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Optimal Engine Design Using Nonlinear Programming and the ...

Optimal Engine Design Using Nonlinear Programming and the ...

expressions set less

expressions set less than or equal to 0 and are denoted by g. Table I shows a formal statement of aconstrained minimization problem .Table 1. Example of Constrained Minimization Problemminimize f (x) (x 1 + 3x 2 +x 3 ) 2 + 4(x 1 -x 2 ) 2subject to:h1(x) = 0 1-x 1 - x 2 - x 3 = 0g1(x) ≤ 0 3 + x 13 - 4x 3 - 6x 2 ≤ 0g2(x) ≤ 0 -x 1 ≤ 0g3(x) ≤ 0 -x 2 ≤ 0g4(x) ≤ 0 -x 3 ≤ 0In shorthand the formal statement of this problem is:min f(x)subj. to : h(x) = 0 (5)g(x) ≤ 0where h(x) = {h 1 (x),h 2 (x),...,hm(x)} is a vector of equality constraints and g(x) ={g 1 (x),g 2 (x),...,g l (x )} is a vector of inequality constraints.2.2.1 The LagrangianA class of algorithms have been developed to solve this problem by minimizing a scalarfunction called the Lagrangian. It is a weighted sum of the objective function and the constraints.The weights are multipliers for each of the constraints. There are l multipliers for the equalityconstraints, λ = ( λ 1 , λ 2 ,... λ l) and m multipliers for the inequality constraints,µ = ( µ 1 , µ 2 ,... µ m ). Hence λ and µ are vectors. The Lagrangian is written asL = f(x) + ∑ l λ i h i (x) + ∑ m µ i g i (x) (6)and the optimization problem is stated asminimize L(x)subject to: µ ≥ 0 (7)λ ≠ 0.Formulation of the Lagrangian transforms a constrained minimization problem into an equivalentunconstrained minimization problem.4

2.2.2 The Karush-Kuhn-Tucker (KKT) ConditionsThe first order necessary condition for the minimization of the Lagrangian is that thegradient of the Lagrangian be equal to zero. (For proof see for example, Dennis et al. [1989]).∇ L = ∇f(x) + µ Τ ∇ h(x) + λ Τ ∇g(x) = 0 T (8)In addition, the following must hold.λ ≠ 0µ ≥ 0λ T h(x) = 0 Tµ T g(x) = 0 T (9)These are called the Karush-Kuhn-Tucker (KKT) conditions. and constitute the first ordernecessary conditions for the constrained optimization problem. One class of algorithms to solvethis problem is called Sequential Quadratic Programming (SQP) and is described briefly below.See Papalambros and Wilde [1988] or Luenberger [1984] for a more detailed discussion of SQPmethods.2.3 Sequential Quadratic Programming (SQP) MethodsSQP methods approximate the gradient of the Lagrangian with a first order Taylor'sexpansion. This is equivalent to approximating the Lagrangian as a quadratic function. Theconstraints are approximated with a linear approximation. An iterative algorithm to solve∇L(x) = 0results in solving a sequence of quadratic programming subproblems; hence the name SQP. Thesolution of the quadratic subproblem yields a step direction for the next iteration in the designspace. In addition, a line search algorithm is usually invoked to determine the best magnitude forthe direction found.2.4 Optimal Engine Design StudiesOptimization algorithms have been in use for some time to determine optimal controlschedules of air-fuel ratio, spark and exhaust gas recirculation as a function of speed and torque tomeet fuel and emissions requirements of the Environmental Protection Agency (EPA) driving cycle(see for example, Auiler et al. [1978]; Rishavy et al. [1977]). Physically based models of thecombustion process in internal combustion engines have been in use for over a decade (Blumberget al. [1980], Heywood [1980]; Kreiger [1980]; Reynolds [1980]) to predict the effects of relevantdesign changes in combustion chamber, valve/port interface, and valve timing (Davis andBorgnakke [1981], Davis et al. [1986, 1988]; Newman et al. [1989]). However, optimal designs5

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