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 ⟨⋅,⋅⟩H and norm ‖ 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.
ReferencesThereby 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.
[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.
