Subsections

• Formulation
• Performance

5 Isometrization of $\alpha$-pinene

Determine the reaction coefficients in the thermal isometrization of $\alpha$-pinene. The linear kinetic model [6] is

$$y_1' = -(\theta_1 + \theta_2) y_1$$

$$y_2' = \theta_1 y_1$$

$$y_3' = \theta_2 y_1 - (\theta_3 + \theta_4 )y_3 + \theta_5 y_5$$ (1)

$$y_4' = \theta_3 y_3$$

$$y_5' = \theta_4 y_3 - \theta_5 y_5,$$

where $\theta_i \ge 0$ are the reaction coefficients. Initial conditions for are known. The problem is to minimize

$$\sum _ {j=1}^8 \Vert y ( \tau_j ; \theta ) - z_j \Vert ^ 2 ,$$ (2)

where $z_j$ are concentration measurements for $y$ at time points $\tau_1, \ldots ,\tau_8$.

Formulation

Our formulation of the $\alpha$-pinene problem as an optimization problem follows [21,3]. We use a $k$-stage collocation method, a uniform partition with $n_h$ subintervals of $[0,\tau_8]$, and the standard [2, pages 247-249] basis representation,

$$\ v_{i} + \sum_{j=1}^{k} \frac{(t - t_i)^j}{j!\ h^{j-1}} w_{ij} , \qquad t \in [t_i,t_{i+1}] ,$$

for the components of the solution $y$ of . The constraints in the optimization problem are the initial conditions in , the continuity conditions, and the collocation equations. The continuity equations at each interior grid point are a set of $5(n_h -1 )$ linear equations. The collocation equations are a set of $5 k n_h$ nonlinear equations obtained by requiring that the collocation approximation satisfy at the collocation points. Data for this problem appears in Table 5.1.

Table 5.1: Isometrization of $\alpha$-pinene data

Variables	$5(k+1) n_h + 5$
Constraints	$5(k+1) n_h$
Bounds	5
Linear equality constraints	$5 n_h$
Linear inequality constraints	0
Nonlinear equality constraints	$5 k n_h$
Nonlinear inequality constraints	0
Nonzeros in $\nabla ^2 f(x)$	$40 (k+1)^2$
Nonzeros in $c'(x)$	$10 k(k+1) n_h$

Performance

We provide results for the AMPL formulation with $k=3$ in Table 5.2. The initial values for the $\theta$ parameters are $\theta_i = 0.0$. The initial basis parameters are chosen so that the collocation approximation is piecewise constant and interpolates the data. The solution and data are shown in Figure 5.1.

Table 5.2: Performance on isometrization problem

Solver	$n_h=25$	$n_h=50$	$n_h=100$	$n_h=200$
LANCELOT	1426.01 s	2720.49 s	$\ddagger$	$\ddagger$
$f$	1.96766e+01$\dagger$	1.93937e+01$\dagger$	$\ddagger$	$\ddagger$
$c$ violation	1.87900e-06$\dagger$	6.09920e-06$\dagger$	$\ddagger$	$\ddagger$
iterations	305	179	$\ddagger$	$\ddagger$
LOQO	28.85 s	6.15 s	6.77 s	16.87 s
$f$	1.98715e+01	1.98721e+01	1.98721e+01	1.98721e+01
$c$ violation	1.3e-11	2.2e-13	7.6e-13	8.4e-13
iterations	389	32	23	21
MINOS	1.98 s	6.74 s	21.66 s	194.84 s
$f$	1.98715e+01	1.98721e+01	1.98721e+01	0.00000e+00$\dagger$
$c$ violation	4.2e-13	4.4e-13	2.3e-12	1.7e+04$\dagger$
iterations	7	8	7	49
SNOPT	3.74 s	13.1 s	48.91 s	235.44 s
$f$	1.98715e+01	1.98721e+01$\dagger$	1.98721e+01	1.98721e+01$\dagger$
$c$ violation	3.9e-13	4.2e-13$\dagger$	6.7e-13	5.1e-13$\dagger$
iterations	13	18	21	37
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

LANCELOT stops with the message step got too small, near the solution for $n_h \leq 50$. MINOS fails completely on $n_h = 200$ with unbounded (or badly scaled) problem, while SNOPT manages a [p]rimal feasible solution, which could not satisfy dual feasibility for both $n_h = 50,\ 200$.

Figure 5.1: Solution and data for the $\alpha$-pinene problem