Lect02

Lect02 - D. Bertsekas EE553 LECTURE 2 LECTURE OUTLINE...

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EE553 LECTURE 2 LECTURE OUTLINE Unconstrained Optimization Local Minima Necessary Conditions for Local Minima Sufficient Conditions for Local Minima The Role of Convexity Quadratic Unconstrained Problems Existence of Optimal Solutions Iterative Descent Methods D. Bertsekas Adapted for USC EE553 by M. Safonov © D. Bertsekas, MIT, Cambridge, MA
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MINIMA AND CONVEXITY Local minima are also global under convexity α f(x*) + (1 - α )f(x) x f( α x* + (1- α )x) x x* f(x) Illustration of why local minima of convex functions are also global. Suppose that f is convex and that x is a local minimum of f . Let x be such that f ( x ) <f ( x ). By convexity, for all α (0 , 1), f ( αx +(1 α ) x ) αf ( x )+(1 α ) f ( x ) ( x ) . Thus, f decreases monotonically on the line segment con- necting x with x and x cannot be a local minimum which is not global. Adapted for USC EE553 by M. Safonov © D. Bertsekas, MIT, Cambridge, MA
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OTHER PROPERTIES OF CONVEX FUNCTIONS f
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Lect02 - D. Bertsekas EE553 LECTURE 2 LECTURE OUTLINE...

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