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Hope to ask about p. 139 of S. Boyd's cvx book:

x is optimal iff x is in X (feasible set) and

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And the book use the following pic to illustrate:

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My Q is: why there is a negative sign '-' in front of the gradient of f0?

I 'guess' the answer is this points to the direction of lower value? However, f0(x) is the lowest value; there is no f0(x') lower than f0(x). What does this '-' sign mean?

Thanks!

sleeve chen
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1 Answers1

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$f_0(x)$ is only the lowest value inside the feasible region. There are lower values but they are not feasible.

Matt L.
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  • Must x be on the boundary of the feasible set C? If not, how to draw this figure? I mean if x is in the middle of the C, how to put gradient of f0? – sleeve chen Mar 31 '14 at 20:06
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    @sleevechen The optimal solution need not be on the boundary of the feasible set. If it isn't, this just means that the constraints defining the feasible set are not active, i.e. they in fact do not constrain the solution. In this case the gradient is zero because the function attains an unconstrained minimum. – Matt L. Apr 02 '14 at 20:29