{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture10

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

This preview shows pages 1–8. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern