Quadratic Programming in Hilbert space. Part III. General case.

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The algorithm described in the previous post can be applied for solving a more general quadratic programming problem [1], [2].

Let $H$ is a Hilbert space with inner product ${\langle \cdot \, , \cdot \rangle}_H$ and norm $\| \cdot \|_H$ and $A$ is a linear positive-definite self-adjoint operator acting in $H$. Let linear manifold $D = D(A)$, $D \subset H$ is the domain of definition of this operator and the element $z$ belongs to $D$.

It is required to minimize functional $J$:
\begin{eqnarray}
   (J, \varphi) =  {\langle A \varphi \, , \varphi \rangle}_H - 2 {\langle z \, , \varphi \rangle}_H \to \min \ , \
\end{eqnarray}

subject to constraints

\begin{eqnarray}
    {\langle h_i , \varphi \rangle}_H &=& u_i \, ,
    \quad 1 \le i \le L \ ,
\\
    {\langle h_{i+L} , \varphi \rangle}_H  &\le& u_{i+L} \, ,
    \quad 1 \le i \le M \, ,
\\  \nonumber
 \varphi  &\in& D \, , \quad  D \subset H \, .
\end{eqnarray}

The elements $h_i \in D, \, 1 \le i \le M+L$ are assumed to be linearly independent (thereby the system of constraints is compatible). 

Let $H_A$ be an energetic space generated by the operator $A$ [2], [3]. This space is a completion of linear manifold $D$ with a scalar product
\begin{eqnarray*}
  [\varphi, \vartheta]_A  =  {\langle A \varphi \, , \vartheta \rangle}_H \ ,  \quad \varphi, \vartheta \in H_A \, ,
\end{eqnarray*}
and corresponding norm $[\varphi, \varphi]_A$.
We introduce elements $\bar z$ and $\bar h_i$ from $H_A$ as solutions of the following equations
\begin{eqnarray*}
A \bar z &=& z,  \quad \bar z \in H_A  \, ,
\\
 A \bar h_i &=& h_i \, , \ \ \,  \bar h_i \in H_A \, , \quad 1 \le i \le M+L \, ,
\end{eqnarray*}

and quadratic functional

\begin{eqnarray}
 (\bar J, \varphi) = \| \varphi - z \|_A^2 \, .
\end{eqnarray}

It is easy to see that

\begin{eqnarray*}
 (J, \varphi) =  (\bar J, \varphi) - \|  z \|_A^2 \, .
\end{eqnarray*}

and constraints (2), (3) are transformed to

\begin{eqnarray}
    [\bar h_i , \varphi]_A &=& u_i \, ,  \quad 1 \le i \le L \ ,
\\
    [\bar h_{i+L} , \varphi]_A  &\le& u_{i+L} \, ,  \ 1 \le i \le M+L \, ,
\\  \nonumber
 \varphi  &\in& H_A \,  , \quad \bar h_i \in H_A.
\end{eqnarray}
Thereby the problem of minimization of functional (1) on linear manihold $D$ subject to constraints (2), (3) is equivalent to the problem of minimization of functional (4) in Hilbert space $H_A$ under constraints (5), (6). This later problem is a problem of finding a projection of element $z$ onto a nonempty convex closed set defined by (5), (6) and it has an unique solution. The solution can be found by using algorithms described in the previous posts.

References

[1] V. Gorbunov, Extremum Problems of Measurements Data Processing, Ilim, 1990 (in Russian).
[2] V.Lebedev, An Introduction to Functional Analysis in Computational Mathematics, Birkhäuser, 1997
[3] V. Agoshkov, P. Dubovski, V. Shutyaev, Methods for Solving Mathematical Physics Problems. Cambridge International Science Publishing Ltd, 2006.