GE330_lect21 - Lecture 21 Nonlinear Programming Material...

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Lecture 21 Nonlinear Programming April 29, 2009

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Material Chapter 18: Section 18.1.1 Lagrangian Method in Section 18.2 Section 18.2.2 1
From Linear to Nonlinear Linear programming: linear objective function, linear con- straints, and continuous variables; Nonlinear programming: nonlinear objective function, non- linear constraints, and continuous variables. A general form of nonlinear programming problem: min(max) f ( x 1 ,x 2 , ··· ,x n ) s.t. g i ( x 1 ,x 2 , ··· ,x n ) (= , )0 i = 1 , ··· ,m, where f and g i are scalar-valued functions for all i = 1 , ··· ,m . We can write the above nonlinear program in a compact form: min(max) f ( x ) s.t. g i ( x ) (= , )0 i = 1 , ··· ,m, where x is a vector in R n . 2

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Deﬁnitions Consider a nonlinear programming problem min(max) f ( x ) s.t. g i ( x ) (= , )0 i = 1 , ··· ,m, A vector x * is called a local minimum ( local maximum ) if x * satisﬁes all constraints and f ( x * ) ( ) f ( x ) for all feasible x that is suﬃciently close to x * . A vector x * is called a global minimum ( global maximum ) if x * satisﬁes all constraints and f ( x * ) ( ) f ( x ) for all feasible x . 3
Global v.s. Local In general, global optima is hard to ﬁnd (NP-hard problem). Therefore, most of the prevalent algorithms resort to local optima. In some special case, a local optimum is also a global opti- mum. For example, in linear programming, a local optimum is also a global due to the linearity of the objective function and constraints. In fact, if the objective function is convex, and the feasible region, i.e., the set of all x satisfying all the constraints, is convex, then it can be proven that any local optimum is also a global optimum. 4

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Convex Sets Given a set X R n , we say X is convex if for any two points x 1 ,x 2 X , we have λx 1 + (1 - λ ) x 2 X for all λ [0 , 1]. A set is convex if for any two points in the set the line segment connecting them falls into the set. Examples: An empty set, a single point, a circle on a plane; A subspace of R n ; The feasible region of a linear program: { x R n | Ax = b,x 0 } . NOTE: The intersection of several convex sets is still convex, but the union of several convex sets may not be convex.
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GE330_lect21 - Lecture 21 Nonlinear Programming Material...

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