Strong relative minimum

A minimal value $ J ( \widetilde{y} ) $ taken by a functional $ J ( y) $ at a curve $ \widetilde{y} ( x) $, $ x _ {1} \leq x \leq x _ {2} $, such that

$$ \tag{1 } J ( \widetilde{y} ) \leq J ( y) $$

for all comparison curves $ y ( x) $ satisfying the condition of being $ \epsilon $- near of zero order, i.e.

$$ \tag{2 } | y ( x) - \widetilde{y} ( x) | \leq \epsilon $$

on the whole interval $ [ x _ {1} , x _ {2} ] $. It is assumed that $ \widetilde{y} ( x) $, $ y ( x) $ satisfy given boundary conditions.

If, along with condition (2), which requires $ \epsilon $- nearness of ordinates, one adds the condition of $ \epsilon $- nearness of the derivatives:

$$ \tag{3 } | y ^ \prime ( x) - \widetilde{y} {} ^ \prime ( x) | \leq \epsilon $$

on the whole interval $ [ x _ {1} , x _ {2} ] $, then one speaks of $ \epsilon $- nearness of first order.

The value taken by the functional $ J ( y) $ at a curve $ \widetilde{y} ( x) $ for which (1) is satisfied for all comparison curves $ y ( x) $ which are $ \epsilon $- near of first order, is called a weak relative minimum.

Since the condition of $ \epsilon $- nearness of zero order selects a broader class of curves than the condition of $ \epsilon $- nearness of first order, every strong minimum is simultaneously a weak minimum (cf. also Weak relative minimum); but not every weak minimum is strong. In this connection the necessary, and also sufficient, conditions of optimality for strong and weak relative minima do not have the same form.

Alongside the idea of a strong relative minimum the idea of an absolute minimum can be introduced. An absolute minimum is the minimal value taken by $ J ( y) $ on the whole set of curves on which it has a meaning. An absolute minimum is global, whereas strong and weak relative minima are local.

An absolute minimum is also a strong relative minimum, but not every strong relative minimum is an absolute minimum.

A variational problem having more than one strong relative minimum is called a multi-extremum problem. For the solution of practical variational problems a strong relative minimum can be found approximately, using the numerical methods of variational calculus (see Variational calculus, numerical methods of).

For problems in which a strong relative minimum is unique, the necessary conditions for optimality of a strong relative minimum are simultaneously sufficient conditions for an absolute minimum. This situation holds, for example, in the theory of optimal control of linear problems of time-optimal control (see Time-optimal control problem), and also for certain other classes of problems in variational calculus.

References

[1]	M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)
[2]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)

References

[a1]	L.E. [L.E. El'sgol'ts] Elsgolc, "Calculus of variations" , Pergamon (1961) (Translated from Russian)
[a2]	L. Cesari, "Optimization - Theory and applications" , Springer (1983)