Optimality Conditions Centre for Complex Dynamic Systems and Control

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Optimality Conditions Centre for Complex Dynamic Systems and Control
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  Optimality Conditions Mar´ıa M. SeronSeptember 2004 Centre for Complex DynamicSystems and Control  Outline 1  Unconstrained OptimisationLocal and Global MinimaDescent DirectionNecessary Conditions for a MinimumNecessary and Sufficient Conditions for a Minimum 2  Constrained OptimisationGeometric Necessary Optimality ConditionsProblems with Inequality and Equality ConstraintsThe Fritz John Necessary ConditionsKarush–Kuhn–Tucker Necessary ConditionsKarush–Kuhn–Tucker Sufficient ConditionsQuadratic Programs Centre for Complex DynamicSystems and Control  Unconstrained Optimisation An unconstrained optimisation problem is a problem of the formminimise  f  ( x  ) ,  (1)without any constraint on the vector  x  .Definition (Local and Global Minima) Consider the problem of minimising f  ( x  )  over  R n  and let   ¯ x   ∈  R n  .If f  (¯ x  )  ≤  f  ( x  )  for all x   ∈  R n  , then   ¯ x is called a   global minimum  .If there exists an   ε -neighbourhood N  ε (¯ x  )  around   ¯ x such that f  (¯ x  )  ≤  f  ( x  )  for all x   ∈  N  ε (¯ x  ) , then   ¯ x is called a   local minimum  .If f  (¯ x  )  <  f  ( x  )  for all x   ∈  N  ε (¯ x  ) , x     ¯ x, for some   ε >  0 , then   ¯ x is called a   strict local minimum  . Centre for Complex DynamicSystems and Control  Local and Global Minima The figure illustrates local and global minima of a function  f   overthe reals. Strict local minimumGlobal minimaLocal minima f  Figure: Local and global minima Clearly, a global minimum is also a local minimum. Centre for Complex DynamicSystems and Control  Descent Direction Given a point  x   ∈  R n  , we wish to determine, if possible, whether ornot the point is a local or global minimum of a function  f  .For differentiable functions, there exist conditions that provide thischaracterisation, as we will see below.We start by characterising  descent directions  .Theorem (Descent Direction) Let f   :  R n  →  R be differentiable at   ¯ x. If there exists a vector d such that  ∇ f  (¯ x  ) d   <  0 , then there exists a   δ >  0  such that f  (¯ x   + λ d  )  <  f  (¯ x  )  for each   λ  ∈  ( 0 ,δ ) , so that d is a   descent direction   of f at   ¯ x. Centre for Complex DynamicSystems and Control
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