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mthsc810-lecture03

mthsc810-lecture03 - MthSc 810 Mathematical Programming...

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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 3 Pietro Belotti Dept. of Mathematical Sciences Clemson University September 1, 2011 Reading for today: Sections 1.4, 2.1, 2.2 Reading for Sep. 6: Sections 2.3-2.7 Convex problems Def.: An optimization problem is convex if ◮ the objective function is convex ◮ all constraints are convex Convex optimization problems are easy : If a problem P is convex, a local optimum x ⋆ of P is also a global optimum of P . (Hint) When modeling an optimization problem, it would be good if we found a convex problem. Nonconvex problems z o pt = min f ( x ) s . t . f i ( x ) ≤ ∀ i = 1 , 2 . . . , m What if either of f i , i = , 1 . . . , m is not convex? In order to solve them, ◮ We can aim for a feasible solution , whose objective function value is an upper bound z ub ≥ z opt. ◮ We can obtain a convex relaxation e.g. by eliminating the nonconvex constraints ⇒ we get an lower bound z lb ≤ z opt. Linear Optimization Linear optimization problems are convex . min c ⊤ x s . t . A x = b x ≥ ◮ f ( x ) = x and g ( x ) =- x are both convex functions: f ( α x ′ + ( 1- α ) x ′′ ) = α x ′ + ( 1- α ) x ′′ = α f ( x ′ ) + ( 1- α ) f ( x ′′ ) g ( α x ′ + ( 1- α ) x ′′ ) =- ( α x ′ + ( 1- α ) x ′′ ) = α g ( x ′ ) + ( 1- α ) g ( x ′′ ) ⇒ c ′ f ( x ) and c ′ g ( x ) are convex too for c ≥ ◮ The objective function is a sum of convex functions...
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