Circuito de combustible del motor

Trust region optimisation, see Conn et al. (2000), makes use of a local model,τ(·), of the

function to be optimised, here the penalised log-likelihood, and iteratively steps closer to the optimal solution by taking steps within a region whereτ(·)is trusted. By convention the

trust region approach is a minimisation algorithm, we therefore minimise the negative pe- nalised log-likelihood as an equivalent to our problem. This requires a local approximation of the negative penalised log-likelihood,−l_p(·). As is typical in the trust region approach, we use a second-order Taylor series to describe the local model. In each iteration of the algorithm we centre the model on the current, fixed, estimate of our spline coefficients,ccc_h

τ(ppph) =−(lp(ccch) +gp(ccch)Tppph+ 1 2ppp

T

hHp(ccch)ppph)

whereppp_h is the proposed vector of parameter values,g_pis the (1×n) gradient vector and Hpis the (n×n) Hessian matrix oflp(·), which can be calculated exactly.

Within the trust region algorithm we iteratively step towards the optimum within a re- gion in which we believe modelτ(·)to be a good approximation of−lp(·). For this reason the trust region approach imposes an upper limit on how large of a step we can take in each iteration. The area in which the algorithm is allowed to move is call the ‘trust region’. Let us denote the radius of this ‘trust region’, centred atccc_h, as∆.

At each iteration,h, of the trust region algorithm we solve the trust region sub-problem:

CHAPTER 5. A SPLINE-BASED INPUT MODEL 93

where|| · ||denotes Euclidean distance. The constraintccc≥000 is used to ensure nonnegativity of the resulting rate function. Note that this constraint is stronger than necessary since negative spline coefficients are possible whilst still maintaining a positive rate function. But the constraint leads to a simple way to force the rate function,λ(t;ccc), to stay nonnegative.

The trust region subproblem (5.3.5) is a convex, quadratic program and thus has a unique solution. To solve it in practice we used the Gurobi Optimization (2018) quadratic solver.

Note that if the true rate function, λc(t), is known to have a cyclic structure, we can

impose this structure upon our spline function by adding constraints of the form

λ(0;ccc) =λ(T;ccc), λ0(0;ccc) =λ0(T;ccc), λ00(0;ccc) =λ00(T;ccc),

to the trust region subproblem. Such constraints can easily be incorporated.

Given a proposed stepppp_hfrom the trust region subproblem, we decide whether to accept or reject the step according to the ratio

ρh=

lp(λ(t;ccch))−lp(λ(t;ccch+ppph))

τ(000)−τ(ppph)

. (5.3.6)

This ratio compares the actual reduction in the penalised log-likelihood to the predicted reduction from the model. A value of ρh close to 1 says that there is good agreement between modelτ(·)and−lp(·). We acceptppphifρh>α, whereα is set by the practitioner. Ifppp_his accepted our new position isccc_h+1=ccch+ppph.

Note that the radius of the trust region∆is adaptive throughout the algorithm. Ifρh is close to 1 and||ppp_h||=∆, then ∆is restricting our step, and we would increase the radius

of the trust region. Alternatively, if there is not a good agreement betweenτ(·)and−lp(·), we restrict the model to the region where the Taylor series approximation is better. Rules on when to change the trust region radius are set using thresholds. For example, we might change∆whenρh<γ =0.25 orρh>β =0.75. As we get closer to the optimum,bcccθ, the size of the trust region shrinks. We stop the algorithm when||ppp_h||<εwhereεis a stopping

value set by the user. Within the trust region approachα, β, γ and ε are decided by the

The trust region algorithm is used within the spline-based method to minimise the neg- ative penalised log-likelihood, and thus optimise the spline-coefficients for a fixed penalty,

θ. One drawback of the trust region algorithm is that it may struggle to converge to the true

optimal spline coefficient values or even stall when the number of knots grows too large. The dimension of the optimisation problem increases with the number of knots used to build the spline function, and thus finding the optimum is harder as the number of knots grows. We must also take into account that modelτ(·)is a second-order approximation, whereas

the spline function is a cubic polynomial. As the number of knots increases the spline func- tion becomes more and more flexible on smaller and smaller intervals. This means that to ensure modelτ(·)is a valid approximation at the pointccch we must take smaller steps, ppph. The smaller the step we take in each iteration the slower the convergence and in some cases the algorithm may even stop before the optimum has been reached. Note that this is a prob- lem in the trust region algorithm; if the spline coefficients do not converge to their optimal value for a chosen number of knots this does not mean that a spline function cannot be fit with that number of knots. It may be possible to adaptively change the parameters of the trust region algorithm, in the same way that the size of the trust region radius,∆, changes, to

ensure convergence occurs. Another possibility is to use a different optimisation approach to find the optimal spline coefficients, we leave this as suggested future work.