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jetoso
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Given a convex set X and a convex function f: X - R, show that for any c from R, the set S={x from X: f(x)<=c} is convex
Any advice about how to prove it?
Any advice about how to prove it?
Last edited:
jetoso said:Given a convex set X and a convex function f: X - R, show that for any c from R, the set S={x from X: f(x)<=c}
Convexity is a mathematical property that describes a function or set as being "curved outwards" or "bowl-shaped". In other words, a convex function or set has no "dents", and any line segment connecting two points on the function or set lies entirely above the function or set's curve.
Convexity is important in optimization because it allows us to determine whether a function has a unique global minimum or maximum. If a function is convex, then the global minimum or maximum can be found by setting the derivative equal to zero and solving for the optimal value.
f is the function that we are trying to prove is convex, and X is the set of values that the function takes. We use f and X to calculate the derivative of the function and to determine the curvature of the function at different points, which helps us determine whether the function is convex.
To prove convexity of S with f and X, we first calculate the first and second derivatives of the function f. Then, we use these derivatives to determine the curvature of the function at different points in the set X. If the function is convex, then the curvature will always be positive. We also check for any "dents" or breaks in the function, as these would indicate non-convexity. If the function passes these tests, then it is considered convex.
No, not all functions are convex. Some functions are concave, meaning they are "curved inwards" or "bowl-shaped" upside down. Other functions may have a combination of convex and concave sections, making them non-convex. It is important to carefully analyze the curvature and behavior of a function before determining its convexity.