Table 2 Backtracking algorithm。

At each pivot xk perform a line search in direction pk δk =<∇f(xk),pk >/ ||pk|| (derivative in search direction)

x s = xk +(λ/|δk|)pk/||pk|| (initial backtracking step) 

if xs lies outside the unit hypercube then project xs onto the boundary fit a parabola through {(xk, f(xk)),(xk,δ k),(xs, f(xs))}find the minimum xm of the parabola (first new pivot candidate) 

if xm lies outside the unit hypercube then project xm onto the boundary Δx =||xm−xk|| (set step size) while f(xm)≥ f(xk)−β|δk|Δx do (backtracking condition) 

fit a cubic polynomial through {(xk, f(xk)),(xk,δ k),(xm, f(xm)),(xs, f(xs))}find the minimum xc of the cubic if xc lies outside the unit hypercube then project xc onto the boundary set xs = xm (update new interpolation point)

 set xm = xc (update new pivot candidate) set 

Δx =||xm−xk|| (update new step size)end 

xk+1 = xm (found new pivot xk+1)

sequent backtracking steps to increase the speed of the cost function minimization even further。 The positive constant β helps to control the tolerance used in the search of the new pivot, and λ, also positive, provides a user-control for the initial backtracking step。 Both of these constants influence the convergence rate and the number of function evaluations。 The values taken in this study are β = 0。01 and λ = 0。5。

The Adaptive Cost Function

An optimal engine performance means that the fuel consumption is minimized subject to prescribed emission targets。 As discussed in the Problem Formulation, this constrained optimization problem can be expressed as an unconstrained optimization problem by minimizing the cost function given in Eq。 (1)。 In this study, the fuel consumption is expressed in terms of the normalized specific fuel consumption SFC/SFC0, as is discussed below in more detail。 The emissions, which constitute the penalty terms, are the nitric oxides, NOx, and the particulates, PM。 Therefore, based on Eq。 (1), the cost function used in this study is given by

where x denotes the engine input, ρ the penalty parameter, and the subscripts 0 denote the target values。 The positive weights C, N and S, and the positive exponents c, n and s, determine the importance of each of the expressions involved。 In this study, all the weights and exponent were set to one except for s = 2。 (As discussed below, the factors of 1 2 are used to set f(xk;ρk) = 1 at the start of each line search。) Notice that the dependence between the engine output quantities, SFC, PM and NOx, and the engine operating parameters, x, are not known explicitly。 As discussed in the Problem Formulation, the penalty parameter ρ determines the closeness of the minimum x∗ ρ = minx∈X f(x;ρ) to the actual solution of the original constrained optimization problem。 This suggests that ρ can be updated after every line search。 This ρ-update uses the appropriate values at the new pivot xk+1 according to the following expression 

Table 3 EPA mandates and target values used in the optimization。

EPA Mandate Target

PMo [g/KW-hr] 0。12 0。03 

(NOx)o [g/KW-hr] 4 3。5 

SFCo [g/KW-hr] – 194。13

With this expression for ρk+1, the cost function in Eq。 (2) satisfies f(xk+1;ρk+1) = 1。 Note that with the choice of ρk+1 = max{ρk,ρ k+1} used in the adaptive steepest descent algorithm (cf。 Table 1), the sequence {ρk} is non-decreasing which gives the penalty terms more and more weight, and thus the optimization parameters approach the constrained minimum。 The EPA mandated values for NOx and PM were chosen to meet the Tier 3 2006 (Blue Sky Series) standards of stationary engines as given in Ref。 [28]。 These mandates are PM0 = 0。12 g/KW-hr and (NOx)0 = 4 g/KW-hr1。 The specific fuel consumption is normalized with the tuning case value of SFC0 = 194。13 g/KW-hr。 These values are summarized in Table 3。 In the simulations the SFC has been determined as