Introduction to Multivariate Optimality

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Introduction to Multivariate Optimality

• Lecture VI

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Introduction to Multivariate Optimality (I)

• Development of the Unconstrained optimum
• The vector form of the Taylor Series Expansion is

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Introduction to Multivariate Optimality (II)

• By similar arguments as discussed in the
  univariate case we can then define

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Introduction to Multivariate Optimality (III)

• Constrained Multivariate Optimum
• The general problem of the constrained
  multivariate optimum can be defined as

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Introduction to Multivariate Optimality (IV)

• Most of the problem in the constrained optimum
  comes in defining a feasible perturbation
• We start from a feasible point and use a Taylor
  expansion of the multivariate constraint

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Introduction to Multivariate Optimality (V)

• A critical part of this discussion is the Jacobian

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Introduction to Multivariate Optimality (VI)

• Given the Taylor series expansion of the vector
  equation, we can see that starting from a
  feasible and stepping to another feasible point
  involves solving the equation

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Introduction to Multivariate Optimality (VI)

• Using this information to solve for a
  perturbation which maintains feasibility requires
  first splitting the dx vector up into an mm
  portion and a m(n-m) portion

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Introduction to Multivariate Optimality (VII)

• Solving for the feasible change

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Introduction to Multivariate Optimality (VIII)

• Returning to the original unconstrained objective
  function and using our familiar Taylor series
  expansion

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Introduction to Multivariate Optimality (IX)

• Substituting for the feasible changes in x from
  the expansion of the constraint matrix, we have

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• The second-order necessary conditions are then
  defined by