# 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