Global minimum optimal conditioning

The KTC comprise both the necessary and sufficient conditions for optimality for smooth convex problems. In the problem (8.25)-(8.26), if the objective fix) and inequality constraint functions gj are convex, and the equality constraint functions hj are linear, then the feasible region of the problem is convex, and any local minimum is a global minimum. Further, if x is a feasible solution, if all the problem functions have continuous first derivatives at x, and if the gradients of the active constraints at x are independent, then x is optimal if and only if the KTC are satisfied at x. ... 

It can be shown [173] that the average time for the system to approach S5 is much smaller then the average escape time and thus the optimal escape paths found from the statistical analysis of the escape trajectories is independent of the initial conditions on the attractor and provides an approximation to the global minimum of the corresponding deterministic control problem. 

For the minimum solution to be global, a sufficient condition must ensure that the objective functional value is the lowest possible. In the context of an optimal control problem, a sufficient condition for the minimum should lead to the inequality... 

To yield the above inequality, a sufficient condition must include at least one inequality, which in fact is another optimal control problem. Thus, for the global minimum in the simplest optimal control problem, a set of sufficient conditions by Mangasarian (1966) requires that... 

The above sufficient condition is weak, since it is applicable only in a sufficiently small vicinity of the optimal control. The satisfaction of this condition does not guarantee a global minimum. Moreover, it is not easy to come up with a suitable a. in an optimal control problem. 

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

However, a local extremum does not necessarily occur at every point that satisfies (11) that is, (11) is not a sufficient condition for optimality. In practice, necessary conditions are used to identify stationary points, which are candidate extrema, whereas sufficient conditions are used to classify the stationary points as local maxima, local minima, or saddle points (inflection points in Ej). Once all local extrema are found, the global extrema can be found by selecting the absolute maximum or minimum. The necessary and sufficient conditions for determining and classifying the stationary points of a function of one variable are summarized in Table 1. These conditions are easily derived using a Taylor series expansion. 

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