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The Unconstrained Nonlinear Programming Solver |
Conditions of Optimality |
Before beginning the discussion, the following notation is presented for easy reference:
is the dimension of (that is, the number of decision variables).
is the iteration (that is, the vector of decision variables).
is the objective function.
is the gradient of the objective function.
is the Hessian matrix of the objective function.
Denote the feasible region as . In unconstrained problems, any point
is a feasible point. Therefore, the set
is the entire
space.
A point is a local solution of the problem if there exists a neighborhood
of
such that
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Further, a point is a strict local solution if strict inequality holds in the preceding case; that is,
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A point is a global solution of the problem if no point in
has a smaller function value than
); that is,
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All the algorithms in the NLPU solver find a local minimum of an optimization problem.
The following conditions hold for unconstrained optimization problems:
First-order necessary conditions: If is a local solution and
is continuously differentiable in some neighborhood of
, then
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Second-order necessary conditions: If is a local solution and
is twice continuously differentiable in some neighborhood of
, then
is positive semidefinite.
Second-order sufficient conditions: If is twice continuously differentiable in some neighborhood of
and
and
is positive definite, then
is a strict local solution.
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