convex function and convex set(please answer 5#)
Please answer #5, where I put my questions more specific. Thank you very much!
Considering a constrained nonlinear programming (NLP) problem\[min \quad f({\bf x}) \quad {\bf x}\in \mathbb{R}^{n} \]\[s.t. \quad g_{i}({\bf x})\leq 0 \quad i=1,2,...,N \]\[\quad\quad h_{j}({\bf x})=0 \quad j=1,2,...,M \]
Where \(\displaystyle g_{i}({\bf x})\) and \(\displaystyle h_{j}({\bf x})\) is twice continuously differentiable. The feasible region \(\displaystyle S=\{{\bf x}|g_{i}({\bf x}),h_{j}({\bf x}),\forall i,j\}\). It is known that if \(\displaystyle g_{i}({\bf x})\) is convex and \(\displaystyle h_{j}({\bf x})\) is affinely linear for \(\displaystyle {\bf x}\in \mathbb{R}^{n}\), \(\displaystyle S\) is a convex set. However, in my problem, \(\displaystyle g_{i}({\bf x})\) and \(\displaystyle h_{j}({\bf x})\) is indefinite for \(\displaystyle {\bf x}\in \mathbb{R}^{n}\). 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 \(\displaystyle g_{i}({\bf x})\), on what condition, \(\displaystyle g_{i}({\bf x})\) is convex in a neighborhood of a point \(\displaystyle {\bf x_{0}}\in\mathbb{R}^{n}\) ? (A guess is that Hessian of \(\displaystyle g_{i}\) at \(\displaystyle {\bf x_{0}}\) is positive semidefinite. Is that the case?)
Just like the image above. The function is indefinite for all \(\displaystyle x\), but is locally convex in the neighborhood of \(\displaystyle x_{0}\), which is \(\displaystyle (x_{1},x_{2})\).
(2)On what condition, a neighborhood in \(\displaystyle S\) of a feasible point \(\displaystyle {\bf x_{0}}\in S\) is a convex set? (I suppose a sufficient condition is that every \(\displaystyle g_{i}({\bf x})\) and \(\displaystyle h_{j}({\bf x})\) is convex in a neighborhood of \(\displaystyle {\bf x_{0}}\). But is that necessary?)
Just like the image above. The set \(\displaystyle S\) is not convex, but is locally convex in the neighborhood of \(\displaystyle {\bf x_{0}}\) (the red triangle set).
Please answer #5, where I put my questions more specific. Thank you very much!
Considering a constrained nonlinear programming (NLP) problem\[min \quad f({\bf x}) \quad {\bf x}\in \mathbb{R}^{n} \]\[s.t. \quad g_{i}({\bf x})\leq 0 \quad i=1,2,...,N \]\[\quad\quad h_{j}({\bf x})=0 \quad j=1,2,...,M \]
Where \(\displaystyle g_{i}({\bf x})\) and \(\displaystyle h_{j}({\bf x})\) is twice continuously differentiable. The feasible region \(\displaystyle S=\{{\bf x}|g_{i}({\bf x}),h_{j}({\bf x}),\forall i,j\}\). It is known that if \(\displaystyle g_{i}({\bf x})\) is convex and \(\displaystyle h_{j}({\bf x})\) is affinely linear for \(\displaystyle {\bf x}\in \mathbb{R}^{n}\), \(\displaystyle S\) is a convex set. However, in my problem, \(\displaystyle g_{i}({\bf x})\) and \(\displaystyle h_{j}({\bf x})\) is indefinite for \(\displaystyle {\bf x}\in \mathbb{R}^{n}\). 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 \(\displaystyle g_{i}({\bf x})\), on what condition, \(\displaystyle g_{i}({\bf x})\) is convex in a neighborhood of a point \(\displaystyle {\bf x_{0}}\in\mathbb{R}^{n}\) ? (A guess is that Hessian of \(\displaystyle g_{i}\) at \(\displaystyle {\bf x_{0}}\) is positive semidefinite. Is that the case?)
Just like the image above. The function is indefinite for all \(\displaystyle x\), but is locally convex in the neighborhood of \(\displaystyle x_{0}\), which is \(\displaystyle (x_{1},x_{2})\).
(2)On what condition, a neighborhood in \(\displaystyle S\) of a feasible point \(\displaystyle {\bf x_{0}}\in S\) is a convex set? (I suppose a sufficient condition is that every \(\displaystyle g_{i}({\bf x})\) and \(\displaystyle h_{j}({\bf x})\) is convex in a neighborhood of \(\displaystyle {\bf x_{0}}\). But is that necessary?)
Just like the image above. The set \(\displaystyle S\) is not convex, but is locally convex in the neighborhood of \(\displaystyle {\bf x_{0}}\) (the red triangle set).
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