# File numerics/src/QP/QP_Solvers.h¶

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Subroutines for the resolution of QP problems.

Functions

void ql0001_(int *m, int *me, int *mmax, int *n, int *nmax, int *mnn, double *c, double *d, double *a, double *b, double *xl, double *xu, double *x, double *u, int *iout, int *ifail, int *iprint, double *war, int *lwar, int *iwar, int *liwar, double *eps)

minimize .5*x’*C*x + d’*x subject to A(j)*x + b(j) = 0 , j=1,…,me A(j)*x + b(j) >= 0 , j=me+1,…,m xl <= x <= xu

Here C must be an n by n symmetric and positive matrix, d an n-dimensional vector, A an m by n matrix and b an m-dimensional vector. The above situation is indicated by iwar(1)=1. Alternatively, i.e. if iwar(1)=0, the objective function matrix can also be provided in factorized form. In this case, C is an upper triangular matrix.

The subroutine reorganizes some data so that the problem can be solved by a modification of an algorithm proposed by Powell (1983).

usage:

call ql0001(m,me,mmax,n,nmax,mnn,c,d,a,b,xl,xu,x,u,iout,ifail, iprint,war,lwar,iwar,liwar)

Definition of the parameters:

y version: 1.5 (june, 1991)

Author
(c): k. schittkowski, mathematisches institut, universitaet bayreuth, 95440 bayreuth, germany, f.r.
Parameters
• m: total number of constraints.
• me: number of equality constraints.
• mmax: row dimension of a. mmax must be at least one and greater than m.
• n: number of variables.
• nmax: row dimension of C. nmax must be greater or equal to n.
• mnn: must be equal to m + n + n.
• c: (nmax,nmax): objective function matrix which should be symmetric and positive definite. If iwar(1) = 0, c is supposed to be the choleskey-factor of another matrix, i.e. c is upper triangular.
• d: (nmax) contains the constant vector of the objective function.
• a: (mmax,nmax): contains the data matrix of the linear constraints.
• b: (mmax) contains the constant data of the linear constraints.
• xl: (n) contain the lower and upper bounds for the variables.
• xu: (n) contain the lower and upper bounds for the variables.
• x: (n) on return, x contains the optimal solution vector.
• u: (mnn) on return, u contains the lagrange multipliers. The first m positions are reserved for the multipliers of the m linear constraints and the subsequent ones for the multipliers of the lower and upper bounds. On successful termination, all values of u with respect to inequalities and bounds should be greater or equal to zero.
• iout: integer indicating the desired output unit number, i.e. all write-statements start with ‘write(iout,… ‘.
• ifail: shows the termination reason. ifail = 0 : successful return. ifail = 1 : too many iterations (more than 40*(n+m)). ifail = 2 : accuracy insufficient to satisfy convergence criterion. ifail = 5 : length of a working array is too short. ifail > 10 : the constraints are inconsistent.
• iprint: output control. iprint = 0 : no output of ql0001. iprint > 0 : brief output in error cases.
• war: (lwar) real working array. the length lwar should be grater than 3*nmax*nmax/2 + 10*nmax + 2*mmax.
• iwar: (liwar): integer working array. the length liwar should be at least n. if iwar(1)=1 initially, then the cholesky decomposition which is required by the dual algorithm to get the first unconstrained minimum of the objective function, is performed internally. otherwise, i.e. if iwar(1)=0, then it is assumed that the user provides the initial fac- torization by himself and stores it in the upper trian- gular part of the array c. a named common-block /cmache/eps must be provided by the user,
• eps: defines a guess for the underlying machine precision.