Exercise 4.13 –(1) Convexity. Show that the following function is concave: f (x) = 1 – (||Ax+b|l, + ||Ax+b||2+ ||Ax+b||,³, with A E R"X", x € R", b E R" and Ax+b#0. Justify your deduction. Hint: you will need to make use of several operations that preserve convexity, including the composition with scalar functions: Let f(x) = h(g(x)) Then
Exercise 4.13 –(1) Convexity. Show that the following function is concave: f (x) = 1 – (||Ax+b|l, + ||Ax+b||2+ ||Ax+b||,³, with A E R"X", x € R", b E R" and Ax+b#0. Justify your deduction. Hint: you will need to make use of several operations that preserve convexity, including the composition with scalar functions: Let f(x) = h(g(x)) Then
College Algebra (MindTap Course List)
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Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
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Transcribed Image Text:Exercise 4.13 –(1) Convexity.
Show that the following function is concave:
f (x) = 1 – (||Ax+b|l, + ||Ax+b||2+ ||Ax+b||m)²,
with A E R"X", xE R", b E R" and Ax+b + 0. Justify your deduction.
Hint: you will need to make use of several operations that preserve convexity, including the
composition with scalar functions:
Let
f(x) = h(g(x))
Then
of
{
h is concave, h is non-decreasing, g is concave
h is concave, h is non-increasing, g is convex
f is concave if
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