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d(k+1)2 = dk3 d(k+1)n = dk(n+1) set x(k+1)0 = xk(n+1), k = k+1, and repeat from step 2. In step 1 the vectors d are a set of coordinate directions. In step 2 f(x) is minimized along the path xk0, xk1, . . . xkn. In step 3 f(x) is minimized in the new conjugate direction. The advantage with this method is that derivatives do not have to be evaluated. X* is the optimized value of the independent variable. n is the number of independent variables involved in the search. With this method an initial guess x00 is assumed and a set of directions d0n are assumed. A series of line searches is performed for each iteration. In the first iteration f(x) is minimized sequentially in directions d01, d02, . . . d0n, starting from point x00 yielding points x01, x02, . . . x0n. The new direction is generated as d0(n+1) = x0n – x0) and f(x) is minimized in this direction to yield a new point x0(n+1). The variable metric method is also known as the quasi Newton method. The original Newton method uses gradients and derivates to find the optimized value of a function. It does this by using the Taylor series to fit a function on the objective function. This involves developing a Hessian matrix (of second derivatives) and making small changes to the independent variables until a convergence criterion is met. The Hessian matrix () is a square matrix of second order partial derivatives of a function. A gradient vector (g), which is a vector of first derivatives of the Taylor function, is also necessary with the Newton methods. The optimum change in the independent variables is: = −g 30PDF Image | Exergoeconomic Analysis and Optimization of ORC
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