What is QUINOPT?

QUINOPT (QUadratic INtegral OPTimisation) is an open-source add-on for YALMIP to compute rigorous upper and lower bounds on the optimal value of optimization problems with infinite-dimensional polynomial quadratic integral inequality constraints. Such problems commonly arise from stability analysis of linear PDEs using the so-called energy method (also known as \(\mathcal{L}^2\) stability), and in bounding time-average properties of turbulent fluid flows using the so-called background method.

In the simplest form, given a bounded interval \([a,b] \subset \mathbb{R}\), a \(k\)-times continuously differentiable function \(u \in C^k([a,b],\mathbb{R})\), and a vector of optimization variables \(\gamma \in \mathbb{R}^s\), QUINOPT computes upper and/or lower bounds on the optimal value of the optimization problem

\[\begin{split}\begin{aligned} \min_{\gamma} \quad &c^T \gamma\\ \text{subject to} \quad &\int_a^b Q_{\gamma}(x,u(x),u'(x),...,u^{(k)}(x)) \,{\rm d}x \geq 0 \quad \forall u(x) \in \mathcal{H} \end{aligned}\end{split}\]

by constructing SDP-representable inner and outer approximations of its feasible set. In the problem above, \(Q_{\gamma}(x,u(x),u'(x),...,u^{(k)}(x))\) is

  • a quadratic polynomial in \(u(x),u'(x),...,u^{(k)}(x)\);
  • a polynomial in \(x\);
  • an affine function of the optimization variable \(\gamma\).

Moreover, \(\mathcal{H}\) is the subspace of functions that satisfy \(m\) homogeneous boundary conditions, i.e.

\[\mathcal{H} := \left\{ u \in C^k([a,b],\mathbb{R}) \quad a_1 u(a) + a_2 u(b) + a_3 u'(a) + \cdots + a_{2k} u^{(k)}(b) = 0\right\},\]

where \(a_0,\,\ldots,\,a_{2k} \in \mathbb{R}^m\) are known vectors.

Note

Inhomogeneous boundary conditions can be “lifted” by changing variables according to \(u(x)=v(x)+p(x)\), where \(p(x)\) is a polynomial of sufficiently high degree satisfying the inhomogeneous boundary conditions.

A particularly simple example of an optimization problem with an integral inequality is to determine the best Poincaré constant, i.e. the largest \(\gamma > 0\) such that

\[\int_0^1 \left[ \vert u'(x)\vert^2 - \gamma \vert u(x)\vert^2 \right] {\rm d}x \geq 0 \quad \forall u \in C^2([0,1],\mathbb{R}),\quad u(0)=0=u(1).\]

Upper and lower bounds on the largest \(\gamma\) are found by QUINOPT upon solving two SDPs.

Note

QUINOPT can also handle problems with more dependent variables, i.e. \(u:[a,b]\to\mathbb{R}^q\), and problems in which the boundary values of the dependent variables and their derivatives appear explicitly in the integrand of the inequality constraint. For more details, see our paper or have a look at the examples.