# D-admissible directions and extermal points

In summary, the D-admissible directions for J are y+Ev where v is any function in D. The possible external points for J on D are those that satisfy the equation G(y;v) = integ (a to b) y(x)v(x) dx = 1, where v is any function in D.
Let X = C[a,b], J(y) = integ(a to b) sin^3(x) + y(x)^2 dx and D={yEX; integ(a to b) y(x)dx = 1}
(a) what are the D-admissible directions for J?
(b) Find all possible (local) external points for J on D?

so far i have:
(let e be epsilon)
lim e->0 J(y+ev) - J(y) / e
= lim e->0 integ (a to b) [ sin^3(x) + (y+ev)^2(x)dx - J(y) ] / e
= lim e->0 integ (a to b) [2ey(x)v(x) + e^2v(x)^2) dx ] / e
= 2 integ(a to b) y(x)v(x) dx
A similar example was done in class, so i just copied it for this question.
G(y) = integ(a to b) y(x) dx = 1
gateaux G(y;v) = integ (a to b) y(x)v(x) dx

is this right so far? how do i go about answering (a) and (b).

i was wondering if the D-admissible directions for J is just
y+Ev E D
so,
integ (a-b) (y+Ev)(x) dx
= 1+ E integ(a-b) v(x) dx

can i simplify any further?

## 1. What are D-admissible directions?

D-admissible directions are directions in which the objective function of a mathematical optimization problem can be improved. These directions satisfy certain conditions such as being feasible and satisfying the Karush-Kuhn-Tucker (KKT) conditions.

## 2. How are D-admissible directions used in optimization problems?

D-admissible directions are used in optimization problems to find the optimal solution. By identifying the directions in which the objective function can be improved, it becomes easier to determine the optimal solution and make progress towards it.

## 3. What are external points in optimization?

External points in optimization refer to points that lie outside the feasible region of a problem. These points are not feasible solutions and are usually used to evaluate the objective function and determine the direction in which it can be improved.

## 4. How do external points relate to D-admissible directions?

External points are important in determining D-admissible directions because they help identify the directions in which the objective function can be improved. By evaluating the objective function at external points, we can determine the direction of steepest descent and make progress towards the optimal solution.

## 5. Can D-admissible directions be used in all optimization problems?

Yes, D-admissible directions can be used in all optimization problems as long as the problem satisfies certain conditions such as convexity and differentiability. These conditions ensure that the objective function can be improved in the identified directions, making it possible to use D-admissible directions in finding the optimal solution.

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