ConvexQP

Problem:

Given

  • an integer \(n\) , the dimension of the ambient space,
  • a SDP matrix :math:` M in mathrm{I!R}^{n times n}`
  • a vector :math:` q in mathrm{I!R}^n`
  • a matrix :math:` A in mathrm{I!R}^{m times n}` of constraints
  • a vector :math:` b in mathrm{I!R}^m`
  • a convex set :math:` {C} in {{mathrm{I!R}}}^m`

the convex QP problem is to find a vector \(z\in{{\mathrm{I\!R}}}^n\) ,

\[\begin{split}\begin{equation*} \begin{array}{lcl} \min & & \frac{1}{2} z^T M z + z^T q \\ s.t & & A z + b \in C \\ \end{array} \end{equation*}\end{split}\]

and is most simple example is when :math:` b= 0 A =I` and we obtain

\[\begin{split}\begin{equation*} \begin{array}{lcl} \min & & \frac{1}{2} z^T M z + Z^T q \\ s.t & & z \in C \\ \end{array} \end{equation*}\end{split}\]

Most of the solver returns

  • the solution vector :math:` z in mathrm{I!R}^n`
  • the vector :math:` u in mathrm{I!R}^m`
  • the multiplier :math:` xi in mathrm{I!R}^m` such that :math:` - xi in partial Psi_C(u) `
  • the vector :math:` w in mathrm{I!R}^n` such that :math:` w =A^T xi `

In the most simple case, we return

  • the solution vector :math:` z = u in mathrm{I!R}^n`
  • the vector :math:` w =xi in mathrm{I!R}^m`

ConvexQP Solvers:

This page gives an overview of the available solvers for Convex QP problems and their required parameters.

For each solver, the input argument are:

  • a ConvexQP problem
  • the unknowns (x,fx)
  • info, the termination value (0: convergence, >0 problem which depends on the solver)
  • a SolverOptions structure, which handles iparam and dparam