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Optimal Engine Design Using Nonlinear Programming and the ...

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 shorth**and** **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 **the**gradient 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 for**the** 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 **the**combustion 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 **and**Borgnakke [1981], Davis et al. [1986, 1988]; Newman et al. [1989]). However, optimal designs5

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