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lecture10 - Yinyu Ye MS&E Stanford MS&E211 Lecture Note#10...

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #10 1 Optimality Conditions for Nonlinear Optimization Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/˜yyye
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #10 2 General Optimization Problems Let the problem have the general mathematical programming (MP) form ( P ) minimize f ( x ) subject to x ∈ F . In all forms of mathematical programming, a feasible solution of a given problem is a vector that satisfies the constraints of the problem, that is, in F . The question: How does one recognize or certify an optimal solution to a generally constrained and objectived optimization problem? Answer: Optimality Condition Theory .
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #10 3 Recall Global and Local Optimizers A global minimizer for (P) is a vector ¯ x such that ¯ x ∈ F and f ( ¯ x ) f ( x ) x ∈ F . Unlike linear programming, sometimes one has to settle for a local minimizer , that is, a vector ¯ x such that ¯ x ∈ F and f ( ¯ x ) f ( x ) x ∈ F ∩ N ( ¯ x ) where N ( ¯ x ) is called a neighborhood of ¯ x . Typically, N ( ¯ x ) = B δ ( ¯ x ) , an open ball centered at ¯ x having suitably small radius δ > 0 . The value of the objective function f at a global minimizer or a local minimizer is also of interest. We call f ( ¯ x ) the global minimum value and a local minimum value , respectively.
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #10 4 Continuously Differentiable Functions The objective and constraint are often specified by functions that are continuously differentiable or in C 1 over certain regions. Sometimes the functions are twice continuously differentiable or in C 2 over certain regions. The theory distinguishes these two cases and develops first-order optimality conditions and second-order optimality conditions .
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #10 5 Motivation from One-Variable Problem Consider a differentiable function f of one variable defined on an interval [ a, e ] . If an interior-point ¯ x is a local/global minimizer, then f x ) = 0; if the left-end-point a is a local minimizer, then f ( a ) 0; if the right-end-point e is a local minimizer, then f ( e ) 0 . To summarize: if ¯ x [ a, e ] is a local minimizer, it must be true f x ) = r a r e , ( r a , r e ) 0 , r a ( x a ) = 0 , r e ( e x ) = 0 for two non-negative numbers r a and r e . These are called the first-order necessary conditions since the first order derivative is used to charaterize the condition, where r a and r e are called Lagrange or dual multipliers for the two inequality constraints x a and x e , respectively; and the last two equations are the complementarity conditions .
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #10 6 x a b c e d f Figure 1: Global and local minimizers of one-varialbe function
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #10 7 Motivation from One-Variable Problem continued If f x ) = 0 , then f x ) 0 is also necessary, which is called the second-order necessary condition , since the second order derivative is used.
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