MIT1_204S10_lec21

MIT1_204S10_lec21 - 1.204 Lecture 21 Nonlinear...

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1.204 Lecture 21 Nonlinear unconstrained optimization: First order conditions: Newton’s method Estimating a logit demand model Nonlinear unconstrained optimization Network equilibrium was a constrained nonlinear Network equilibrium was a constrained nonlinear optimization problem Nonnegativity constraints on flows Equality constraints on O-D flows Other variations (transit, variable demand) have inequality constraints In these two lectures we examine unconstrained nonlinear optimization problems No constraints of any sort on the problem; we just find the global minimum or maximum of the function Lagrangians can be used to write constrained problems as unconstrained problems, but it’s usually best to handle the constraints explicitly for computation 1
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Solving systems of nonlinear equations One way to solve for max z(x), where x is a One way to solve for max z(x), where x is a vector, is to find the first derivatives, set them equal to zero, and solve the resulting system of nonlinear equations This is the simplest approach and, if the problem is convex (any line between two points on the boundary of the feasible space stays entirely in boundary of the feasible space stays entirely in the feasible space), it is ‘good enough’ We will estimate a binary logit demand model with this approach in this lecture We’ll use a true nonlinear unconstrained minimization algorithm in the next lecture, which is a better way Solving nonlinear systems of equations is hard Press, Numerical Recipes : “There are There are no no good, good, : general methods for solving systems of more than one nonlinear equation. Furthermore, it is not hard to see why (very likely) there never will be any good, general methods.” “Consider the case of two dimensions, where we want to solve simultaneously” want to solve simultaneously f(x, y)= 0 g(x, y)= 0 2
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Example of nonlinear system From Press Example, continued f and g are two functions Zero contour lines divide plane in regions where functions are positive or negative Solutions to f(x,y)=0 and g(x,y)=0 are points in common between these contours f and g have no relation to each other, in general To find all common points, which are the solutions to the nonlinear equations, we must map the full zero contours of both functions Zero contours in general consist of a an unknown number of disjoint closed curves For problems in more than two dimensions, we need to find points common to n unrelated zero contour hypersurfaces, each of dimension n-1 Root finding is impossible without insight We must know approximate number and location of solutions a priori From Press 3 f neg f pos f pos f pos g neg g pos g pos g neg g pos f = 0 f = 0 M g = y x Solution of two nonlinear equations in two unknowns No root here Two roots here Figure by MIT OpenCourseWare.
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4 Nonlinear minimization is easier There are
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This note was uploaded on 12/04/2011 for the course ESD 1.204 taught by Professor Georgekocur during the Spring '10 term at MIT.

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MIT1_204S10_lec21 - 1.204 Lecture 21 Nonlinear...

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