Stationary point unconstrained

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

If no active constraints occur (so x is an unconstrained stationary point), then (8.32a) must hold for all vectors y, and the multipliers A and u are zero, so V L = V /. Hence (8.32a) and (8.32b) reduce to the condition discussed in Section 4.5 that if the Hessian matrix of the objective function, evaluated at x, is positive-definite and x is a stationary point, then x is a local unconstrained minimum of/. 

We discuss in this paper the unconstrained optimization of stationary points of a smooth function fix) in many variables. The emphasis is on methods useful for calculating molecular electronic energies and for determining molecular equilibrium and transition state structures. The discussion is general and practical aspects concerning computer implementations are not treated. 

A key idea in developing necessary and sufficient optimality conditions for nonlinear constrained optimization problems is to transform them into unconstrained problems and apply the optimality conditions discussed in Section 3.1 for the determination of the stationary points of the unconstrained function. One such transformation involves the introduction of an auxiliary function, called the Lagrange function L(x,A, p), defined as... 

The transformed unconstrained problem then becomes to find the stationary points of the Lagrange function... 

Minimizing an unconstrained convex objective function In this case, any local minimum will also be a global minimum. Furthermore, if the objective function is differentiable, any stationary point (i.e., a point at which all the first-order derivatives vanish) will be a global minimum. 

Maximizing an unconstrained concave objective function Any stationary point will be a global maximum, and any local maximum will also be a global maximum. 

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