- #1

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- 806

In Optimization theory and nonlinear programming [Sun, Yuan] the following is discussed at section 8.2.

Consider the optimisation of [itex]f:\mathbb R^n\to\mathbb R[/itex] with constraints [itex]g_j:\mathbb R^n\to\mathbb R, g_j(x) = 0, j=1,\ldots , m[/itex].

A feasible set [itex]X[/itex] is a subset of [itex]\mathbb R^n[/itex] which contains all those points that satisfy the system of constraints. The following definition is given.

A feasible direction at [itex]x^*\in X[/itex] is a straight line section containing [itex]x^*[/itex] that consists of ONLY feasible points. Formally, there exists a direction [itex]h\in\mathbb R^n\setminus\{0\}[/itex] (may assume to be normed) with

[tex]

\exists \delta >0, \forall \eta (0\leq \eta\leq \delta \implies x^* + \eta h\in X)

[/tex]

Only this definition is given, so one naturally asks, do such feasible directions even exist, in general?

Further development is done via the language of sequences.

Call a direction [itex]h\neq 0[/itex] to be a sequential feasible direction at [itex]x^*\in X[/itex] if there exist sequences [itex]h_k\to h[/itex] such that for every vanishing sequence [itex]\eta _k >0[/itex] it holds that [itex]x^* + \eta_k h_k\in X[/itex].

Evidently, the sequence converges to [itex]x^*[/itex], but the following is problematic for me.

Set [itex]x_k := x^* + \eta_kh_k[/itex]. If we set [itex]\eta _k := \|x_k-x^*\|[/itex] then

[tex]

\frac{x_k-x^*}{\|x_k-x^*\|} \xrightarrow[k\to\infty]{} h?!\tag{E}

[/tex]

What I understand is that we have a sequence of normed elements, so if it converges to anything, the limit has to be nonzero, but why does it converge?

Let's give some context. Suppose [itex]x^*\in X[/itex] is not a local maximum point (subject to said constraints), then by definition

[tex]

\forall k\in\mathbb N,\exists x_k\in X : \|x^*-x_k\|\leq \frac{1}{k}\quad \&\quad f(x_*) < f(x_k)

[/tex]

So, we can construct a sequence of feasible points that converges to [itex]x^*[/itex]. May we assume without loss of generality, this sequence has a limiting direction? More formally, if we write [itex]x_k = x^* + \eta_ kh_k, k\in\mathbb N[/itex], may we assume [itex]\exists \lim _k h_k\in\mathbb R ^n\setminus\{0\}[/itex]?

So, like I said, probably something trivial, but I really don't understand why we may assume the argument that culminates in (E).