I was looking for a clear definition of quasiconvex functions and came across Figure 3.10 in Nonlinear Programming by Bazaraa, Sherali, and Shetty 2006. I understand a property of of quasiconvex functions is that they are unimodal around the minimum. Figure 3.10a is this case shows two minimums which clearly violates the unimodality property and even violates the definition listed in the book. Could someone kindly comment whether this figure is correct or not?
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A quasi convex function is not necessarily unimodal. (a) is not quasiconvex. The picture is wrong. – copper.hat Mar 25 '20 at 17:23
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Could you kindly refer a source on when quasiconvex functions are not unimodal? According to Boyd and Vandenberghe' Convex Optimization this is one of the ways quasiconvex functions are defined, at least in the one dimensional sense. – Mohamed Abdelhamid Mar 25 '20 at 17:31
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A function is quasi convex iff the level sets are convex. So the indicator function of $[0,1]$ is quasi convex but not unimodal. – copper.hat Mar 25 '20 at 17:34
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I have never seen unimodality as a requirement for quasi convexity. I would be surprised if Stephen has defined it in any other way? – copper.hat Mar 25 '20 at 17:39
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Maybe I misunderstood, but see page 94 in https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf. He wrote a function is called quasiconvex (or unimodal) and then goes on with the definition of the sublevel sets. – Mohamed Abdelhamid Mar 25 '20 at 17:45
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Wow. I am very surprised. – copper.hat Mar 25 '20 at 17:51
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It's also in the definition of quasiconvex functions on Wikipedia. – Mohamed Abdelhamid Mar 25 '20 at 17:53
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The definition is the one I have (always) seen. However, there are lots of quasi convex functions that are not unimodal. – copper.hat Mar 25 '20 at 17:53
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One last question please, could you kindly refer to a formal definition of unimodality? I think what I understand is a simplified version of the truth. – Mohamed Abdelhamid Mar 25 '20 at 18:00
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Good catch :-). I suspect that equating quasi convexity and unimodality was unintended. – copper.hat Mar 25 '20 at 18:00
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There are various definitions, the one I would go with is in Tim's 2nd last comment in https://math.stackexchange.com/q/149476/27978. – copper.hat Mar 25 '20 at 18:01
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However, just to address your question, (a) is neither unimodal not quasi convex. – copper.hat Mar 25 '20 at 18:03
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Great, many thanks. – Mohamed Abdelhamid Mar 25 '20 at 18:04
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I also came across this link: https://math.stackexchange.com/a/1454610/578642. Plotting the function in 3D makes the concept crystal clear. – Mohamed Abdelhamid Mar 25 '20 at 18:07
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Nice. I am quite surprised by the (unimodal) in the B&V definition. – copper.hat Mar 25 '20 at 18:30