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lecture11

# lecture11 - Yinyu Ye MS&E Stanford MS&E211 Lecture Note#11...

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 1 Applications of KKT Conditions and 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 #11 2 What do the Lagrange multipliers mean? All inactive constraints have zero Lagrange multiplier . In general, the Lagrange multiplier on a given active constraint is the rate of change in the OV as the RHS of the constraint increases , ceteris paribus. In general, when the RHS of an active constraint changes, both the OV and OS will change. When you convert maximization to minimization, the Lagrange multiplier of the former is the negative Lagrange multiplier of the latter.
Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 3 Sensitivity analysis of nonlinear optimization In either (LEP) z ( b ) := minimize f ( x ) subject to A x = b , or (LIP) z ( b ) := minimize f ( x ) subject to A x b , we have z ( b ) = y . This is also true when the constraints are nonlinear under some constraint qualification conditions.

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 4 GP Example: Box Design I minimize x · y + y · z + z · x subject to x · y · z = 1 , x, y, z 0 . Optimality Conditions: y + z λ · y · z, x + z λ · x · z, x + y λ · x · y, x ( y + z λ · y · z ) = 0 , y ( x + z λ · x · z ) = 0 , z ( x + y λ · x · y ) = 0 . The problem has a minimizer and the KKT point is unique: x = y = z = 1 , and λ = 2 .
Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 5 GP Example: Box Design II Consider the RHS is increased by : minimize x · y + y · z + z · x subject to x · y · z = 1 + , x, y, z 0 . New Optimality Solution: x = y = z = (1 + ) 1 / 3 , and New Optimality Objective Value: 3 · (1 + ) 2 / 3 , which is about 3 + 2 when is small.

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 6 What do the Complementarity conditions mean? Complementarity condition occurs when inequalities present, either for constraints or variables. For inequality constraints, it basically says that all inactive constraints have zero Lagrange multiplier . For non-negative variables, it basically says that all positive variables have zero Lagrange multipliers or tight Lagrange conditions.
Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 7 Second Order Optimality Conditions The fundamental concept of the first order necessary condition in Optimization is: in order to have ¯ x

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lecture11 - Yinyu Ye MS&E Stanford MS&E211 Lecture Note#11...

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