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Math Help - Locally convex set

  1. #1
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    Locally convex set

    Considering a constrained nonlinear programming (NLP) problem
    min \quad f({\bf x}) \quad {\bf x}\in \mathbb{R}^{n}
    s.t. \quad g_{i}({\bf x})\leq 0 \quad i=1,2,...,N
    \quad\quad\quad\quad h_{j}({\bf x})=0 \quad j=1,2,...,M

    Where g_{i}({\bf x}) and h_{j}({\bf x}) is twice continuously differentiable. The feasible region  S=\{{\bf x}|g_{i}({\bf x}),h_{j}({\bf x}),\forall i,j\}. It is known that if g_{i}({\bf x}) is convex and h_{j}({\bf x}) is affinely linear for {\bf x}\in \mathbb{R}^{n}, S is a convex set. However, in my problem, g_{i}({\bf x}) and h_{j}({\bf x}) is indefinite for {\bf x}\in \mathbb{R}^{n}. So I would like to ask if there is any theory may answer the following two questions:

    (1)For any twice continuously differentiable but indefinite function g_{i}({\bf x}), on what condition, g_{i}({\bf x}) is convex in a neighborhood of a point {\bf x_{0}}\in\mathbb{R}^{n} ? (A guess is that Hessian of g_{i} at {\bf x_{0}} is positive semidefinite. Is that the case?)
    Locally convex set-1.png
    Just like the image above. The function is indefinite for all x, but is locally convex in the neighborhood of x_{0}, which is (x_{1},x_{2}).

    (2)On what condition, a neighborhood in S of a feasible point {\bf x_{0}}\in S is a convex set? (I suppose a sufficient condition is that every g_{i}({\bf x}) and h_{j}({\bf x}) is convex in a neighborhood of {\bf x_{0}}. But is that necessary?)

    Locally convex set-2.png
    Just like the image above. The set S is not convex, but is locally convex in the neighborhood of {\bf x_{0}} (the red triangle set).
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  2. #2
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    Re: Locally convex set

    The questions can also be put like the following. I believe these three questions make it easier for you to answer exactly.

    (1) Given a twice continuously differentiable function f(x),x\in\mathbb{R}, it can be justified that f''(x) is not always positive for \forall x\in\mathbb{R}. However, if f''(x_0)>0, is f(x) ("locally") convex in some epsilon distance around x_0? (As shown in the 1st picutre in #1)

    (2) Given a twice continously differentiable function f({\bf x}),{\bf x}\in\mathbb{R}^{n}, it can be justified that Hessian Matrix of f({\bf x}) is not always postive definite for \forall x\in\mathbb{R}^{n}. However, if Hessian of f({\bf x}) at {\bf x_0} is positve definite, is f({\bf x}) ("locally") convex in some epsilon neighborhood of {\bf x_0}?

    (3) Given a region S defined by g_{i}({\bf x})\leq 0 \quad i=1,2,...,N and h_{j}({\bf x})=0 \quad j=1,2,...,M and {\bf x}\in\mathbb{R}^{n} (usually S defines the feasible region of a general constrained optimization problem), where every g_{i}({\bf x}) and h_{j}({\bf x}) is twice continously differentiable. Here g_{i}({\bf x}) is not convex for \forall {\bf x}\in\mathbb{R}^{n}, h_{j}({\bf x}) is not affinely linear, so S is not a convex set "as a whole". But for a feasible point {\bf x_0}\in S, on what condition (I would like to know condition about g_{i}({\bf x}) and h_{j}({\bf x}), not just the "at + (1-t)b" definition of convexity set), a neighborhood of {\bf x_0} in S is ("locally") convex? (As shown in the 2nd picture in #1)
    As to this question, if this kind of condition exists, Hessian of g_{i}({\bf x_0}) and h_{j}({\bf x_0}) is probably involved, as I guessed.

    Thanks very much!
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