nC (or the number of their columns is not equal to 1),
the internal space for transposed copies of matrices $H$ or $A$ is too small.

If the flag E is set to on and the error code iE is not zero then iE indicates the qpOASES error code, see the include file MessageHandling.hpp from qpOASES library.

Inputs

uQP	Input reference to quadratic programming problem	Reference
uH	Input reference to Hessian matrix H	Reference
uG	Input reference to gradient vector G	Reference
uA	Input reference to constraint matrix A	Reference
uLB	Input reference to lower bound vector LB	Reference
uUB	Input reference to upper bound vector LB	Reference
uLBA	Input reference to lower constraints’ bound vector LB	Reference
uUBA	Input reference to upper constraints’ bound vector LB	Reference
uXopt	Input reference to primal optimal solution	Reference
uYopt	Input reference to dual optimal solution	Reference
unWSR	Maximum number of initial working set recalculations	Long (I32)
utime	Maximum allowed CPU time in seconds for the whole initialisation	Double (F64)
VAR	Indiates that matrices H and A are time varying	Bool
INIT	Calls init() function instead of hotstart() in each block execution	Bool
HLD	If HLD = on then nothing is computed	Bool

Parameters

nVmax	Maximum number of optimization variables nV	Long (I32)
nCmax	Maximum number of optimization constraints nC	Long (I32)
hessianType	Hessian matrix type	Long (I32)
printLevel	Print level	Long (I32)
enableRamping	Enable ramping	Bool
enableFarBounds	Enable use of far bounds	Bool
enableFlippingBounds	Enable use of flipping bounds	Bool
enableRegularisation	Enable regularisation of semidefinite Hessian matrix	Bool
enableFullLITests	Enable use of condition-hardened linear independence tests	Bool
enableNZCTests	Enable nonzero curvature tests	Bool
enableDriftCorrection	Frequency of drift corrections (0 = off)	Long (I32)
enableCholeskyRefact	Frequency of full refactorisation of projected Hessian (0 = off)	Long (I32)
enableEqualities	Equalities shall be always treated as active constraints	Bool
terminationTolerance	Termination tolerance	Double (F64)
boundTolerance	If upper and lower limits differ less than this tolerance, they are regarded equal, i.e. as equality constraint	Double (F64)
boundRelaxation	Initial relaxation of bounds to start homotopy and initial value for far bounds.	Double (F64)
epsNum	Numerator tolerance for ratio tests	Double (F64)
epsDen	Denominator tolerance for ratio tests	Double (F64)
maxPrimalJump	Maximum allowed jump in primal variables in nonzero curvature tests	Double (F64)
maxDualJump	Maximum allowed jump in dual variables in linear independence tests	Double (F64)
initialRamping	Start value for ramping strategy	Double (F64)
finalRamping	Final value for ramping strategy	Double (F64)
initialFarBounds	Initial size of Far Bounds	Double (F64)
growFarBounds	Factor to grow Far Bounds	Double (F64)
initialStatusBounds	Initial status of bounds at first iteration	Long (I32)
epsFlipping	Tolerance of squared entry of Cholesky diagonal which triggers flipping bounds	Double (F64)
numRegularisationSteps	Maximum number of successive regularisation steps	Long (I32)
epsRegularisation	Scaling factor of identity matrix used for Hessian regularisation	Double (F64)
numRefinementSteps	Maximum number of iterative refinement steps	Long (I32)
epsIterRef	Early termination tolerance for iterative refinement	Double (F64)
epsLITests	Tolerance for linear independence tests	Double (F64)
epsNZCTests	Tolerance for nonzero curvature tests	Double (F64)

Outputs

yQP	Output reference to quadratic programming problem	Reference
yH	Output reference to Hessian matrix H	Reference
yG	Output reference to gradient vector G	Reference
yA	Output reference to constraint matrix A	Reference
yLB	Output reference to lower bound vector LB	Reference
yUB	Output reference to upper bound vector LB	Reference
yLBA	Output reference to lower constraints’bound vector LB	Reference
yUBA	Output reference to upper constraints’ bound vector LB	Reference
yXopt	Output reference to primal optimal solution	Reference
yYopt	Output reference to dual optimal solution	Reference
ynWSR	Number of performed initial working set recalculations	Long (I32)
ytime	Elapsed CPU time in seconds for the whole initialisation	Double (F64)
objval	Optimal objective function value	Double (F64)
E	Error indicator	Bool
iE	Error code	Long (I32)

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