Quadratic Programming Solver

A.3 Quadratic Programming Solver

A convenience wrapper function called qps_mips is provided to make it trivial to set up and solve linear programming (LP) and quadratic programming (QP) problems of the following form:

1 min --xTHx + cTx x 2

(A.11)

subject to

l ≤ Ax ≤ u (A.12 ) xmin ≤ x ≤ xmax. (A.13 )

Instead of a function handle, the objective function is speciﬁed in terms of the paramters $H$ and $c$ of quadratic cost coeﬃcients. Internally, qps_mips passes mips the handle of a function that uses these paramters to evaluate the objective function, gradients and Hessian.

The calling syntax for qps_mips is similar to that used by quadprog from the Matlab Optimization Toolbox.

[x, f, exitflag, output, lambda] = qps_mips(H, c, A, l, u, xmin, xmax, x0, opt);

Alternatively, the input arguments can be packaged as ﬁelds in a problem struct and passed in as a single argument, where all ﬁelds except H, c, A and l are optional.

[x, f, exitflag, output, lambda] = qps_mips(problem);

Aside from H and c, all input and output arguments correspond exactly to the same arguments for mips as described in Tables A-1 and A-2.

As with mips and fmincon, the primary diﬀerence between the calling syntax for qps_mips and quadprog is that the linear constraints are speciﬁed in terms of a single doubly-bounded linear function ( $l ≤ Ax ≤ u$ ) as opposed to separate equality constrained ( $Aeqx = beq$ ) and upper bounded ( $Ax ≤ b$ ) functions.

Matpower also includes another wrapper function qps_matpower that provides a consistent interface for all of the QP and LP solvers it has available. This interface is identical to that used by qps_mips with the exception of the structure of the opt input argument. The solver is chosen according to the value of opt.alg. See the help for qps_matpower for details.

Several examples of using qps_matpower to solve LP and QP problems can be found in t_qps_matpower.m.

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