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Optimal Engine Design Using Nonlinear Programming and the ...

of use, its explicit algebraic expressions **and** its continued proliferation into **the** companyoperations.2. Optimization MethodsThe terms objective, constraint, variable, parameter, vector, **and** feasible domain haveexplicit definitions given below.The objective is **the** quantity to be optimized (minimized or maximized). It can be anexplicit algebraic function or it can be an output of ano**the**r computer program .A (design)variable is any quantity allowed to vary during **the** search for **the** optimumobjective. At least **the** objective function or one of **the** constraints should depend on a variable;o**the**rwise it is not relevant to **the** problem statement.A parameter is any quantity appearing in **the** problem statement which is fixed during **the**optimization. For example, **the** values of **the** bounds appearing in **the** constraint set are parameters.A constraint bounds **the** set of variables in some way. Examples are: upper **and** lowerbounds on variables, equality relationships among variables, upper **and** lower bounds on explicitalgebraic expressions relating design variables or upper **and** lower bounds on outputs of a model.The set of variable values bounded by **the** set of constraints is called **the** feasible domain.A vector is simply a set of scalars. The set of variables is a vector; **the** set of equalityconstraints is a vector; **the** set of inequality constraints is a vector. Also recall from multivariablecalculus that **the** gradient of a scalar is a vector; **and** that **the** gradient of a vector is a matrix.2.1 Unconstrained OptimizationThe goal of optimization is to minimize or maximize a single function f, which depends onone or more independent variables. The value of those variables at that minimum or maximum **and****the** value of f is termed **the** optimal solution. The calculation of gradients in **the** design variablespace in search of a minimum is **the** essence of **the** algorithms of interest here. For acomprehensive, underst**and**able introduction to optimization see Chapter 10 of Numerical Recipes- The Art of Scientific Computing, [Press, et.al.,1987].The classical statement of an unconstrained optimization problem is to minimize (ormaximize) a function f which depends upon a vector, x, of n variables where x = {x 1 ,x 2 , ......,x n ,} . The statement of **the** problem for x is:minimize f(x)x = {x 1 ,x 2 , ......, x n ,} ε R n (1)For a single variable problem, (x = {x}) recall from calculus that **the** first order necessarycondition for a minimum is df/dx = 0. Also recall that **the** value of d 2 f/dx 2 at this value of x2

sufficiently determines whe**the**r **the** function is a minimum or maximum; it is called **the** secondorder sufficiency condition . Similarly, for a function of n variables **the** first order necessarycondition is that **the** gradient of f(x) be equal to zero. That is:∂ f = 0∂x 1∇f(x) = ∂ f = 0 (2)∂x 2. .. .∂ f = 0∂x nThis results in a set of n equations that must be solved. Newton's method is a straightforwardalgorithm to solve such a set of equations. The following steps describe **the** algorithm .Step 1.Step 2.Pick a starting point (a guess of **the** solution) x 0 **and** set an iteration counterk = 0. (Superscript indicates iteration number).Calculate a step size, d k , to move in x whered k = -[D(∇f(x) ) k ] -1 (∇f(x) ) k (3)**and** D(∇f(x) )is **the** Jacobian of **the** gradient of f(x) **and** **the** Hessian of f(x).Step 3.Calculate a new value of x usingx k+1 = x k + d k (4)Step 4. If a convergence criteria is satisfied (e.g. || x k+1 - x k || ≤ ε) stop;o**the**rwise increment k by one **and** go to 2.For quadratic functions, it can be shown that this method converges in one step from **the** startingpoint [Dennis et al., 1989].2.2 Constrained OptimizationMost design problems have many constraints imposed. For example, engine designproblems include geometric constraints on **the** engine package, **and** performance criteria constraintson **the** vehicle. The constraints can be in **the** form of an equality ( e.g., π (bore) 2 x (stroke) - 4(volume) = 0) or an inequality (e.g., (maximum piston speed) - 25m/sec ≤ 0). Each equalityconstraint equation is valued at 0 **and** is usually denoted by h. Similarly, inequalities are3

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